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Iterative methods for discrete nonlinear Riemann-Hilbert problems
Journal of Computational North-Holland
and Applied Mathematics
143
46 ( 1993) 143-163
CAM 1333
Iterative methods for discrete nonlinear Riemann-Hilbert problems Elias Wegert Fachbereich Mathematik, Bergakademie Freiberg, Germany Received 14 October Revised 18 February
1991 1992
Abstract Wegert, E., Iterative methods for discrete nonlinear and Applied Mathematics 46 ( 1993 ) 143- 163.
Riemann-Hilbert
problems, Journal of Computational
Let D denote the complex unit disk, T := dD, and let F : T x 6t2 + R be a given function. The paper presents iterative methods for the computation of functions w = u + iv which are holomorphic in IID and satisfy the boundary relation F (t, u(t), v(t) ) = 0 for all t E T. It is shown that an appropriate discretization of this boundary value problem can be obtained by replacing the class of holomorphic functions with the set of polynomials of degree not exceeding n and imposing the boundary condition at a grid of 2n uniformly distributed points on the circle T. The proposed iterative methods for the solution of the discrete as well as the nondiscrete problems converge quadratically in a neighborhood of the solution. Error estimates for the iterative solutions are given in the scale of the Sobolev spaces Wt.
Keywords: Newton type method; Riemann-Hilbert
problem;
interpolation;
collocation
method.
1. Introduction This contribution is devoted to the computational aspect of one classical question in complex analysis: how can one find a function w = u + iv, holomorphic in the complex unit disk IID,if it is known that the boundary values of w satisfy a given nonlinear relation
on the unit circle? Recent heightening of interest in this class of boundary value problems was brought about by revealing their close connection with polynomial hulls [ 1,4,5,14,15] and ,problems of H”optimization [ 11,12 ] ; see also [ 22 1. Correspondence to: Prof. E. Wegert, Fachbereich O-9200 Freiberg,
Germany.
Mathematik, Bergakademie e-mail: [email protected].
0377-04271931% 06.00 @ 1993 -
Elsevier Science Publishers
Freiberg,
B.V. All rights reserved
B.-v.-Cotta-Strage
2,
E. Wegert /Nonlinear Riemann-Hilbert problems
144
In the present paper we propose iterative methods of Newton type which are appropriate to attack the above problem with different types of side conditions. Each iterative step requires the solution of a linear Riemann-Hilbert problem. Discretization is necessary to implement the methods on a computer. Several workers observed that in similar situations a “straightforward” discretization on a grid of equidistant points may result in iterative methods which do not converge (see [24], for an explanation). In order to avoid this difficulty Wegmann introduced in his paper [23] the concept of “discrete linear Riemann-Hilbert problems”. We show that Wegmann’s approach extends to a more general class of problems. In this context an additional “regularizing condition” on the highest Fourier coefficients appears, which has to be adapted to the coefficients of the problem. For sufficiently smooth data the resulting discrete Newton iteration converges locally quadratically to a (polynomial) solution of the discrete nonlinear Riemann-Hilbert problem. The proof rests on a Newton-Kantorovich type theorem for uniformly differentiable operators. In each step of the iteration a discrete linear Riemann-Hilbert problem is to be solved, which requires not more than 0 (N log2N) computational complexity (two FFTs and one solution of a Toeplitz system with three right-hand sides).
2. Function spaces and superposition operators Let Z,, R, R,, @,D and T stand for the nonnegative integers, the real numbers, the positive real numbers, the complex numbers, the complex unit disk and the complex unit circle, respectively. Whenever the notations t and r are used, it is supposed that t E T, z E R and t = eir. We denote by Ek,k E Z,, the Sobolev space W!(T) of real-valued functions defined on the unit circle. The corresponding Sobolev space of complex-valued functions is referred to as E& The norms in Ek and Et are denoted by ]I . Ilk. Recall that I] . Ilk is induced by the inner product
2
(f,g)k:=
(1 +j2+...+j29
sg?
j=-cc
where fjand gj are the Fourier coefficients off and g, respectively, and the bar indicates complex conjugation. If k 2 1, the norm of the product of two functions f and g can be estimated by
Ilf dk The harmonic particular,
cllf Ilk Ikllk*
G
extension
of a function
u defined
on U to the unit disk is again denoted
2n u(O):=
&
J
u ( eir ) dz.
0
For an even positive TN :=
{tj
integer N = 2n we introduce I=
eijnln:
j = O,...,N-
1).
a grid UN of equidistant
points on U:
by U. In
E. Wegert/Nonlinear Riemann-Hilbert problems It is well known that each sequence (fj)iN_c of real numbers function f from the set EN of trigonometric polynomials
is uniquely
145
interpolated
at TN by a
n-l
EN:=
{
C(ajcosjr
UO+~,COSTZT+
+
bjsinjr):
aj,bj E W},
j=O
and hence the projector PN, which assigns to each function f defined on % the function that interpolates f at the grid TN, is well defined. Obviously, EN + iEN =
Cjtj:
Cj E @,Cn
=
PNf E EN
C-n
The linear space EN equipped with the norm of Ek is denoted by Ek. We summarize a few wellknown facts about trigonometric interpolation. By a universal constant we always mean a constant that does not depend on N. Proposition 2.1. Let k, m E H+, 1 < k < m. (i) The operators PN : Ek + Ek are uniformly bounded with respect to N. (ii) For all f E Ek the interpolation error tends to zero: $im IlPNf - f (iii)
ilk =
0.
There exists a universal constant C > 0 such that IIPNf -
f Ilk G
CNk-‘Ylf Ilm
Although the result is certainly well known, we lack a convenient reference for the third assertion. We have a proof which is based on explicit computations with the Fourier coefficients of f and PNf.
In what follows we need some results concerning boundedness, continuity and differentiability of a special kind of superposition operator. Let F : T x R2 + R be a given function. The superposition operator CD generated by F assigns to a pair of functions (u, v) defined on U the function t H F(t,u(t),v(t)). Proposition 2.2. Let k, m E Z+, k 2 1, and suppose that F E Ck+m (T x W2). (i) Zf m = 0, the superposition operator Q, : Ek x Ek + Ek is bounded and uniformly continuous on bounded subsets of Ek x Ek. (ii) Zf m = 1, the operator 0 : Ek x Ek -+ Ek is uniformly differentiable on bounded subsets, i.e., for all bounded subsets X0 of Ek x Ek and all E > 0 there exists a 6 > 0 such that
The derivative D@ at x = (u, v) acts on h = (u, 47) by D@(x)h
= D,F(.,u,v).u+
D,F(.,u,v).G.
E. Wegert /Nonlinear Riemann-Hilbert problems
146
(iii) If m = 2, the superposition operator @ : Ek x Ek --t Ek IS twice continuously differentiable. The derivative of @ is Lipschitz continuous on bounded subsets, i.e., for each bounded subset X0 of Ek x Ek there exists a positive number C such that for all x, y E X0, llD@ (x) - D@ (~)~IL(@xE~) d Cl/x
-Yiik.
Here u H f .u stands for the operator of multiplication with the function f. The proof is mainly based on the continuous embedding I+‘: (8) -+ C(T) and the Lagrange formula, written in the form 1
V(Y)-v,(x)-yl'(XHY-xl
[w(x+~(Y-x))-~~
=
(.x1) dfl(y - xl.
J
0
As references
on this subject we recommend
3. Linear Riemann-Hilbert
[2] and [ 17, Chapter 21.
problems
A linear Riemann-Hilbert problem consists in finding all functions the unit disk D, which satisfy a boundary condition a(t)u(t)
+ b(t)v(t)
= c(t),
w = u + iv, holomorphic
Vt E T.
in
(1)
Here the real-valued functions a, b and c are given on 8. We seek the solutions of ( 1) in the class H” rl E& k > 1, of holomorphic functions with boundary function in E{. Usually we denote by w the boundary function rather than the holomorphic function itself. We always assume that the function a + ib never vanishes on T. The solvability of the linear Riemann-Hilbert problem depends significantly on the winding number K : = wind(a + ib ) of a + ib about the origin, called the index of the problem. For a nonnegative index the (real) dimension of the solution subspace equals 2~ + 1. The most important case pertains to problems with index zero. Standard references for linear Riemann-Hilbert problems are [ 6,131. (Note that in [ 13 ] the index is defined as 2~.) What is needed in the following can also be found more conveniently in
[al. Proposition 3.1. Let a, b,c E Ek, (a + ib) (t) # 0, wind(a + ib) = 0. Then the solution w = u + iv E H”‘ n Ek of ( 1) is the affine subspace {w = W (~22+ d ): d E U!}, where W and G are given by the following formulas: p :=
(a2 + b2)-l12,
iz := pa, b := pb, C:= pc,
W := -i exp( (H + iZ) arg(E + ib)),
w^:= (H + iI)&.
Here I denotes the identity and H stands for the singular integral cotangent kernel (the negative of the operator of complex conjugation): 2n
Hv(e”)
1
:= G
s
0
v(e’“)cot(i(a-r))do.
operator
with the Hilbert
147
E. Wegert /Nonlinear Riemann-Hilbert problems
Recall that if w = u + iv is holomorphic in D and has a boundary function in L*(U), then u = HU + u(0) and ‘u = -Hu + v (0). For all k E Z, the norm of H in Ek equals one. To eliminate the arbitrariness of d, one can impose a side condition on the solution. For instance, it is possible to fix the value of w at z = 1: w(1)
= W =: U + iv.
Of course this condition subject to a(l)U Alternatively, the origin:
+ b(l)V
(2)
must be compatible
with the boundary
i.e., U and I/ must be
= c(1).
one can require that the real and the imaginary
au(O) + Pv(O)
condition,
part of w satisfy a linear relation at
= y.
(3)
We shall mainly discuss the problem (1) with the side condition (3), with only a few remarks about the problem ( 1 ), (2). While the solution to ( 1 ), (2) is always unique, ( 1 ), (3) has only a unique solution if the compatibility condition pcos6
- u:sin6 # 0,
(4)
where 277 6
:=
&
I
arg(a + ib) (e”) dz,
(5)
0
is satisfied.
In this case the solution of (l), d = q(cycos6 + psin6) Bcos6 - asin
(3) is given by the formulas
of Proposition
3.1, with
- y ’
where 2n
rj :=
I
?exp(-H(arg(a
+ ib))) dr.
0
The corresponding d
=
value of d for the solution
S(l)-
G(l) lG(l)l
of ( 1), (2) is
-ReG(l).
By rotating and shifting the complex w-plane, the side condition (3) can be transformed to the “normal form” u (0) = 0. However, we do not use this simplification now, because we shall later perform a similar transformation in a different context. In order to write the Riemann-Hilbert problem ( 1), (3) as an operator equation, we let Xk : = Ek x Ek x Iw and define the operator A : Xk --) Xk by A(u,v,l)
= (au + bw, u-
Hv -n,au(O)
+ pv(O)).
E. Wegert /Nonlinear Riemann-Hilbert problems
148
It is not hard to see that A (u, V, A) = (c, 0, y ) if and only if the function u + iv is a solution of the Riemann-Hilbert problem (I), (3) and A = u(0). Rewriting the equation A(u,v,l) = (f,g,v) in form of a linear Riemann-Hilbert problem, we infer from Proposition 3.1 that A is invertible as a continuous linear operator, provided the compatibility condition (4) is satisfied. We stress the fact that the Riemann-Hilbert problem is invariant with respect to rotations of the w-plane, but the operator A does not have this property.
4. Discrete linear Riemann-Hilbert For our present purpose condition (3) in the form
problems
it is more convenient
to write the boundary
relation
(1) and the side
Im(f (t)w (t)) = c(t),
(6)
Re(vw(O)) =
(7)
Y,
where f is a given complex-valued function and v, is a prescribed complex number. In order to find an appropriate discretization of (6), (7), we replace the unit disk T by the grid TN, and look for all polynomials
(P:P(t)
pa$:=
which fulfil the discretized
Im(f(t)p(t))
= ePjtj), j=O
boundary
= c(t),
(8)
condition
vt E UN,
(9)
and the side condition
Re(vpo) =
(10)
Y.
