Error estimates for quasi-compact Markov operators

Nonlinear Analysis 42 (2000) 85 – 95

www.elsevier.nl/locate/na

Error estimates for quasi-compact Markov operators Jiu Ding a;∗ , Fern Y. Hunt b aDepartment

of Mathematics, The University of Southern Mississippi, Hattiesburg, MS 39406-5045, USA bMathematical and Computational Sciences Division, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA Received 2 March 1998; accepted 17 July 1998

Keywords: Markov operators; Quasi-compactness

1. Introduction Studying statistical properties of chaotic deterministic dynamical systems leads to the concept of Frobenius–Perron operators which has been investigated from both theoretical and numerical aspects in the past several decades [1, 11]. In particular, attempting to prove Ulam’s conjecture on the convergence of a piecewise constant approximations scheme for computing a xed density of the Frobenius–Perron operator associated with a chaotic dynamical system has stimulated the construction of some numerical methods for solving general Markov operator equations. Among them are the higher-order Markov nite approximations that were rst proposed in [4]. Based on the pioneering work of Li [13] on the convergence of Ulam’s method for a class of piecewise C 2 and stretching mappings of an interval, the convergence of the class of Markov nite approximations methods has been proved for solving the xed point problem of Frobenius–Perron operators for the class of piecewise C 2 and stretching mappings [4]. A rst attempt for the convergence rate analysis of such methods was given in [9] with the spectral approximations technique from [2], and a more direct approach using the Lasota–Yorke inequality and the Dunford integral of bounded linear operators was presented in [3]. A recent work on the error estimate is [5], based on the previous approach in [3]. ∗

Corresponding author. Tel.: 001-601-266-5623; fax: 001-601-266-5818. E-mail address: [email protected] (J. Ding)

0362-546X/00/$ - see front matter ? 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 9 8 ) 0 0 3 3 5 - 6

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Corresponding to a nonsingular transformation de ned on a - nite measure space, the Frobenius–Perron operator is a special Markov operator. Markov operators describe the evolution of density functions in dynamical systems, since they map densities to densities (see [11] for more details on Markov operators). It is thus natural to approximate them with nite dimensional Markov operators from the viewpoint of constructing structure-preserving numerical methods. Ulam’s method and higher-order Markov nite approximations methods satisfy this requirement. Hence the numerical analysis of such methods is important. The purpose of this paper is to give an error estimate for Markov nite approximations to a class of quasi-compact Markov operators. Speci cally, we prove the convergence of Markov nite approximations, and we give error estimates for such methods, using the property of quasi-compactness and the concept of strongly stable convergence of a sequence of bounded linear operators. In the next section we prove the convergence of the numerical method, and the error estimate will be given in Section 3, and we conclude in Section 4. 2. The convergence We borrow some basic de nitions and results from [11]. Let (X; ; ) be a - nite 1 measure space, R and let L (X ) be the Banach space of complex -integrable 1functions with kfk1 = |f| d. Denote by D the set of all nonnegative functions f ∈ L (X ) such that kfk1 = 1. Any f ∈ D will be called a density. An operator P : L1 (X ) → L1 (X ) is called a Markov operator if P is linear and PD ⊂ D. It is clear that for f ∈ L1 (X ) Z Z Pf d = f d; and kPk1 = 1. Moreover, if Pf = f for real f ∈ L1 (X ), then Pf+ = f+ and Pf− = f− , where f+ = max{f; 0} and f− = max{−f; 0}. Suppose V is a dense subspace of L1 (X ) and there is a norm k kV de ned on V for which V is a Banach space and the closed unit ball of (V; k kV ) is compact in L1 (X ). In this paper, we consider a class of Markov operators P : L1 (X ) → L1 (X ) such that P : (V; k kV ) → (V; k kV ) is well de ned and is quasi-compact, that is, there is a positive integer r and a compact operator K : (V; k kV ) → (V; k kV ) such that kP r − KkV ¡1:

(1)

The existence of a xed density of P under a mild condition is from a simple compactness argument as the following result shows. Theorem 2.1. If P is a quasi-compact Markov operator as above such that kP n kV are uniformly bounded; then for any f ∈ D ∩ V , n−1

lim

n→∞ ∗

1X i P f = f∗ ; n i=0

where f ∈ V is a ÿxed density of P.