It turns out that the polynomial add the “regularizing condition”
p is not uniquely
determined
by (9) and ( lo), and therefore
we
(11)
Re(vzh) = Y,
for the leading coefficient p,, of p. As will be proved, the problem (9)-( 11) is well-posed for all sufficiently large N, provided the coefficients f, q,, y satisfy certain compatibility conditions. The triple (f, p, w) E Et x C2 is said to be in [Fif f is a nonvanishing function with the winding number zero and Re(q.%)
Re(cy_%) # 0,
# 0,
where f,:=
exp
&Jarg(_/-(eir))dr 0
.
(12)
149
E. Wegert /Nonlinear Riemann-Hilbert problems Obviously, [F is an open subset of Ek x C2. If (f, q, I+U)E F and f is a polynomial exceeding n - 1,
of degree not
n-1 f(t)
=
(13)
Cfitj, j=O
we Write (f, f$?,ry) E [FN. Note that if (f, ~JJ,w) E [FN, then f0 # 0 and fo = f0/]f0]. Following we start with the investigation of (9)-( 11) for (f, ~1,v/) E FN. (9)-( 11) has a unique solution p E pi,
Lemma 4.1. If (f, y, y ) E [FN, the problem p = f.
(H
+
il)pNc + xsinnT+ lfl2
[23]
given bY
Yf,
(14)
-#K)(eiT)cosnrdr,
(15)
with 2n lhl2 x
‘=
7
+
Re(yf0)
;;I$;;
0 2a
‘I=
1 Re(qf0)’
(&) Im(glf0) 1 + Re(qf0) % so
(e”) -k x sin nr dz If (eiT)12
(16)
’
Proof. The boundary condition (9) and the additional conditions (lo), ( 11) form a linear system of 2n + 2 (real) equations for the n + 1 (complex) coefficients of the polynomial p, and hence it is sufficient to prove the uniqueness of the solution. Inserting the representation ( 13 ) and the ansatz (8 ) into the boundary condition Im(f(t)p(t))
= c(t)
= P,Vc(t),
v’t E TN,
(17)
we see that on both sides of ( 17) stand trigonometric polynomials Comparing the Fourier coefficients we get that on all of T, Im(f(t)p(t))
= c(t)
Because f is a polynomial and from ( 18) we infer that
= P,@(t) + Re(foPn)
that vanishes
&c(t)
+ Re(Top,)
nowhere
sinnr
lfW2 Applying the Hilbert and this yields that
operator
W + iI)
+ yy
IfI
with
Re(Top0 ) y
:=
l&32
.
n.
(18)
sinnr. on @ the function
p/f
is holomorphic
in D
(19)
vt EU.
to ( 19) we find the real part of p/f
PNc + x sinnT
x := Retfop,),
)
of degree not exceeding
up to the constant
Re(p0/ fo ),
E. Wegert /Nonlinear Riemann-Hilbert problems
150
Computing the numbers x and y from and the representation (14)-(1(j). 0 For a given positive following estimates:
number
( 11) and ,( lo),
we obtain the uniqueness
of the solution
K we denote by IFS the set of all (f, q, w) E [FN which satisfy the
llfllk G K> infIf(t)l
1 K
2 -3
Re(q,fo)
Re(wfo)
2 i,
> k.
Lemma
4.2. For each K > 0 there exists a universal constant C > 0 such that for all (f, q, v/) E iF$ the solution of (9)-( 11) fulfils the estimate
lIpIlk d C(llpNcllk + IYI + NkI1/I). Proof. From the representation
( 14) one can read off the estimate
k’llk d C(llpNellk + Nklxl + IYl). Using this result and the inequality 2n
cosnrdr
(PNc)(e"))
d CNPklJ PNcII~,
.I 0
resulting from the definition 1x1 < C (ITI +
of the norm in W2k(T), one verifies that
N-kIIPNCllk)
>
lyl
G
c
(Ivl
+
IFI +
q
IlpNcllk).
In the next step the case where (f, 9, ty) is close to the set Es is considered. Lemma
4.3. Let K > 0 be given.
Then there exist universal positive constants E and C with the (9)- ( 11) is uniquely solvable for all (f, ~1,y ) E F with
following property. The problem dist((f,yl,
w),F$)
(20)
< E,
where the distance is measured in the norm of Ei x C2. The solutions p E 5’; fulfil the estimate lIpIlk G C
(IIpNcltk
+
I?1 +
(21)
Nklyl).
Proof. If (f, q, w) is subject to (20), one can choose
Ilf -
g//k
d
&>
Iv
-
01*1 d E,
(g, p*. w*) E F$ such that
Iv - v*j < c.
On letting E sufficiently small, we can ensure that (g, 9, w) E Fz,K. By Lemma Riemann-Hilbert problems for the polynomials p(j), h(g(t)p(‘)(t)) Re(pp(j)) 0
= c(t) + Im((g(t) = y,
Re(y/p,?)
= i?,
-f
(t))p”-l’(t)),
vt E
TN,
4.2, the discrete
E. Wegert/Nonlinear Riemann-Hilbert problems are uniquely solvable. We start the iteration sequence (p(j)). On account of
Re(q(Pf+‘) we can estimate
-Py’))
= 0,
with p co) = 0 and prove the convergence
Re(t,v(Pc+‘)
by Lemma 4.2 and Proposition
llp(j+i) _p(j)llk
< CllPNIm((f < C21(Im(f
-PA”))
of the
= 0,
2.1
- g)(p(j) - g)(p(j)
< C311f - g11&“’
151
-p(j-‘)))(lk
-p(j-i))(lk
- p”-“Ilk.
Since C is a universal constant, the right-hand side is not greater than i lip(j) - p(j-‘)Il,, provided that E is sufficiently small. Because the fixed point p* of the iteration is a solution p = p* of (9)-( 11 ), the existence portion proved. The estimate solution of (9)- ( 1 1 ), a similar reasoning as above yields that p+ 11, which proves uniqueness. 0 IIP’ - P*ll 6 ; lip* Theorem 4.4. Fix (f*,q*,tg*) E F.Then there exists a natural number No and universal positive constants E and C such that the following holds. Zf N E 22,, N >/ NOand Ilf - f*llk
G E,
Iv, - v*l d 6,
then the discrete Riemann-Hilbert solution can be estimated by
(22)
Iw - w*l =s 6,
problem (9)-( 11) is uniquely solvable in the class IIp$. The
lIPIlk6 C (Ilplvcllk+ 11’1+ Nk\yJ). Proof. If e is sufficiently small, any triple (f, q, u/) which satisfies the estimate The function g defined by
(22) belongs to IF.
g:=exp((H+iZ)argf) fulfils the boundary
relation
Re(f (t)g(t))
= If (t)g(t)l,
Vt E U,
(23)
and we have g(0) = fo. By virtue of (23) the discrete boundary can be rewritten in the form
Note that for sufficiently to (22).
small E the IV!-norms
condition
of Ig/f I are uniformly
for the polynomial
bounded
p
for all f subject
E. Wegert /Nonlinear Riemann-Hilbert problems
152
In order to apply Lemma 4.3 we approximate g*(t)
=
g by polynomials.
The Taylor series
ggjt'
j=O
of the holomorphic function series yields a function
g’ : = exp( (H + il) arg f * ) E Et converges
in E,$. Truncating
this
n-l = Cgjtj,
g;(t)
j=O
and using the continuity we can estimate
properties
of the superposition
operator
and the mapping properties
of H,
llg - &Ilk G llg - g*llk + llg’ - &Ilk G Cllf - f*llk + llg’ - SJlk. If E is sufficiently small and N is sufficiently large, Lemma 4.3 applies to the triple provided that (f, ~1,w) is subject to (22). This completes the proof. 0 Remark 4.5. The assertion
of Theorem
(g, v, v/)
4.4 remains in force if one seeks the solution p in the class
n-1 PN
:=
(Pn
cosn7
+
Cpjtj>,
j=O
rather than in lP$. This follows from the observation p(t)
=
that the functions (24)
&jt'
j=O
and n-1
p(t)
= pn cosnz
+
(25)
Cpjtj
j=O
coincide
on the grid TN and that the norms of these functions
in Eh are equivalent.
In what follows we prefer to work with the class PN since the real part u and the imaginary ‘u of p E PN belong to Ef;, i.e.,
part
n-1 U(t)
=
Uo
+
uncosn7+
C(iijCOSj7+
u^jSinjs),
(26)
Gjsinjr).
(27)
j=l n-l v(t)
=
210 +
21n cosn7+
C(Vjcosjt+
j=l Unfortunately PN contains functions which are not holomorphic in D. Nevertheless, since the functions p and i in (24) and (25) have the same values on the grid TN, each solution p E PN
153
E. Wegert /Nonlinear Riemann-Hilbert problems of a discrete orthogonality
Riemann-Hilbert problem generates a solution p E “ft. Moreover, on account of the system (tj ) in IV! (U) we have for any function w E H” n E&
llw -Pll!i
of the
a ; II&z-“Ilk = + Il&J”llk,
and hence
This relation shows that each error estimate which will be obtained in the sequel for functions j? E PN is also valid for p E Pi, apart from the factor 3. With (26), (27) the discrete boundary value problem for p E PN can be written in the form
a(t)u(t) a40
+
+ b(t)vtt)
Pvo =
(28)
vt E UN,
= c(t),
(29)
Y,
Gu, + lJv, = j?.
Theorem
(30) 1 < k < m, a, b, c E Em, wind(a
Let k, m E E,,
+ ib) = 0, and assume that
pcos6
- cusin6 # 0,
(31)
pcos6
- Zsin6
(32)
# 0.
Then the solutions WN E PN of (28)-(30) with p = 0 converge in Ek to the solution w of (I), (3). If m > k, there exists a universal constant C such that (IWN
-
<
CNk-M(lw\\m.
Since the and w-plane, we assume By comparing Fourier 7 = is equivalent x Ek [w and =
Sections
2
the that = 1 coefficients the operator
problems are a = verifies that QNA(u,v,A)
with respect this case = QN(c,O,Y),
rotations
of
problem (28)-(30) where (u,v,A) E
(PNu,PNv,A)
3 for
definition
of
and PN).
denote by
the restriction
QNA to
n im QN = Ef; x Ek x R. If (f, g, v ) E im QN, the equation AN (u, v,A) = (f, g, v ) is equivalent to the discrete linear Riemann-Hilbert a(t);(t)
+ b(t);(t)
= f(t)
au0 + /IGo = V - ag(O),
problem - a(t)g(t),
vt E TN,
Gn = 0,
for the function u + iv E PN. Here (i-i,??) is related to (u,v,A) by u = u + g, v = 6, A = U(0). Therefore Theorem 4.4 in connection with Remark 4.5 tells us that AN : Xk n im QN + Xk n im QN is invertible for all sufficiently large N and that the norms of A;’ are uniformly bounded with respect to N.
154
E. Wegert /Nonlinear Riemann-Hilbert problems
Let now w = u + iv and wN = UN + ivN E PN denote the SOlUtiOnS Of the nondiscrete and the discrete problem, respectively. We use a standard argument to prove the convergence of XN := (uN,vN,uN(~)) to x := (u,v,u(O)) in X“. Since QNAXN = QNAx, we have XN - QNX = A;‘QNA(x and because jlAi’Q~Al[ llx~ -
An application
is uniformly
of Proposition
2.1 completes
x :=
y :=
11~~- QNX]~ < Cllx - QNX[[. Further,
the proof.
in force (without
Im(f(l)P(t))
of the problem 4.1 are
bounded,
XII d IIXN- QNxII + IIQNx - XII G (1 + C)llx - Q~xll.