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Proof. Let T = P r − K. Then (I − T )−1 =

∞ X

Tn

n=0

is a bounded linear operator from V onto itself since kT kV ¡1. Let n−1

An =

1X i P n i=0

be the sequence of Cesaro’s sums. Then An = (I − T )−1 KAn + (I − T )−1 (An − P r An ):

(2)

Now An − P r An =

I + P + · · · + P r−1 (I − P n ): n

Since kP n kV are bounded 1 + kPkV + · · · + kP r−1 kV (1 + kP n kV ) = 0: n→∞ n

lim kAn − P r An kV ≤ lim

n→∞

Choose a density f ∈ V: Then lim (I − T )−1 (An − P r An )f = 0:

n→∞

Since (I − T )−1 K is compact and kAn fkV are bounded, the sequence (I − T )−1 KAn f has a cluster point. Thus, from Eq. (2), An f has a cluster point f∗ ∈ V . By the Kakutani–Yosida Theorem [7] lim An f = f∗

n→∞

and f∗ ∈ V is a xed density of P. In this paper, we only consider a special class of quasi-compact Markov operators for the simplicity of presentation. Let X = [0; 1] and let m denote the Lebesgue measure de ned on the Borel algebra of [0; 1]. Given a complex function f, set 1 _ 0

f=

sup

n X

0=x0 ¡x1 ¡···¡xn =1 i=1

|f(xi ) − f(xi−1 )|;

and for f ∈ L1 (0; 1), de ne (1 ) 1 _ _ f = inf g: g(x) = f(x); x ∈ [0; 1] m − a:e: : 0

0

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In other words, the variation of an equivalence class f˜ ∈ L1 (0; 1) is the in mum of the ˜ Let variations of all f ∈ f. ) ( 1 _ 1 f¡∞ BV (0; 1) = f ∈ L (0; 1): 0

with the norm kfkBV = kfk1 +

1 _

f;

∀f ∈ BV (0; 1):

0

Then V = (BV (0; 1); k kBV ) is a Banach space which is dense under the L1 norm, and any closed ball of BV (0; 1) is strongly compact in L1 (0; 1), due to the famous Helly Theorem. Now consider a class of Markov operators P : L1 (0; 1) → L1 (0; 1) that satisfy the condition that P maps BV (0; 1) into itself and there exist k ≥ 1; 0¡a¡1, and b0 ¡∞ such that 1 _ 0

Pkf ≤ a

1 _

f + b0 kfk1 ;

f ∈ BV (0; 1):

(3)

0

Then, from the de nition of k kBV , we have kP k fkBV ≤ akfkBV + bkfk1 ;

f ∈ BV (0; 1);

(4)

where b = 1 − a + b0 . Thus, by the Ionescu–Tulcea and Marinescu Theorem (see, e.g., [1] or [10]), P : BV (0; 1) → BV (0; 1) is quasi-compact. Moreover, for any f ∈ BV (0; 1), write n = ik + j with 0 ≤ j¡k, we see that kP n fkBV are uniformly bounded by kP n fkBV = kP ik+jfkBV ≤ k(P k )i fkBV + kP jfkBV ≤ ai kfkBV + b

i−1 X

al kfk1 + kP jfkBV

l=0

≤ kfkBV +

b kfk1 + kP jfkBV : 1−a

Thus, Banach’s uniform boundedness principle implies that kP n kBV are uniformly bounded. Therefore, P has a xed density f∗ ∈ BV (0; 1) by Theorem 2.1. It can be shown (see, e.g., [1]) that for a class of piecewise C 1 and stretching mappings S : [0; 1] → [0; 1] such that the function 1=|S 0 | is of bounded variation, the corresponding Frobenius–Perron operator PS : L1 (0; 1) → L1 (0; 1) satis es that there are constants 0¡¡1; C¿0, and b¿0 such that for any f ∈ BV (0; 1) and any n ≥ 1, kPSn fkBV ≤ C n kfkBV + bkfk1 : Hence, choosing k such that C k ¡1, Eq. (4) is satis ed.

(5)

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In [4, 13, 14], some Markov nite approximations of P were proposed to compute the xed density f∗ . We describe Ulam’s piecewise constant scheme and the piecewise linear Markov approximations method as follows. Divide [0; 1] into n equal parts I i = [xi−1 ; xi ] with length h = 1=n. Denote by
0n and
1n , respectively, the corresponding spaces of piecewise constant functions and continuous piecewise linear functions. Note that
0n ⊂ BV (0; 1) and
1n ⊂ BV (0; 1). A basis for
0n is ei0 (x) =  I i (x);

i = 1; : : : ; n;

where  A is the characteristic function of A, and a basis for
1n consists of   x − xi 1 ; i = 0; 1; : : : ; n; ei (x) = w h where w(x) = (1 − |x|) [−1; 1] (x): Let fi =

1 h

Z Ii

f dm

be the average value of f over I i . Now de ne Qn0 : L1 (0; 1) → L1 (0; 1) by Qn0 f =

n X

fi ei0

(6)

i=1

and Qn1 : L1 (0; 1) → L1 (0; 1) by Qn1 f = f1 e01 +

n−1 X fi + fi+1 1 ei + fn en1 : 2

(7)

i=1

Let Pnj = Qnj P for j = 0; 1. In Ulam’s method, a xed density fn0 of Pn0 is computed to approximate f∗ , while in the piecewise linear method, we solve Pn1 fn1 = fn1 instead. Some useful properties of Pnj are summarized in the following proposition; the proof is referred to [4, 5, 13]. Proposition 2.1. (i) Qnj and Pnj are Markov operators; thus Pnj has a ÿxed density in
jn . (ii) limn→∞ kQn0 f − fk1 = 0 for f ∈ L1 (0; 1). (iii) limn→∞ kQn1 f − fkBV = 0 for f ∈ BV (0; 1). (iv) kQnjfkBV ≤ kfkBV for all f ∈ BV (0; 1). From now on we assume that the Markov operator P satis es Eq. (3) with k = 1, and we assume that f∗ is the unique xed density of P. From the above proposition, for all f ∈ BV (0; 1) kPnjfkBV ≤ akfkBV + bkfk1 :

(8)

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Lemma 2.1. Suppose that {fn } ⊂ BV (0; 1) is a sequence of ÿxed points of a sequence of operators Pn = Qn P for which Eq. (8) is true. Then, kfn − f∗ kBV ≤

1 (bkfn − f∗ k1 + kf∗ − Qn f∗ kBV ): 1−a

(9)

Proof. Write f∗ − fn = Pn (f∗ − fn ) + f∗ − Qn f∗ : Then, using Eq. (8), we obtain kfn − f∗ kBV = kPn (f∗ − fn ) + f∗ − Qn f∗ kBV ≤ akfn − f∗ kBV + bkfn − f∗ k1 + kf∗ − Qn f∗ kBV : Thus Eq. (9) follows. Theorem 2.2. Let {fnj } be a sequence of ÿxed densities of Pnj in
jn . Then lim kfnj − f∗ k1 = 0:

n→∞

(10)

Moreover, lim kfn1 − f∗ kBV = 0:

n→∞

(11)

Proof. From Eq. (8), kfnj kBV = kPnjfnj kBV ≤ akfnj kBV + bkfnj k = akfnj kV + b; from which we see that fnj are uniformly bounded in the BV-norm. Hence, a subsequence fnji converge to some density g ∈ BV (0; 1) under the L1 -norm. A standard argument (see, e.g., [4]) shows that g = f∗ and lim kfnj − f∗ k1 = 0:

n→∞

This proves Eq. (10), and Eq. (11) is from Lemma 2.1 and the fact [5] that lim kf∗ − Qn1 f∗ kBV = 0:

n→∞

3. The convergence rate analysis Now we use the technique of spectral approximations that was developed in the monograph [2] for the error estimation of the piecewise linear Markov approximations method. Although the convergence of Ulam’s method in L1 (0; 1) has been proved for the class of Markov operators studied in this paper, it seems that so far the best error