Remark 4.7. The result remains discretization
p(l)
- QN~),
= c(t), = W =: U + iv,
0 condition
the first compatibility
(31))
for the
\Jt E TN, Re(vPn)
( 1)) (2). The corresponding
= Y,
values of x and y in the representation
( 14) of Lemma
2n
IhI Rekh)
s
(PNC)(e”) cos nrdr,
+xsinnr 1J2nfhc(eir) 5 If (ei*)12 cot(Lz)dT 0
If(
b(l)U-a(l)V
2
.
0
5. Nonlinear Riemann-Hilbert
problems
Generalizing the linear Riemann-Hilbert problem of Section 3 we now look for all holomorphic functions w = u +- iv E H”” fl Et which fulfil a nonlinear boundary relation F(t,u(t),v(t))
= 0,
vt E lr.
(33)
In what follows it is assumed that F E C’ (T x rW2)and that the gradient of F with respect to u and v never vanishes on the set M:=
{(t,u+iv)EUxC:
We study the problem
(33) together
au(O) + Pv(0) although the additional w(l)
= Y,
F(t,u,v)
= O}.
with the side condition (34)
condition
= W=:U+iV
(35)
(with F ( 1, U, I’) = 0) is more convenient to investigate the global solvability of the problem. The reader who is interested in results of this kind and in related questions may consult [ 5,16,18,20].
155
E. Wegert/Nonlinear Riemann-Hilbert problems
Here we suppose that a solution w* = U* + iv* E H” n IV: of (33) exists. With any pair of functions (u, Y ) we associate the functions a and b given by a(t)
b(t) := &F(t,u(t),v(t)),
:= &F(Cu(t),v(t)),
and we let a0 and bo be the functions associated with (u*, II* ) . It is assumed that wind(ac + ibo ) = 0. The latter is not a significant restriction, since it can always be achieved by appropriate transformations. We remark that under the above assumptions w* E EE if F E Ck ( [ 191, see [ 31 for a related result in Holder spaces). In order to establish a Newton method for the iterative solution of (33), (34) we introduce the operator A : Xk + Xk by A(u,v,A)
= (F(.,u,v),u-Hw-il,au(O)+pv(O)-y).
It follows immediately from this definition that A (u, u,A) = 0 is equivalent ;1 = u (0). In the next lemma we list some relevant properties of A.
to (33),
(34), and
Lemma 5.1. Let F E Ck+l, k > 1, and define So by (5) with a = ao, b = bo. (i) The operator A : Xk + Xk is continuously differentiable and its derivative DA(x)
at
x = (u,v,A) isgiven by DA(x)(@,il)
= (a6 + bG, 6 - HG - ;1,G(O) + @7(O)).
(ii) Let wind(ao + ibo) = 0. If j3 cos 60 - CYsin 60 # 0, then there exists a neighborhood X,k c Xk ofx* := (u*,v*, u* (0)) such that DA(x) is invertible fir all x E X,$ and the norms ofDA(x)-’ in L ( Xk ) are uniformly bounded on Xt . (iii) ZfF E C k+2, the operator A : Xk + Xk is twice continuously differentiable. For each bounded subset Xt c Xk there exists a positive number C such that for all x, y E Xk,
Proof. The assertions (i) and (iii) follow immediately from Proposition 2.2. In order to prove (ii), we let a := DUF(.,u,w), b : = D, F (+, u, u ) and define 6 by (5). Since, under the assumptions of (ii), 6 depends continuously on u and U, /? cos 6 - u: sin 6 is bounded away from zero whenever x = (u, V, A) is sufficiently close to x*. The equation DA(x)(CC& is equivalent
= (f,g,u)
to the linear Riemann-Hilbert
a2 + bv^ = f - ag,
problem
G(O) + p??(o) = V - ag(O),
for the holomorphic function 6 = G + iv^, where 6 = U - g, v^ = V, i = U(0) = G(O) + g (0). Therefore Proposition 3.1 yields the invertibility of DA(x), and a straightforward estimation of the terms in the representation of the solution gives the uniform boundedness of the inverse operators
DA(x)-‘.
0
Newton’s method xj+i following form.
= Xj - DA(xj)-'
A(xj)
for the equation
A(x) = 0 can be written in the
E. Wegert /Nonlinear Riemann-Hilbert problems
156
Step 1. Choose j = 1,
a (not necessarily
holomorphic)
gi := Ui -NV,
fi := F(.,u1,v1),
complex-valued
- Ui(O),
function
wi = u1 + iv1 . Let
Vi := /X4,(0) + /3V,(O) -y.
Step 2. Put bj := D,F(., uj,Uj),
aj := D,F(.,uj,Vj), Solve the linear Riemann-Hilbert aju + bjv = cj,
Cj :=
fi - U.jgj.
problem
cUu(0) + /IV(O) =
Vj.
Let
:=
f;+1
Vj+l
Uj-U-gj,
uj+l :=
F(.,uj+l,Vj+I),
Put j := j + 1 and repeat the procedure In each iterative
step the operator
I=
Vj - 21,
gj+l
:= 0,
Vj+l I= 0.
beginning with the second step.
H has to be computed
only twice although the computation of Since gj = 0, all iteratives Wj are is automatically satisfied beginning with the second
DA (x)-‘A (x) alone would require three complex conjugations. holomorphic iteration. Theorem
for j > 2. Also the side condition
5.2.
Let F E Ck+‘, k > 1; let ao, bo, So have the same meaning as in Lemma 5.1, assume that wind(ae + ibo) = 0, and /? cos 6. - cysin 60 # 0. (i) Zf F E C k+ l (U x R*), then for each real number q E (0,l) there exists a positive number E such thatfor j = 1,2,..., l/w1 -
W*(lk
< & *
IIwj+l
- W*llk
d
q11wj - W*llk*
(ii) ZfF E Ck+*(T x R2),th ere exist positive numbers E and C such that for j = 1,2,. IlWl -
W*llk <
E
*
Proof. We study the convergence
Xk of x* mentioned Xj+l-X*
in Lemma
. .,
IIWj+l- W*llk < CllWj -W*IIiof the sequence xj = (Uj, vj, U, (0) ). If Xj lies in the neighborhood 5.1 (ii), we have -
=
Xj -Xx*
=
DA(xj)-'(A(x*)-A(Xj)-DA(x,)(x*
=
DA(x,)-'(A(x*)-A(Xj)-DA(X*)(X*-xj))
D~4(xj)-1(A(X,)-A(X*)) -xj))
+DA(xj)-'(DA(x*)-DA(X,))(X*-xi). By Lemma
5.l(ii), IIA(x*) lIDA
CO := supXEX; (IDA(x)-‘11 is finite. Since, by the first assertion -A(xj) -DA(xj)ll
-DA(X*)(X* G &,
-Xj)ll
d
&l/x,
-x*1/,
of this lemma,
157
E. Wegert /Nonlinear Riemann-Hilhert problems provided
that ]]Xj - x*1( is sufficiently Ilxj+l
-X*1(
G
small, we get that
4llXj --*Il.
If J(xi -x*(1 Q E for sufficiently small E, the sequence (Xj) remains in the set Xt and Xj converges to x*. If F E Ck +* (T x lR* ) , local quadratic convergence of the Newton method is ensured by assertions 0 (i) and (iii) of Lemma 5.1; (see [25, Proposition 5.11, for instance). Remark 5.3. In order to construct a Newton method for Riemann-Hilbert problems with the side condition w (1) = IV, we first of all observe that this relation is locally equivalent to the condition au(l)
+ /3V(l)
= y := au
+ pv,
if only ab ( 1) - /?a ( 1) # 0. After replacing the operator B(u,w,A)
:= (F(.,u,v),u-Hv
-&(uu(l)
A by
+ pv(l)
a repetition of the above considerations shows that Theorem tion) remains in force under these modifications.
6. Discrete nonlinear Riemann-Hilbert
-y), 5.2 (without
(36) the compatibility
condi-
problems
We work under the same hypotheses as in the preceding section. The aim, of this section is to show that for all sufficiently large N the discrete nonlinear Riemann-Hilbert problem
F(t,un(t),#(t))
= 0,
auN(0)
= y,
+ pv?o)
Vt E UN,
Eiu; + IJ?J,N =0
(37) (38) (39)
has a solution wN = uN + iv N E PN and that wN converges to the solution w’ of the corresponding nondiscrete problem. In (39) U: and UC stand for the coefficients of cos no in the Fourier series of uN and wN, respectively. Theorem 6.1. Let F E Ck+‘, k 2 1; let ao, bo,& have the same meaning as in Lemma 5.1, and assume that wind (a0 + ibe) = 0. If p cos 60 - LYsin 60 # 0 and pcos 60 - Gsin 60 # 0, then the
discrete Riemann-Hilbert problem (37)-(39) has a solution wN E PN for all sufjciently large N. This solution is unique in a neighborhood of w+. There exists a universal constant C such that the defect between wN and the solution W* of the nondiscrete problem (33), (34) can be estimated by
lIWN- w*llk < c llPj@* - t”*l(k. The proof of Theorem differentiable operators.
6.1 rests on the following Newton-Kantorovich
(40) type result for uniformly
E. Wegert/Nonlinear Riemann-Hilbert problems
158
Lemma 6.2. Let X be a Banach space and U an open subset of X. Assume that the operator
B : U + X is uniformly differentiable on U and has a uniformly continuous derivative, i.e., there exists a monotone nondecreasing function q : R+ - R+ such that for all E > 0, x,y E U,
(lx -yll < V(E) * IIDB(x) -DB(y)ll
X,YE
llx-~11
U,
LetxOEU,d:=
G q(e)
< -.&
llDB(x)(y-x)-B(y)
(41) +B(x)ll
G dlx--Il.
(42)
If DB(xo) is invertible, K := IjDB(xo)-‘11 and
dist(xc,dU).
llB(xo)ll
*
6 E,
min(O
(&))
(43)
I
then the sequence (xj)i”,O defined by Xj+]
:=
-DB(xo)-‘B(xj)
xj
lies in U and converges to a solution x* of the equation B(x) = 0. The error can be estimated by IlXj - x*[[
6 2-‘+‘llxl
2-i+1KIIB(~o)ll.
- xoll <
(44)
Proof. First of all we remark that [lx1 -x011
6
llDB(x~)-‘II
= WIWXO)II
< imin(d,v
(A)).
(45)
that xj+l and xj belong to U, and that
Under the assumption Ilxj+l
llB(x~)ll
-xjll
G rl(&)
- xj+l
= xj+l
llxj-xoll
9
(46)
d q (A),
the representation xji2
-xi-DB(Xo)-‘(B(xj+l)-B(xj))
= DB(xo)-‘(DB(xo)(xj+l
-xi)
- B(Xj+l) + B(xj))
leads to the estimate
IlXj+2-Xj+lII
G IlDB(x~)-'ll
(IDB(xj)(xj+l -Xj) -B(xj+l)
+IIDB(xo)-‘II On account
of (41), Ilxj+Z -xj+lll
IIDB(xj) -DB(xo)lI
+ B(xj)ll
IJxj+l -Xjjl.
(42) and (46) we obtain that s K (&
+ &)
Ilxj+l
-xjll
= 4 Ilxj+l
(47)
-xjll.