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bound in the L1 -norm is O(ln n=n) [10], and the same technique in [10] applied to the piecewise linear method can only give the same order of the error bound as for the piecewise constant method. However, with the help of quasi-compactness in the space of functions of bounded variation, we are able to estimate the convergence rate in the BV -norm for the method of piecewise linear Markov approximations. Since P : (BV (0; 1); k kBV ) → (BV (0; 1); k kBV ) is quasi-compact and kP n kBV are uniformly bounded, (P), the spectrum of P, is contained in the closed unit disk with nitely many spectral points on the disk boundary. Each of these is an isolated eigenvalue and is in fact an order one pole of the resolvent of P [7]. In particular, the eigenvalue 1 of P is a pole of order 1. Now it is possible to employ the spectral approximation technique of [2]. The concept that we shall use here is that of strongly stable convergence. Let Tn be a sequence of bounded linear operators that approximate a bounded linear operator T on a Banach space X with norm k k. We say that the sequence Tn stably converges to T if Tn strongly converges to T , that is, limn→∞ Tn x = Tx for all x ∈ X , and there is a constant M and a positive integer N such that kTn−1 k ≤ M for all n ≥ N . Let  be an isolated eigenvalue of T of algebraic multiplicity l¡∞, and let be a closed Jordan curve around  such that the closed region enclosed by contains no spectral points of P but . Then the bounded linear operator Z 1 (z − T )−1 dz E ≡ E(T ) = 2i is well de ned and is a projection associated with , and dim(R(E)) = l. Now let Tn be a sequence of bounded linear operators that approximate T such that ∩ (Tn ) = ∅ for n large enough. Then we can de ne the corresponding projections Z 1 (z − Tn )−1 dz: En ≡ E(Tn ) = 2i Deÿnition 3.1 (Chatelin [2]). We say that Tn is a sequence of strongly stable approximations of T in , if (i) Tn − z stably converges to T − z for each z ∈ − {}, (ii) dim(R(En )) = l for n large enough. The following lemma is a key result for our error estimate. Its proof is referred to [2]. Lemma 3.1. If Tn is a sequence of strongly stable approximations of T in , then for n large enough, the quantity k(1 − E)fn k for fn ∈ R(En ) and kfn k = 1 is at least of the order n = k(T − Tn )Ek, that is, kfn − Efn k = O(k(T − Tn )Ek):

(12)

Now we apply the above result to estimate the convergence rate of the method of Markov nite approximations for solving the quasi-compact Markov operator equation

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Pf = f in BV (0; 1). Basically, we need to show that the sequence of Markov nite approximations Pnj of P are indeed strongly stable approximations of P in BV (0; 1) associated with the eigenvalue  = 1 if Eq. (3) is satis ed with k = 1. For this purpose, we shall use the following lemma, which appeared in the L1 (X ) context in approximating the invariant measures of randomly perturbed dissipative maps [8] and which has been proved for BV (0; 1) in [3, 5]. In the following, let Pn = Pnj for xed j = 0; 1, and let E = E(P) and En = E(Pn ) as de ned above. Note that dim R(E) = 1. Lemma 3.2. For n large enough, the eigenspace of Pn corresponding to the eigenvalue  = 1 is one dimensional and it is the same as R(En ). Moreover, there is an open disk centered at 1 such that its boundary encloses no spectral points of P and Pn except for 1. Since 0¡a¡1, we can choose small enough from the above lemma such that |z|¿a for all z ∈ . The following lemma is a basis for Lemma 3.4 which shows that Pn − z stably converges to P − z for each z ∈ − {1}. Lemma 3.3. Let z ∈ − {1}. Then there is a number K(z) such that for n large enough, k(Pn − z)−1 fk1 ≤ K(z);

∀f ∈ BV (0; 1);

kfkBV = 1:

Proof. If the lemma is not true, then there is a subsequence of {n}, which is still denoted as {n} for the simplicity of notation, and a sequence of fn ∈ BV (0; 1) with kfn kBV = 1, such that lim kgn k1 = ∞;

n→∞

where gn = (Pn − z)−1 fn . Let g˜n =

gn kgn k1

fn and fn˜ = : kgn k1

Then fn˜ = (Pn − z)g˜n and lim kfn˜ kBV = lim

n→∞

n→∞

1 = 0: kgn k1

Since z g˜n = Pn g˜n − fn˜ , |z|kg˜n kBV = kPn g˜n − fn˜ kBV ≤ kPn g˜n kBV + kfn˜ kBV ≤ akg˜n kBV + bkg˜n k1 + kfn˜ kBV = akg˜n kBV + b + kfn˜ k1 from which we have kg˜n kBV ≤

b + kfn˜ k1 ≤ C; |z| − a

∀n;