Consequently the first relation in (46) holds with j replaced by j + 1. Using (45) one proves by induction that xj belongs to U for all j and that both estimates (46) are valid during the iteration. From (47) we get the convergence of the sequence (xi) and the error estimate (44). Obviously in the limit x* is a solution of the equation B(x) = 0. Proof of Theorem 6.1. S$ce the problem assume that G = 1 and j? = 0.
is invariant
under
rotations
of the w-plane,
we may
159
E. Wegert /Nonlinear Riemann-Hilbert problems
isasolutionofA(x*) = 0. Let A : Xk + Xk be defined as in Section 5. Then x* : = (u*,v*,u*(O)) We denote by Xt the &neighborhood of x* in X k. By virtue of Lemma 5.1 (ii), the reasoning already used in the proof of Theorem 4.6 yields that the restrictions TN (x ) : = QNDA (x ) 1Xk n im QN of QNDA(x) to Xk n im QN are invertible as continuous linear operators on Xk n im QN for all x E X,k, and that the norms of their inverses SN (x ) : = (TN (x ) )-l are bounded (uniformly with respect to x E Xj and N) if only 6 is sufficiently small and N is sufficiently large. In order to apply Lemma 6.2 we let X := Xk n im QN, U := {x E X: 1x - QNX*II < 1). On account of Proposition 2.2 the operator B : = QNA : X + X fullils the assumptions (41) and (42) of Lemma 6.2. Moreover, since llQ,vll is uniformly bounded, the same function q will do for all N. Let x0 : = QNx*. Then
IIQNA(xo)II = IIQNA(QNx*) - Q~A(x*)ll d SUPIPAII IIQNIIIIQNx* - x*ll, where the supremum is to be taken over a closed ball in Xk which contains QNX* and x*. Since 2.2 the QNX* converges to x*, this ball does not depend on N, and by virtue of Proposition supremum is finite. Consequently, condition (43) is satisfied if only N is sufficiently large. An application of Lemma 6.2 proves the convergence of xj to an element xN which is a solution of QNA(x) = 0. With xN = (uN,uN,AN) we find a solution wN = uN + iVN of the discrete Riemann-Hilbert problem. Finally, the error estimate (40) follows from
llXN - X’II G INN - xoll + IIxoSince DQNA (xN ) is invertible
x*ll
(considered
< 2 11x1 -x0(1
+ 11x0 - x*11 < C ~QNx*
as an operator
-x*11.
on Xk n im QN ), the solution
xN
is unique in a certain ball centered at x N. Because the norms of (DQNA(x~))-~ are uniformly bounded, we can choose the radius of this ball to be independent of N. This, in connection with the convergence of xN to x* yields the uniqueness of the solution xN in a neighborhood of x* for sufficiently large N. 0 Writing Newton’s method for solving the operator equation QNA(x~) = 0 (where xN E Xk n imQ~) in terms of discrete Riemann-Hilbert problems, we get the following algorithm. Step 1. Choose an initial solution
uy, $
E EN. Put j = 1 and let
gi := ur - H?J,N - uY(O),
fi := F(+~,vfV),
Vi := auY(O) + /37J~(O, -7.
Step 2. Let
bj := D,F(.,ur,~r),
aj := DuF(.,ur,~r),
and solve the linear discrete Riemann-Hilbert aj(l)U(t) au(O)
+
bj(t)v(t)
+ BV (0)
= Vj,
=
Cj(t),
Cj Z= fj - ajgj,
problem vt
E %N,
un = 0.
Let N
‘j+l h+1
Z=
Ur-
U-
:=
F(.,u~N+~,~?+,),
gj,
N ‘j+l
:= gj+l
Put j := j + 1 and repeat the second step.
v,%l, :=
09
Vj+l
I=
0.
160
E. Wegert /Nonlinear Riemann-Hilbert problems
The convergence
of the sequence
Theorem 6.3. Let a~, bo, 80 have the 0, p cos a0 - a sin do # 0, sin 60 # 0, (i) If F E @+I(% x R*), k 3 number E such that for all j = 1,2,.
<& lb? - WNllk
*
wjN := ur + iv,! is the subject of the next theorem. same meaning as in Lemma 5.1, assume that wind(ao + ibo) = and let WY E PN. 1, then for each real number q E (0, 1) there exists a positive . . and all N B NO,
- WNllkd IIwjN+l
q IIw/” - zJyl,+
(ii) ZfF E Ck+*(T x R*), th ere exist positive numbers E and C such that for j = 1,2, . . . and all N 3 NO,
114 - WNllk< E *
- wq;. Ilw,N,l- WNl(/(d c \\wy
uy
Proof . Let X? := (uy, VT, (0)). By virtue of Theorem 6.1 the element xy belongs to a certain neighborhood Xj of x’ for all N B NO if only E is sufficiently small. The operators &A : Xk + Xk are continuously differentiable. If F E Ck+*, the derivative of QNA is uniformly (with respect to N) Lipschitz continuous on Xj. From Theorem 4.4 and Remark 4.5 we infer that the inverses SN(X) of the operators DQNA(x) : Xk n imQ~ --f Xk n im QN are well-defined for all x E Xj if 6 is sufficiently small and N is sufficiently large. Moreover the norms of SN (x) are uniformly bounded. Since the above iterative method is equivalent to N xj+l
= x,! - SN(X~)QNA(X~),
the proof can be linished
along the lines of the proof of Theorem
6.1.
0
Remark 6.4. By replacing the operator QNA by QNB, with B from (36), one obtains analogous results for the solvability of the discrete Riemann-Hilbert problem with the side condition w ( 1) = W and for the convergence of the corresponding Newton iteration. In this case the first compatibility condition can be omitted.
7. Implementation and test results The Newton method converges only locally and hence its practical application requires the knowledge of an appropriate initial solution. We recommend to couple the iterative method with an imbedding method, i.e., one has to construct a homotopy path which connects the problem to be solved with a problem whose solution is known. For Riemann-Hilbert problems with closed restriction curves the homotopy path must be chosen carefully, in order to avoid that the family contains problems which are “holomorphically nontraceable” (see [ 201). The simplest way to implement one of the iterative methods on a computer is a “straightforward” discretization of the algorithm described in Section 5. By “straightforward” we mean that the explicit formulas for the solution of the linear Riemann-Hilbert problem are used and that the operator H is replaced by the (negative of the) discrete Wittich operator, which acts on the values of a function on the grid TN. The Wittich operator can be efficiently computed by means of two FFTs (see [ 71). Unfortunately, this way of discretization may disturb the convergence of the iteration. It is known for quite similar problems that after a period of quadratic “convergence” the error
161
E. Wegert /Nonlinear Riemann-Hilbert problems
(possibly) decreases linearly and may finally even increase. A detailed analysis and an explanation of this effect for numerical conformal mapping is given in [24]. On the other hand, the discrete iterative method proposed in Section 6 is ready for implementation, provided that fast algorithms for the solution of discrete Riemann-Hilbert problems are available. In [ 23 1, Wegmann recommended either to use a conjugate gradient method (with computational cost 0 (N log N) ) or to reduce the problem to the solution of a system with a Toeplitz matrix (and three right-hand sides). For the latter class several fast algorithms are published; we used the asymptotically fast O(Nlog2N) algorithm developed by Heinig and Jankowski, which works without the assumption of strong nonsingularity of the matrix ( [ 9, Section 51). The same paper contains an 0 (N) algorithm for parallel computers. We have tested both methods, the “discretized” of Section 5 and the “discrete” of Section 6, in particular for the example F(t,u,v):=
v-U2-C(p,t)(c(p,t)-1),
where 1 -PCOST dw’T):=~(
with the side condition solution w*(z)
+
1_2pcosT+p2
=
psinz >y
u (0) = p. For parameter
values p E [0,1) this problem
has the exact
“:“,z’.
The condition number increases as p + 1; if p = 1, the solution has a pole at z = 1. Figures 1 and 2 show the logarithm of the (approximate) sup-norm of the error log,, l]wr - w* 1) (different markers) and the increment log,, ]lwy+ 1 - wf’ll (solid lines) during the iteration for the parameter value p = 0.8. Although the accuracy of both methods is comparable, the “discrete method” has better performance than the “discretized method”, due to the absence of the linear error branch. Remarkably we did not observe the increasing error branch which should appear after a large number of iterations. This is in contrast to the results discussed in our earlier paper [ 211 for a
N=64: +,N=,28: X,N=Z6:
N=64: +, N=lZg: x, N=2.56:o, N=5,2: *
o, N=5,2: *
Fig. 1. Logarithm of error and increment cretized method.
for dis-
Fig. 2. Logarithm
of error and increment method.
for discrete
162
E. Wegert /Nonlinear
_80’-
-?-
Riemann-Hilbert
problems
_20d.-.-A_
30
40
50
N=64: +,N=128:x,N=256:o,N=512:
m
70
*
Fig. 3. Logarithm of error and increment for method in [21]. quite similar iterative method for Riemann-Hilbert problems given in parametric form. In order to draw a direct comparison we have applied this earlier method to the above example. The restriction curves have been parametrized by pr(s)
= is-s*-c((p,t)(c(p,t)
- 1).
Figure 3 illustrates the results of our computations. So far we have no explanation completely different “long time behavior” shown in Figs. 1 and 3.
for the
Acknowledgements I wish to thank Harald Heidler from FORTRAN code of the “discrete method” and suggesting several improvements.
the Technische Universittit Chemnitz for writing the and two anonymous referees for correcting some errors
References 111 H. Alexander and J. Wermer, Polynomial hulls with convex fibers, Math. Ann. 27 (1985) 99-109. 121 J. Appell and P.P. Zabrejko, Nonlinear Superposition Operators (Cambridge Univ. Press, Cambridge, 1990). 131 E.M. Cirka, Regularity of boundaries of analytic sets, Mat. Sb. 117 (1982) 291-334 (in Russian). 141 F. ForstneriE, Analytic disks with boundaries in a maximal real submanifold of C*, Ann. Inst. Fourier (Grenoble) 37 (1987) l-44. is1 F. ForstneriE, Polynomial hulls of sets fibered over the circle, Indiana Univ. Math. J. 37 (1988) 869-889. [61 F.D. Gakhov, Boundary Value Problems (Nauka, Moscow, 3rd ed., 1977, in Russian). 171 M.H. Gutknecht, Fast algorithms for the conjugate periodic function, Computing 22 ( 1979) 79-91. 181 M.H. Gutknecht, Numerical conformal mapping methods based on function conjugation, in: L.N. Trefethen, Ed., Numerical Conformal Mapping (North-Holland, Amsterdam, 1986) 31-77; also: J. Comput. Ap@ Math. 14 (l&2) (1996) 31-77. G. Heinig and P. Jankowski, Parallel and superfast algorithms for Hankel systems of equations, Numer. [91 Math. 58 (1990) 109-127. 1101 J.W. Helton, Optimization over spaces of analytic functions and the Corona problem, J. Operator Theory 15 (1986) 359-375.
E. Wegert /Nonlinear Riemann-Hilbert problems
163
[ 111 J.W. Helton, Operator Theory, Analytic Functions, Matrices, and Electrical Engineering (Amer. Mathematical Sot., Providence, RI, 1987). [ 121 J.W. Helton and D.E. Marshall, Frequency domain design and analytic selections, Indiana Univ. Math. J. 39 (1990) 157-184. [ 13 ] N.I. Muskhelishvili, Singular Integral Equations (Noordhoff, Groningen, 1946). [ 14 ] Z. Slodkowski, Polynomial hulls with convex sections and interpolating spaces, Proc. Amer. Math. Sac. 294 (1986) 367-377. [ 151 Z. Slodkowski, Polynomial hulls with convex fibers and complex geodesics, J. Funct. Anal. 94 ( 1990) 156-176. [ 161 A.I. Snirel’man, A degree of quasi linear-like mapping and the nonlinear Hilbert problem, Mat. Sb. 89 (113) (1972) 366-389 (in Russian). [ 171 T. Valent, Boundary Value Problems of Finite Elasticity, Springer Tracts Nat. Philos. 31 (Springer, New York, 1988). [ 181 L. von Wolfersdorf, A class of nonlinear Riemann-Hilbert problems for holomorphic functions, Math. Nachr. 116 (1984) 89-107. [ 191 E. Wegert, Geometrische Methoden fiir nichtlineare Randwertaufgaben vom Riemann-Hilbertschen Typus, Dissertation B, Bergakademie Freiberg, 1988. [20] E. Wegert, Boundary value problems and extremal problems for holomorphic functions, Complex Variables Theory Appl. 11 (1989) 233-256. [ 2 1 ] E. Wegert, An iterative method for solving nonlinear Riemann-Hilbert problems, J. Comput. Appl. Math. 29 (3) (1990) 311-327. [22] E. Wegert, Boundary value problems and best approximation by holomorphic functions, J. Approx. Theory 61 ( 1990) 322-334. [23] R. Wegmann, Discrete Riemann-Hilbert problems, interpolation of simply closed curves, and numerical conformal mapping, J. Comput. Appl. Math. 23 (3) (1988) 323-352. [24] R. Wegmann, Discretized versions of Newton type iterative methods for conformal mapping, J. Comput. Appl. Math. 29 (2) (1990) 207-224. [25] E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol. Z (Springer, New York, 1986).
and Applied Mathematics
143
46 ( 1993) 143-163
CAM 1333
Iterative methods for discrete nonlinear Riemann-Hilbert problems Elias Wegert Fachbereich Mathematik, Bergakademie Freiberg, Germany Received 14 October Revised 18 February
1991 1992
Abstract Wegert, E., Iterative methods for discrete nonlinear and Applied Mathematics 46 ( 1993 ) 143- 163.
Riemann-Hilbert
problems, Journal of Computational
Let D denote the complex unit disk, T := dD, and let F : T x 6t2 + R be a given function. The paper presents iterative methods for the computation of functions w = u + iv which are holomorphic in IID and satisfy the boundary relation F (t, u(t), v(t) ) = 0 for all t E T. It is shown that an appropriate discretization of this boundary value problem can be obtained by replacing the class of holomorphic functions with the set of polynomials of degree not exceeding n and imposing the boundary condition at a grid of 2n uniformly distributed points on the circle T. The proposed iterative methods for the solution of the discrete as well as the nondiscrete problems converge quadratically in a neighborhood of the solution. Error estimates for the iterative solutions are given in the scale of the Sobolev spaces Wt.
Keywords: Newton type method; Riemann-Hilbert
problem;
interpolation;
collocation
method.
1. Introduction This contribution is devoted to the computational aspect of one classical question in complex analysis: how can one find a function w = u + iv, holomorphic in the complex unit disk IID,if it is known that the boundary values of w satisfy a given nonlinear relation
on the unit circle? Recent heightening of interest in this class of boundary value problems was brought about by revealing their close connection with polynomial hulls [ 1,4,5,14,15] and ,problems of H”optimization [ 11,12 ] ; see also [ 22 1. Correspondence to: Prof. E. Wegert, Fachbereich O-9200 Freiberg,
Germany.
Mathematik, Bergakademie e-mail: [email protected].
0377-04271931% 06.00 @ 1993 -
Elsevier Science Publishers
Freiberg,
B.V. All rights reserved
B.-v.-Cotta-Strage
2,
E. Wegert /Nonlinear Riemann-Hilbert problems
144
In the present paper we propose iterative methods of Newton type which are appropriate to attack the above problem with different types of side conditions. Each iterative step requires the solution of a linear Riemann-Hilbert problem. Discretization is necessary to implement the methods on a computer. Several workers observed that in similar situations a “straightforward” discretization on a grid of equidistant points may result in iterative methods which do not converge (see [24], for an explanation). In order to avoid this difficulty Wegmann introduced in his paper [23] the concept of “discrete linear Riemann-Hilbert problems”. We show that Wegmann’s approach extends to a more general class of problems. In this context an additional “regularizing condition” on the highest Fourier coefficients appears, which has to be adapted to the coefficients of the problem. For sufficiently smooth data the resulting discrete Newton iteration converges locally quadratically to a (polynomial) solution of the discrete nonlinear Riemann-Hilbert problem. The proof rests on a Newton-Kantorovich type theorem for uniformly differentiable operators. In each step of the iteration a discrete linear Riemann-Hilbert problem is to be solved, which requires not more than 0 (N log2N) computational complexity (two FFTs and one solution of a Toeplitz system with three right-hand sides).
2. Function spaces and superposition operators Let Z,, R, R,, @,D and T stand for the nonnegative integers, the real numbers, the positive real numbers, the complex numbers, the complex unit disk and the complex unit circle, respectively. Whenever the notations t and r are used, it is supposed that t E T, z E R and t = eir. We denote by Ek,k E Z,, the Sobolev space W!(T) of real-valued functions defined on the unit circle. The corresponding Sobolev space of complex-valued functions is referred to as E& The norms in Ek and Et are denoted by ]I . Ilk. Recall that I] . Ilk is induced by the inner product
2
(f,g)k:=
(1 +j2+...+j29
sg?
j=-cc
where fjand gj are the Fourier coefficients off and g, respectively, and the bar indicates complex conjugation. If k 2 1, the norm of the product of two functions f and g can be estimated by
Ilf dk The harmonic particular,
cllf Ilk Ikllk*
G
extension
of a function
u defined
on U to the unit disk is again denoted
2n u(O):=
&
J
u ( eir ) dz.
0
For an even positive TN :=
{tj
integer N = 2n we introduce I=
eijnln:
j = O,...,N-
1).
a grid UN of equidistant
points on U:
by U. In
E. Wegert/Nonlinear Riemann-Hilbert problems It is well known that each sequence (fj)iN_c of real numbers function f from the set EN of trigonometric polynomials
is uniquely
145
interpolated
at TN by a
n-l
EN:=
{
C(ajcosjr
UO+~,COSTZT+
+
bjsinjr):
aj,bj E W},
j=O
and hence the projector PN, which assigns to each function f defined on % the function that interpolates f at the grid TN, is well defined. Obviously, EN + iEN =
Cjtj:
Cj E @,Cn
=
PNf E EN
C-n
The linear space EN equipped with the norm of Ek is denoted by Ek. We summarize a few wellknown facts about trigonometric interpolation. By a universal constant we always mean a constant that does not depend on N. Proposition 2.1. Let k, m E H+, 1 < k < m. (i) The operators PN : Ek + Ek are uniformly bounded with respect to N. (ii) For all f E Ek the interpolation error tends to zero: $im IlPNf - f (iii)
ilk =
0.
There exists a universal constant C > 0 such that IIPNf -
f Ilk G
CNk-‘Ylf Ilm
Although the result is certainly well known, we lack a convenient reference for the third assertion. We have a proof which is based on explicit computations with the Fourier coefficients of f and PNf.
In what follows we need some results concerning boundedness, continuity and differentiability of a special kind of superposition operator. Let F : T x R2 + R be a given function. The superposition operator CD generated by F assigns to a pair of functions (u, v) defined on U the function t H F(t,u(t),v(t)). Proposition 2.2. Let k, m E Z+, k 2 1, and suppose that F E Ck+m (T x W2). (i) Zf m = 0, the superposition operator Q, : Ek x Ek + Ek is bounded and uniformly continuous on bounded subsets of Ek x Ek. (ii) Zf m = 1, the operator 0 : Ek x Ek -+ Ek is uniformly differentiable on bounded subsets, i.e., for all bounded subsets X0 of Ek x Ek and all E > 0 there exists a 6 > 0 such that
The derivative D@ at x = (u, v) acts on h = (u, 47) by D@(x)h
= D,F(.,u,v).u+
D,F(.,u,v).G.
E. Wegert /Nonlinear Riemann-Hilbert problems
146
(iii) If m = 2, the superposition operator @ : Ek x Ek --t Ek IS twice continuously differentiable. The derivative of @ is Lipschitz continuous on bounded subsets, i.e., for each bounded subset X0 of Ek x Ek there exists a positive number C such that for all x, y E X0, llD@ (x) - D@ (~)~IL(@xE~) d Cl/x
-Yiik.
Here u H f .u stands for the operator of multiplication with the function f. The proof is mainly based on the continuous embedding I+‘: (8) -+ C(T) and the Lagrange formula, written in the form 1
V(Y)-v,(x)-yl'(XHY-xl
[w(x+~(Y-x))-~~
=
(.x1) dfl(y - xl.
J
0
As references
on this subject we recommend
3. Linear Riemann-Hilbert
[2] and [ 17, Chapter 21.
problems
A linear Riemann-Hilbert problem consists in finding all functions the unit disk D, which satisfy a boundary condition a(t)u(t)
+ b(t)v(t)
= c(t),
w = u + iv, holomorphic
Vt E T.
in
(1)
Here the real-valued functions a, b and c are given on 8. We seek the solutions of ( 1) in the class H” rl E& k > 1, of holomorphic functions with boundary function in E{. Usually we denote by w the boundary function rather than the holomorphic function itself. We always assume that the function a + ib never vanishes on T. The solvability of the linear Riemann-Hilbert problem depends significantly on the winding number K : = wind(a + ib ) of a + ib about the origin, called the index of the problem. For a nonnegative index the (real) dimension of the solution subspace equals 2~ + 1. The most important case pertains to problems with index zero. Standard references for linear Riemann-Hilbert problems are [ 6,131. (Note that in [ 13 ] the index is defined as 2~.) What is needed in the following can also be found more conveniently in
[al. Proposition 3.1. Let a, b,c E Ek, (a + ib) (t) # 0, wind(a + ib) = 0. Then the solution w = u + iv E H”‘ n Ek of ( 1) is the affine subspace {w = W (~22+ d ): d E U!}, where W and G are given by the following formulas: p :=
(a2 + b2)-l12,
iz := pa, b := pb, C:= pc,
W := -i exp( (H + iZ) arg(E + ib)),
w^:= (H + iI)&.
Here I denotes the identity and H stands for the singular integral cotangent kernel (the negative of the operator of complex conjugation): 2n
Hv(e”)
1
:= G
s
0
v(e’“)cot(i(a-r))do.
operator
with the Hilbert
147
E. Wegert /Nonlinear Riemann-Hilbert problems
Recall that if w = u + iv is holomorphic in D and has a boundary function in L*(U), then u = HU + u(0) and ‘u = -Hu + v (0). For all k E Z, the norm of H in Ek equals one. To eliminate the arbitrariness of d, one can impose a side condition on the solution. For instance, it is possible to fix the value of w at z = 1: w(1)
= W =: U + iv.
Of course this condition subject to a(l)U Alternatively, the origin:
+ b(l)V
(2)
must be compatible
with the boundary
i.e., U and I/ must be
= c(1).
one can require that the real and the imaginary
au(O) + Pv(O)
condition,
part of w satisfy a linear relation at
= y.
(3)
We shall mainly discuss the problem (1) with the side condition (3), with only a few remarks about the problem ( 1 ), (2). While the solution to ( 1 ), (2) is always unique, ( 1 ), (3) has only a unique solution if the compatibility condition pcos6
- u:sin6 # 0,
(4)
where 277 6
:=
&
I
arg(a + ib) (e”) dz,
(5)
0
is satisfied.
In this case the solution of (l), d = q(cycos6 + psin6) Bcos6 - asin
(3) is given by the formulas
of Proposition
3.1, with
- y ’
where 2n
rj :=
I
?exp(-H(arg(a
+ ib))) dr.
0
The corresponding d
=
value of d for the solution
S(l)-
G(l) lG(l)l
of ( 1), (2) is
-ReG(l).
By rotating and shifting the complex w-plane, the side condition (3) can be transformed to the “normal form” u (0) = 0. However, we do not use this simplification now, because we shall later perform a similar transformation in a different context. In order to write the Riemann-Hilbert problem ( 1), (3) as an operator equation, we let Xk : = Ek x Ek x Iw and define the operator A : Xk --) Xk by A(u,v,l)
= (au + bw, u-
Hv -n,au(O)
+ pv(O)).
E. Wegert /Nonlinear Riemann-Hilbert problems
148
It is not hard to see that A (u, V, A) = (c, 0, y ) if and only if the function u + iv is a solution of the Riemann-Hilbert problem (I), (3) and A = u(0). Rewriting the equation A(u,v,l) = (f,g,v) in form of a linear Riemann-Hilbert problem, we infer from Proposition 3.1 that A is invertible as a continuous linear operator, provided the compatibility condition (4) is satisfied. We stress the fact that the Riemann-Hilbert problem is invariant with respect to rotations of the w-plane, but the operator A does not have this property.
4. Discrete linear Riemann-Hilbert For our present purpose condition (3) in the form
problems
it is more convenient
to write the boundary
relation
(1) and the side
Im(f (t)w (t)) = c(t),
(6)
Re(vw(O)) =
(7)
Y,
where f is a given complex-valued function and v, is a prescribed complex number. In order to find an appropriate discretization of (6), (7), we replace the unit disk T by the grid TN, and look for all polynomials
(P:P(t)
pa$:=
which fulfil the discretized
Im(f(t)p(t))
= ePjtj), j=O
boundary
= c(t),
(8)
condition
vt E UN,
(9)
and the side condition
Re(vpo) =
(10)
Y.
It turns out that the polynomial add the “regularizing condition”
p is not uniquely
determined
by (9) and ( lo), and therefore
we
(11)
Re(vzh) = Y,
for the leading coefficient p,, of p. As will be proved, the problem (9)-( 11) is well-posed for all sufficiently large N, provided the coefficients f, q,, y satisfy certain compatibility conditions. The triple (f, p, w) E Et x C2 is said to be in [Fif f is a nonvanishing function with the winding number zero and Re(q.%)
Re(cy_%) # 0,
# 0,
where f,:=
exp
&Jarg(_/-(eir))dr 0
.
(12)
149
E. Wegert /Nonlinear Riemann-Hilbert problems Obviously, [F is an open subset of Ek x C2. If (f, q, I+U)E F and f is a polynomial exceeding n - 1,
of degree not
n-1 f(t)
=
(13)
Cfitj, j=O
we Write (f, f$?,ry) E [FN. Note that if (f, ~JJ,w) E [FN, then f0 # 0 and fo = f0/]f0]. Following we start with the investigation of (9)-( 11) for (f, ~1,v/) E FN. (9)-( 11) has a unique solution p E pi,
Lemma 4.1. If (f, y, y ) E [FN, the problem p = f.
(H
+
il)pNc + xsinnT+ lfl2
[23]
given bY
Yf,
(14)
-#K)(eiT)cosnrdr,
(15)
with 2n lhl2 x
‘=
7
+
Re(yf0)
;;I$;;
0 2a
‘I=
1 Re(qf0)’
(&) Im(glf0) 1 + Re(qf0) % so
(e”) -k x sin nr dz If (eiT)12
(16)
’
Proof. The boundary condition (9) and the additional conditions (lo), ( 11) form a linear system of 2n + 2 (real) equations for the n + 1 (complex) coefficients of the polynomial p, and hence it is sufficient to prove the uniqueness of the solution. Inserting the representation ( 13 ) and the ansatz (8 ) into the boundary condition Im(f(t)p(t))
= c(t)
= P,Vc(t),
v’t E TN,
(17)
we see that on both sides of ( 17) stand trigonometric polynomials Comparing the Fourier coefficients we get that on all of T, Im(f(t)p(t))
= c(t)
Because f is a polynomial and from ( 18) we infer that
= P,@(t) + Re(foPn)
that vanishes
&c(t)
+ Re(Top,)
nowhere
sinnr
lfW2 Applying the Hilbert and this yields that
operator
W + iI)
+ yy
IfI
with
Re(Top0 ) y
:=
l&32
.
n.
(18)
sinnr. on @ the function
p/f
is holomorphic
in D
(19)
vt EU.
to ( 19) we find the real part of p/f
PNc + x sinnT
x := Retfop,),
)
of degree not exceeding
up to the constant
Re(p0/ fo ),
E. Wegert /Nonlinear Riemann-Hilbert problems
150
Computing the numbers x and y from and the representation (14)-(1(j). 0 For a given positive following estimates:
number
( 11) and ,( lo),
we obtain the uniqueness
of the solution
K we denote by IFS the set of all (f, q, w) E [FN which satisfy the
llfllk G K> infIf(t)l
1 K
2 -3
Re(q,fo)
Re(wfo)
2 i,
> k.
Lemma
4.2. For each K > 0 there exists a universal constant C > 0 such that for all (f, q, v/) E iF$ the solution of (9)-( 11) fulfils the estimate
lIpIlk d C(llpNcllk + IYI + NkI1/I). Proof. From the representation
( 14) one can read off the estimate
k’llk d C(llpNellk + Nklxl + IYl). Using this result and the inequality 2n
cosnrdr
(PNc)(e"))
d CNPklJ PNcII~,
.I 0
resulting from the definition 1x1 < C (ITI +
of the norm in W2k(T), one verifies that
N-kIIPNCllk)
>
lyl
G
c
(Ivl
+
IFI +
q
IlpNcllk).
In the next step the case where (f, 9, ty) is close to the set Es is considered. Lemma
4.3. Let K > 0 be given.
Then there exist universal positive constants E and C with the (9)- ( 11) is uniquely solvable for all (f, ~1,y ) E F with
following property. The problem dist((f,yl,
w),F$)
(20)
< E,
where the distance is measured in the norm of Ei x C2. The solutions p E 5’; fulfil the estimate lIpIlk G C
(IIpNcltk
+
I?1 +
(21)
Nklyl).
Proof. If (f, q, w) is subject to (20), one can choose
Ilf -
g//k
d
&>
Iv
-
01*1 d E,
(g, p*. w*) E F$ such that
Iv - v*j < c.
On letting E sufficiently small, we can ensure that (g, 9, w) E Fz,K. By Lemma Riemann-Hilbert problems for the polynomials p(j), h(g(t)p(‘)(t)) Re(pp(j)) 0
= c(t) + Im((g(t) = y,
Re(y/p,?)
= i?,
-f
(t))p”-l’(t)),
vt E
TN,
4.2, the discrete
E. Wegert/Nonlinear Riemann-Hilbert problems are uniquely solvable. We start the iteration sequence (p(j)). On account of
Re(q(Pf+‘) we can estimate
-Py’))
= 0,
with p co) = 0 and prove the convergence
Re(t,v(Pc+‘)
by Lemma 4.2 and Proposition
llp(j+i) _p(j)llk
< CllPNIm((f < C21(Im(f
-PA”))
of the
= 0,
2.1
- g)(p(j) - g)(p(j)
< C311f - g11&“’
151
-p(j-‘)))(lk
-p(j-i))(lk
- p”-“Ilk.
Since C is a universal constant, the right-hand side is not greater than i lip(j) - p(j-‘)Il,, provided that E is sufficiently small. Because the fixed point p* of the iteration is a solution p = p* of (9)-( 11 ), the existence portion proved. The estimate solution of (9)- ( 1 1 ), a similar reasoning as above yields that p+ 11, which proves uniqueness. 0 IIP’ - P*ll 6 ; lip* Theorem 4.4. Fix (f*,q*,tg*) E F.Then there exists a natural number No and universal positive constants E and C such that the following holds. Zf N E 22,, N >/ NOand Ilf - f*llk
G E,
Iv, - v*l d 6,
then the discrete Riemann-Hilbert solution can be estimated by
(22)
Iw - w*l =s 6,
problem (9)-( 11) is uniquely solvable in the class IIp$. The
lIPIlk6 C (Ilplvcllk+ 11’1+ Nk\yJ). Proof. If e is sufficiently small, any triple (f, q, u/) which satisfies the estimate The function g defined by
(22) belongs to IF.
g:=exp((H+iZ)argf) fulfils the boundary
relation
Re(f (t)g(t))
= If (t)g(t)l,
Vt E U,
(23)
and we have g(0) = fo. By virtue of (23) the discrete boundary can be rewritten in the form
Note that for sufficiently to (22).
small E the IV!-norms
condition
of Ig/f I are uniformly
for the polynomial
bounded
p
for all f subject
E. Wegert /Nonlinear Riemann-Hilbert problems
152
In order to apply Lemma 4.3 we approximate g*(t)
=
g by polynomials.
The Taylor series
ggjt'
j=O
of the holomorphic function series yields a function
g’ : = exp( (H + il) arg f * ) E Et converges
in E,$. Truncating
this
n-l = Cgjtj,
g;(t)
j=O
and using the continuity we can estimate
properties
of the superposition
operator
and the mapping properties
of H,
llg - &Ilk G llg - g*llk + llg’ - &Ilk G Cllf - f*llk + llg’ - SJlk. If E is sufficiently small and N is sufficiently large, Lemma 4.3 applies to the triple provided that (f, ~1,w) is subject to (22). This completes the proof. 0 Remark 4.5. The assertion
of Theorem
(g, v, v/)
4.4 remains in force if one seeks the solution p in the class
n-1 PN
:=
(Pn
cosn7
+
Cpjtj>,
j=O
rather than in lP$. This follows from the observation p(t)
=
that the functions (24)
&jt'
j=O
and n-1
p(t)
= pn cosnz
+
(25)
Cpjtj
j=O
coincide
on the grid TN and that the norms of these functions
in Eh are equivalent.
In what follows we prefer to work with the class PN since the real part u and the imaginary ‘u of p E PN belong to Ef;, i.e.,
part
n-1 U(t)
=
Uo
+
uncosn7+
C(iijCOSj7+
u^jSinjs),
(26)
Gjsinjr).
(27)
j=l n-l v(t)
=
210 +
21n cosn7+
C(Vjcosjt+
j=l Unfortunately PN contains functions which are not holomorphic in D. Nevertheless, since the functions p and i in (24) and (25) have the same values on the grid TN, each solution p E PN
153
E. Wegert /Nonlinear Riemann-Hilbert problems of a discrete orthogonality
Riemann-Hilbert problem generates a solution p E “ft. Moreover, on account of the system (tj ) in IV! (U) we have for any function w E H” n E&
llw -Pll!i
of the
a ; II&z-“Ilk = + Il&J”llk,
and hence
This relation shows that each error estimate which will be obtained in the sequel for functions j? E PN is also valid for p E Pi, apart from the factor 3. With (26), (27) the discrete boundary value problem for p E PN can be written in the form
a(t)u(t) a40
+
+ b(t)vtt)
Pvo =
(28)
vt E UN,
= c(t),
(29)
Y,
Gu, + lJv, = j?.
Theorem
(30) 1 < k < m, a, b, c E Em, wind(a
Let k, m E E,,
+ ib) = 0, and assume that
pcos6
- cusin6 # 0,
(31)
pcos6
- Zsin6
(32)
# 0.
Then the solutions WN E PN of (28)-(30) with p = 0 converge in Ek to the solution w of (I), (3). If m > k, there exists a universal constant C such that (IWN
-
<
CNk-M(lw\\m.
Since the and w-plane, we assume By comparing Fourier 7 = is equivalent x Ek [w and =
Sections
2
the that = 1 coefficients the operator
problems are a = verifies that QNA(u,v,A)
with respect this case = QN(c,O,Y),
rotations
of
problem (28)-(30) where (u,v,A) E
(PNu,PNv,A)
3 for
definition
of
and PN).
denote by
the restriction
QNA to
n im QN = Ef; x Ek x R. If (f, g, v ) E im QN, the equation AN (u, v,A) = (f, g, v ) is equivalent to the discrete linear Riemann-Hilbert a(t);(t)
+ b(t);(t)
= f(t)
au0 + /IGo = V - ag(O),
problem - a(t)g(t),
vt E TN,
Gn = 0,
for the function u + iv E PN. Here (i-i,??) is related to (u,v,A) by u = u + g, v = 6, A = U(0). Therefore Theorem 4.4 in connection with Remark 4.5 tells us that AN : Xk n im QN + Xk n im QN is invertible for all sufficiently large N and that the norms of A;’ are uniformly bounded with respect to N.
154
E. Wegert /Nonlinear Riemann-Hilbert problems
Let now w = u + iv and wN = UN + ivN E PN denote the SOlUtiOnS Of the nondiscrete and the discrete problem, respectively. We use a standard argument to prove the convergence of XN := (uN,vN,uN(~)) to x := (u,v,u(O)) in X“. Since QNAXN = QNAx, we have XN - QNX = A;‘QNA(x and because jlAi’Q~Al[ llx~ -
An application
is uniformly
of Proposition
2.1 completes
x :=
y :=
11~~- QNX]~ < Cllx - QNX[[. Further,
the proof.
in force (without
Im(f(l)P(t))
of the problem 4.1 are
bounded,
XII d IIXN- QNxII + IIQNx - XII G (1 + C)llx - Q~xll.
Remark 4.7. The result remains discretization
p(l)
- QN~),
= c(t), = W =: U + iv,
0 condition
the first compatibility
(31))
for the
\Jt E TN, Re(vPn)
( 1)) (2). The corresponding
= Y,
values of x and y in the representation
( 14) of Lemma
2n
IhI Rekh)
s
(PNC)(e”) cos nrdr,
+xsinnr 1J2nfhc(eir) 5 If (ei*)12 cot(Lz)dT 0
If(
b(l)U-a(l)V
2
.
0
5. Nonlinear Riemann-Hilbert
problems
Generalizing the linear Riemann-Hilbert problem of Section 3 we now look for all holomorphic functions w = u +- iv E H”” fl Et which fulfil a nonlinear boundary relation F(t,u(t),v(t))
= 0,
vt E lr.
(33)
In what follows it is assumed that F E C’ (T x rW2)and that the gradient of F with respect to u and v never vanishes on the set M:=
{(t,u+iv)EUxC:
We study the problem
(33) together
au(O) + Pv(0) although the additional w(l)
= Y,
F(t,u,v)
= O}.
with the side condition (34)
condition
= W=:U+iV
(35)
(with F ( 1, U, I’) = 0) is more convenient to investigate the global solvability of the problem. The reader who is interested in results of this kind and in related questions may consult [ 5,16,18,20].
155
E. Wegert/Nonlinear Riemann-Hilbert problems
Here we suppose that a solution w* = U* + iv* E H” n IV: of (33) exists. With any pair of functions (u, Y ) we associate the functions a and b given by a(t)
b(t) := &F(t,u(t),v(t)),
:= &F(Cu(t),v(t)),
and we let a0 and bo be the functions associated with (u*, II* ) . It is assumed that wind(ac + ibo ) = 0. The latter is not a significant restriction, since it can always be achieved by appropriate transformations. We remark that under the above assumptions w* E EE if F E Ck ( [ 191, see [ 31 for a related result in Holder spaces). In order to establish a Newton method for the iterative solution of (33), (34) we introduce the operator A : Xk + Xk by A(u,v,A)
= (F(.,u,v),u-Hw-il,au(O)+pv(O)-y).
It follows immediately from this definition that A (u, u,A) = 0 is equivalent ;1 = u (0). In the next lemma we list some relevant properties of A.
to (33),
(34), and
Lemma 5.1. Let F E Ck+l, k > 1, and define So by (5) with a = ao, b = bo. (i) The operator A : Xk + Xk is continuously differentiable and its derivative DA(x)
at
x = (u,v,A) isgiven by DA(x)(@,il)
= (a6 + bG, 6 - HG - ;1,G(O) + @7(O)).
(ii) Let wind(ao + ibo) = 0. If j3 cos 60 - CYsin 60 # 0, then there exists a neighborhood X,k c Xk ofx* := (u*,v*, u* (0)) such that DA(x) is invertible fir all x E X,$ and the norms ofDA(x)-’ in L ( Xk ) are uniformly bounded on Xt . (iii) ZfF E C k+2, the operator A : Xk + Xk is twice continuously differentiable. For each bounded subset Xt c Xk there exists a positive number C such that for all x, y E Xk,
Proof. The assertions (i) and (iii) follow immediately from Proposition 2.2. In order to prove (ii), we let a := DUF(.,u,w), b : = D, F (+, u, u ) and define 6 by (5). Since, under the assumptions of (ii), 6 depends continuously on u and U, /? cos 6 - u: sin 6 is bounded away from zero whenever x = (u, V, A) is sufficiently close to x*. The equation DA(x)(CC& is equivalent
= (f,g,u)
to the linear Riemann-Hilbert
a2 + bv^ = f - ag,
problem
G(O) + p??(o) = V - ag(O),
for the holomorphic function 6 = G + iv^, where 6 = U - g, v^ = V, i = U(0) = G(O) + g (0). Therefore Proposition 3.1 yields the invertibility of DA(x), and a straightforward estimation of the terms in the representation of the solution gives the uniform boundedness of the inverse operators
DA(x)-‘.
0
Newton’s method xj+i following form.
= Xj - DA(xj)-'
A(xj)
for the equation
A(x) = 0 can be written in the
E. Wegert /Nonlinear Riemann-Hilbert problems
156
Step 1. Choose j = 1,
a (not necessarily
holomorphic)
gi := Ui -NV,
fi := F(.,u1,v1),
complex-valued
- Ui(O),
function
wi = u1 + iv1 . Let
Vi := /X4,(0) + /3V,(O) -y.
Step 2. Put bj := D,F(., uj,Uj),
aj := D,F(.,uj,Vj), Solve the linear Riemann-Hilbert aju + bjv = cj,
Cj :=
fi - U.jgj.
problem
cUu(0) + /IV(O) =
Vj.
Let
:=
f;+1
Vj+l
Uj-U-gj,
uj+l :=
F(.,uj+l,Vj+I),
Put j := j + 1 and repeat the procedure In each iterative
step the operator
I=
Vj - 21,
gj+l
:= 0,
Vj+l I= 0.
beginning with the second step.
H has to be computed
only twice although the computation of Since gj = 0, all iteratives Wj are is automatically satisfied beginning with the second
DA (x)-‘A (x) alone would require three complex conjugations. holomorphic iteration. Theorem
for j > 2. Also the side condition
5.2.
Let F E Ck+‘, k > 1; let ao, bo, So have the same meaning as in Lemma 5.1, assume that wind(ae + ibo) = 0, and /? cos 6. - cysin 60 # 0. (i) Zf F E C k+ l (U x R*), then for each real number q E (0,l) there exists a positive number E such thatfor j = 1,2,..., l/w1 -
W*(lk
< & *
IIwj+l
- W*llk
d
q11wj - W*llk*
(ii) ZfF E Ck+*(T x R2),th ere exist positive numbers E and C such that for j = 1,2,. IlWl -
W*llk <
E
*
Proof. We study the convergence
Xk of x* mentioned Xj+l-X*
in Lemma
. .,
IIWj+l- W*llk < CllWj -W*IIiof the sequence xj = (Uj, vj, U, (0) ). If Xj lies in the neighborhood 5.1 (ii), we have -
=
Xj -Xx*
=
DA(xj)-'(A(x*)-A(Xj)-DA(x,)(x*
=
DA(x,)-'(A(x*)-A(Xj)-DA(X*)(X*-xj))
D~4(xj)-1(A(X,)-A(X*)) -xj))
+DA(xj)-'(DA(x*)-DA(X,))(X*-xi). By Lemma
5.l(ii), IIA(x*) lIDA
CO := supXEX; (IDA(x)-‘11 is finite. Since, by the first assertion -A(xj) -DA(xj)ll
-DA(X*)(X* G &,
-Xj)ll
d
&l/x,
-x*1/,
of this lemma,
157
E. Wegert /Nonlinear Riemann-Hilhert problems provided
that ]]Xj - x*1( is sufficiently Ilxj+l
-X*1(
G
small, we get that
4llXj --*Il.
If J(xi -x*(1 Q E for sufficiently small E, the sequence (Xj) remains in the set Xt and Xj converges to x*. If F E Ck +* (T x lR* ) , local quadratic convergence of the Newton method is ensured by assertions 0 (i) and (iii) of Lemma 5.1; (see [25, Proposition 5.11, for instance). Remark 5.3. In order to construct a Newton method for Riemann-Hilbert problems with the side condition w (1) = IV, we first of all observe that this relation is locally equivalent to the condition au(l)
+ /3V(l)
= y := au
+ pv,
if only ab ( 1) - /?a ( 1) # 0. After replacing the operator B(u,w,A)
:= (F(.,u,v),u-Hv
-&(uu(l)
A by
+ pv(l)
a repetition of the above considerations shows that Theorem tion) remains in force under these modifications.
6. Discrete nonlinear Riemann-Hilbert
-y), 5.2 (without
(36) the compatibility
condi-
problems
We work under the same hypotheses as in the preceding section. The aim, of this section is to show that for all sufficiently large N the discrete nonlinear Riemann-Hilbert problem
F(t,un(t),#(t))
= 0,
auN(0)
= y,
+ pv?o)
Vt E UN,
Eiu; + IJ?J,N =0
(37) (38) (39)
has a solution wN = uN + iv N E PN and that wN converges to the solution w’ of the corresponding nondiscrete problem. In (39) U: and UC stand for the coefficients of cos no in the Fourier series of uN and wN, respectively. Theorem 6.1. Let F E Ck+‘, k 2 1; let ao, bo,& have the same meaning as in Lemma 5.1, and assume that wind (a0 + ibe) = 0. If p cos 60 - LYsin 60 # 0 and pcos 60 - Gsin 60 # 0, then the
discrete Riemann-Hilbert problem (37)-(39) has a solution wN E PN for all sufjciently large N. This solution is unique in a neighborhood of w+. There exists a universal constant C such that the defect between wN and the solution W* of the nondiscrete problem (33), (34) can be estimated by
lIWN- w*llk < c llPj@* - t”*l(k. The proof of Theorem differentiable operators.
6.1 rests on the following Newton-Kantorovich
(40) type result for uniformly
E. Wegert/Nonlinear Riemann-Hilbert problems
158
Lemma 6.2. Let X be a Banach space and U an open subset of X. Assume that the operator
B : U + X is uniformly differentiable on U and has a uniformly continuous derivative, i.e., there exists a monotone nondecreasing function q : R+ - R+ such that for all E > 0, x,y E U,
(lx -yll < V(E) * IIDB(x) -DB(y)ll
X,YE
llx-~11
U,
LetxOEU,d:=
G q(e)
< -.&
llDB(x)(y-x)-B(y)
(41) +B(x)ll
G dlx--Il.
(42)
If DB(xo) is invertible, K := IjDB(xo)-‘11 and
dist(xc,dU).
llB(xo)ll
*
6 E,
min(O
(&))
(43)
I
then the sequence (xj)i”,O defined by Xj+]
:=
-DB(xo)-‘B(xj)
xj
lies in U and converges to a solution x* of the equation B(x) = 0. The error can be estimated by IlXj - x*[[
6 2-‘+‘llxl
2-i+1KIIB(~o)ll.
- xoll <
(44)
Proof. First of all we remark that [lx1 -x011
6
llDB(x~)-‘II
= WIWXO)II
< imin(d,v
(A)).
(45)
that xj+l and xj belong to U, and that
Under the assumption Ilxj+l
llB(x~)ll
-xjll
G rl(&)
- xj+l
= xj+l
llxj-xoll
9
(46)
d q (A),
the representation xji2
-xi-DB(Xo)-‘(B(xj+l)-B(xj))
= DB(xo)-‘(DB(xo)(xj+l
-xi)
- B(Xj+l) + B(xj))
leads to the estimate
IlXj+2-Xj+lII
G IlDB(x~)-'ll
(IDB(xj)(xj+l -Xj) -B(xj+l)
+IIDB(xo)-‘II On account
of (41), Ilxj+Z -xj+lll
IIDB(xj) -DB(xo)lI
+ B(xj)ll
IJxj+l -Xjjl.
(42) and (46) we obtain that s K (&
+ &)
Ilxj+l
-xjll
= 4 Ilxj+l
(47)
-xjll.
Consequently the first relation in (46) holds with j replaced by j + 1. Using (45) one proves by induction that xj belongs to U for all j and that both estimates (46) are valid during the iteration. From (47) we get the convergence of the sequence (xi) and the error estimate (44). Obviously in the limit x* is a solution of the equation B(x) = 0. Proof of Theorem 6.1. S$ce the problem assume that G = 1 and j? = 0.
is invariant
under
rotations
of the w-plane,
we may
159
E. Wegert /Nonlinear Riemann-Hilbert problems
isasolutionofA(x*) = 0. Let A : Xk + Xk be defined as in Section 5. Then x* : = (u*,v*,u*(O)) We denote by Xt the &neighborhood of x* in X k. By virtue of Lemma 5.1 (ii), the reasoning already used in the proof of Theorem 4.6 yields that the restrictions TN (x ) : = QNDA (x ) 1Xk n im QN of QNDA(x) to Xk n im QN are invertible as continuous linear operators on Xk n im QN for all x E X,k, and that the norms of their inverses SN (x ) : = (TN (x ) )-l are bounded (uniformly with respect to x E Xj and N) if only 6 is sufficiently small and N is sufficiently large. In order to apply Lemma 6.2 we let X := Xk n im QN, U := {x E X: 1x - QNX*II < 1). On account of Proposition 2.2 the operator B : = QNA : X + X fullils the assumptions (41) and (42) of Lemma 6.2. Moreover, since llQ,vll is uniformly bounded, the same function q will do for all N. Let x0 : = QNx*. Then
IIQNA(xo)II = IIQNA(QNx*) - Q~A(x*)ll d SUPIPAII IIQNIIIIQNx* - x*ll, where the supremum is to be taken over a closed ball in Xk which contains QNX* and x*. Since 2.2 the QNX* converges to x*, this ball does not depend on N, and by virtue of Proposition supremum is finite. Consequently, condition (43) is satisfied if only N is sufficiently large. An application of Lemma 6.2 proves the convergence of xj to an element xN which is a solution of QNA(x) = 0. With xN = (uN,uN,AN) we find a solution wN = uN + iVN of the discrete Riemann-Hilbert problem. Finally, the error estimate (40) follows from
llXN - X’II G INN - xoll + IIxoSince DQNA (xN ) is invertible
x*ll
(considered
< 2 11x1 -x0(1
+ 11x0 - x*11 < C ~QNx*
as an operator
-x*11.
on Xk n im QN ), the solution
xN
is unique in a certain ball centered at x N. Because the norms of (DQNA(x~))-~ are uniformly bounded, we can choose the radius of this ball to be independent of N. This, in connection with the convergence of xN to x* yields the uniqueness of the solution xN in a neighborhood of x* for sufficiently large N. 0 Writing Newton’s method for solving the operator equation QNA(x~) = 0 (where xN E Xk n imQ~) in terms of discrete Riemann-Hilbert problems, we get the following algorithm. Step 1. Choose an initial solution
uy, $
E EN. Put j = 1 and let
gi := ur - H?J,N - uY(O),
fi := F(+~,vfV),
Vi := auY(O) + /37J~(O, -7.
Step 2. Let
bj := D,F(.,ur,~r),
aj := DuF(.,ur,~r),
and solve the linear discrete Riemann-Hilbert aj(l)U(t) au(O)
+
bj(t)v(t)
+ BV (0)
= Vj,
=
Cj(t),
Cj Z= fj - ajgj,
problem vt
E %N,
un = 0.
Let N
‘j+l h+1
Z=
Ur-
U-
:=
F(.,u~N+~,~?+,),
gj,
N ‘j+l
:= gj+l
Put j := j + 1 and repeat the second step.
v,%l, :=
09
Vj+l
I=
0.
160
E. Wegert /Nonlinear Riemann-Hilbert problems
The convergence
of the sequence
Theorem 6.3. Let a~, bo, 80 have the 0, p cos a0 - a sin do # 0, sin 60 # 0, (i) If F E @+I(% x R*), k 3 number E such that for all j = 1,2,.
<& lb? - WNllk
*
wjN := ur + iv,! is the subject of the next theorem. same meaning as in Lemma 5.1, assume that wind(ao + ibo) = and let WY E PN. 1, then for each real number q E (0, 1) there exists a positive . . and all N B NO,
- WNllkd IIwjN+l
q IIw/” - zJyl,+
(ii) ZfF E Ck+*(T x R*), th ere exist positive numbers E and C such that for j = 1,2, . . . and all N 3 NO,
114 - WNllk< E *
- wq;. Ilw,N,l- WNl(/(d c \\wy
uy
Proof . Let X? := (uy, VT, (0)). By virtue of Theorem 6.1 the element xy belongs to a certain neighborhood Xj of x’ for all N B NO if only E is sufficiently small. The operators &A : Xk + Xk are continuously differentiable. If F E Ck+*, the derivative of QNA is uniformly (with respect to N) Lipschitz continuous on Xj. From Theorem 4.4 and Remark 4.5 we infer that the inverses SN(X) of the operators DQNA(x) : Xk n imQ~ --f Xk n im QN are well-defined for all x E Xj if 6 is sufficiently small and N is sufficiently large. Moreover the norms of SN (x) are uniformly bounded. Since the above iterative method is equivalent to N xj+l
= x,! - SN(X~)QNA(X~),
the proof can be linished
along the lines of the proof of Theorem
6.1.
0
Remark 6.4. By replacing the operator QNA by QNB, with B from (36), one obtains analogous results for the solvability of the discrete Riemann-Hilbert problem with the side condition w ( 1) = W and for the convergence of the corresponding Newton iteration. In this case the first compatibility condition can be omitted.
7. Implementation and test results The Newton method converges only locally and hence its practical application requires the knowledge of an appropriate initial solution. We recommend to couple the iterative method with an imbedding method, i.e., one has to construct a homotopy path which connects the problem to be solved with a problem whose solution is known. For Riemann-Hilbert problems with closed restriction curves the homotopy path must be chosen carefully, in order to avoid that the family contains problems which are “holomorphically nontraceable” (see [ 201). The simplest way to implement one of the iterative methods on a computer is a “straightforward” discretization of the algorithm described in Section 5. By “straightforward” we mean that the explicit formulas for the solution of the linear Riemann-Hilbert problem are used and that the operator H is replaced by the (negative of the) discrete Wittich operator, which acts on the values of a function on the grid TN. The Wittich operator can be efficiently computed by means of two FFTs (see [ 71). Unfortunately, this way of discretization may disturb the convergence of the iteration. It is known for quite similar problems that after a period of quadratic “convergence” the error
161
E. Wegert /Nonlinear Riemann-Hilbert problems
(possibly) decreases linearly and may finally even increase. A detailed analysis and an explanation of this effect for numerical conformal mapping is given in [24]. On the other hand, the discrete iterative method proposed in Section 6 is ready for implementation, provided that fast algorithms for the solution of discrete Riemann-Hilbert problems are available. In [ 23 1, Wegmann recommended either to use a conjugate gradient method (with computational cost 0 (N log N) ) or to reduce the problem to the solution of a system with a Toeplitz matrix (and three right-hand sides). For the latter class several fast algorithms are published; we used the asymptotically fast O(Nlog2N) algorithm developed by Heinig and Jankowski, which works without the assumption of strong nonsingularity of the matrix ( [ 9, Section 51). The same paper contains an 0 (N) algorithm for parallel computers. We have tested both methods, the “discretized” of Section 5 and the “discrete” of Section 6, in particular for the example F(t,u,v):=
v-U2-C(p,t)(c(p,t)-1),
where 1 -PCOST dw’T):=~(
with the side condition solution w*(z)
+
1_2pcosT+p2
=
psinz >y
u (0) = p. For parameter
values p E [0,1) this problem
has the exact
“:“,z’.
The condition number increases as p + 1; if p = 1, the solution has a pole at z = 1. Figures 1 and 2 show the logarithm of the (approximate) sup-norm of the error log,, l]wr - w* 1) (different markers) and the increment log,, ]lwy+ 1 - wf’ll (solid lines) during the iteration for the parameter value p = 0.8. Although the accuracy of both methods is comparable, the “discrete method” has better performance than the “discretized method”, due to the absence of the linear error branch. Remarkably we did not observe the increasing error branch which should appear after a large number of iterations. This is in contrast to the results discussed in our earlier paper [ 211 for a
N=64: +,N=,28: X,N=Z6:
N=64: +, N=lZg: x, N=2.56:o, N=5,2: *
o, N=5,2: *
Fig. 1. Logarithm of error and increment cretized method.
for dis-
Fig. 2. Logarithm
of error and increment method.
for discrete
162
E. Wegert /Nonlinear
_80’-
-?-
Riemann-Hilbert
problems
_20d.-.-A_
30
40
50
N=64: +,N=128:x,N=256:o,N=512:
m
70
*
Fig. 3. Logarithm of error and increment for method in [21]. quite similar iterative method for Riemann-Hilbert problems given in parametric form. In order to draw a direct comparison we have applied this earlier method to the above example. The restriction curves have been parametrized by pr(s)
= is-s*-c((p,t)(c(p,t)
- 1).
Figure 3 illustrates the results of our computations. So far we have no explanation completely different “long time behavior” shown in Figs. 1 and 3.
for the
Acknowledgements I wish to thank Harald Heidler from FORTRAN code of the “discrete method” and suggesting several improvements.
the Technische Universittit Chemnitz for writing the and two anonymous referees for correcting some errors
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[ 111 J.W. Helton, Operator Theory, Analytic Functions, Matrices, and Electrical Engineering (Amer. Mathematical Sot., Providence, RI, 1987). [ 121 J.W. Helton and D.E. Marshall, Frequency domain design and analytic selections, Indiana Univ. Math. J. 39 (1990) 157-184. [ 13 ] N.I. Muskhelishvili, Singular Integral Equations (Noordhoff, Groningen, 1946). [ 14 ] Z. Slodkowski, Polynomial hulls with convex sections and interpolating spaces, Proc. Amer. Math. Sac. 294 (1986) 367-377. [ 151 Z. Slodkowski, Polynomial hulls with convex fibers and complex geodesics, J. Funct. Anal. 94 ( 1990) 156-176. [ 161 A.I. Snirel’man, A degree of quasi linear-like mapping and the nonlinear Hilbert problem, Mat. Sb. 89 (113) (1972) 366-389 (in Russian). [ 171 T. Valent, Boundary Value Problems of Finite Elasticity, Springer Tracts Nat. Philos. 31 (Springer, New York, 1988). [ 181 L. von Wolfersdorf, A class of nonlinear Riemann-Hilbert problems for holomorphic functions, Math. Nachr. 116 (1984) 89-107. [ 191 E. Wegert, Geometrische Methoden fiir nichtlineare Randwertaufgaben vom Riemann-Hilbertschen Typus, Dissertation B, Bergakademie Freiberg, 1988. [20] E. Wegert, Boundary value problems and extremal problems for holomorphic functions, Complex Variables Theory Appl. 11 (1989) 233-256. [ 2 1 ] E. Wegert, An iterative method for solving nonlinear Riemann-Hilbert problems, J. Comput. Appl. Math. 29 (3) (1990) 311-327. [22] E. Wegert, Boundary value problems and best approximation by holomorphic functions, J. Approx. Theory 61 ( 1990) 322-334. [23] R. Wegmann, Discrete Riemann-Hilbert problems, interpolation of simply closed curves, and numerical conformal mapping, J. Comput. Appl. Math. 23 (3) (1988) 323-352. [24] R. Wegmann, Discretized versions of Newton type iterative methods for conformal mapping, J. Comput. Appl. Math. 29 (2) (1990) 207-224. [25] E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol. Z (Springer, New York, 1986).