J. Ding, F.Y. Hunt / Nonlinear Analysis 42 (2000) 85 – 95

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where C is a constant. Thus, there is a subsequence g˜ni such that lim kg˜ni − gk1 = 0

i→∞

for some g ∈ BV (0; 1) and kgk1 = limi→∞ kg˜ni k1 = 1. Now from kzg − Pgk1 ≤ |z|kg − g˜ni k1 + k(z − Pni )g˜ni k1 +kPni (g˜ni − g)k1 + kPni g − Pgk1 → 0; we see that z ∈ (P), a contradiction to the fact that ∩ (P) = {1}. Lemma 3.4. Given any z ∈ − {1}, there is a number M (z) such that for n large enough, k(Pn − z)−1 kBV ≤ M (z): Proof. Given n and f ∈ BV (0; 1) with kfkBV = 1. Let gn = (Pn − z)−1 f: By Lemma 3.3, kgn k1 ≤ K(z). From (Pn − z)gn = f |z|kgn kBV = kPn gn − fkBV ≤ kPn gn kBV + 1 ≤ akgn kBV + bkgn k1 + 1 ≤ akgn kBV + bK(z) + 1: Hence, kgn kBV ≤

bK(z) + 1 ≡ M (z): |z| − a

Lemmas 3.2 and 3.4 show that {Pn } is a sequence of strongly stable approximations of P in by De nition 3.1. The following lemma was rst proved in [3] in the context of Frobenius–Perron operators. Lemma 3.5. Ef = f∗ for any density f ∈ BV (0; 1). Proof. Since R(E) = N (P − 1) = span{f∗ } is one dimensional, we see that Ef = cf∗ , where  Z Z 1 Z 1 1 Ef dm = (z − P)−1 dz f dm c= 2i 0 0 # Z "Z 1 1 −1 (z − P) f dm dz: = 2i 0

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J. Ding, F.Y. Hunt / Nonlinear Analysis 42 (2000) 85 – 95

Let g(z) = (z − P)−1 f. Then (z − P)g(z) = f, and Z

1 0

Z f dm = z

1 0

Z =z from which c=

1 2i

Z

1 0

Z g(z) dm −

1 0

Z g(z) dm −

1 0

Pg(z) dm Z g(z) dm = (z − 1)

1 0

g(z) dm;

1 dz = 1: z−1

Therefore, Ef = f∗ . Now, Lemma 3.1 leads to the main result of this section. Theorem 3.1. Let fn1 ∈
1n be a sequence of ÿxed densities of Pn1 from the method of piecewise linear Markov ÿnite approximations that approximate the unique ÿxed density f∗ of P. Then for n large enough, kfn1 − f∗ kBV = O(kf∗ − Qn1 f∗ kBV ): In particular, if f∗ ∈ C 2 [0; 1], then   1 : kfn1 − f∗ kBV = O n

(13)

(14)

Proof. To show Eq. (13), it is enough to note that Efn1 = f∗ and (P − Pn1 )Ef = Pf∗ − Qn1 Pf∗ = f∗ − Qn1 f∗ for all densities f. Eq. (14) is from Proposition 3.1 in [5]. W1 Remark 3.1. Since (see [5]) 0 (f∗ − Qn0 f∗ ) 6→ 0 in general unless f∗ is piecewise constant, there is no similar error estimate to Eq. (14) for Ulam’s method. It is believed that in general Ulam’s method does not converge under the BV -norm. As a consequence of the above theorem, we have the following error estimate for computing absolutely continuous invariant measures. Let S : [0; 1] → [0; 1] be a nonsingular transformation, that is, m(A) = 0 implies that m(S −1 (A)) = 0. The operator PS : L1 (0; 1) → L1 (0; 1) de ned by Z d f dm PS f(x) = dx S −1 ([0; x]) is called the Frobenius–Perron operator associated with S. It is well known that PS is a Markov operator and its xed density gives rise to an absolutely continuous

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invariant measure under S. From the Lasota–Yorke theorem [12] and Theorem 3.1, we immediately have Corollary 3.1. Suppose S is piecewise C 2 and satisÿes the condition that inf |S 0 |¿2. If f∗ ∈ C 2 [0; 1] is the unique ÿxed density of PS . Then the approximate densities fn from the method of piecewise linear Markov approximations satisfy   1 ∗ : kfn − f kBV = O n 4. Conclusions In this paper, we presented an alternative approach to estimating the convergence rate of approximate xed densities from piecewise linear Markov nite approximations for a class of Markov operators P : L1 (0; 1) → L1 (0; 1) that satisfy the variation inequality (3). This approach is based on the technique of strongly stable convergence in spectral approximations of bounded linear operators. Under a minor revision, our result can also be applied to the case of multi-dimensional regions. Also the idea is applicable to more general cases that Eq. (3) is satis ed for k¿1 or P has more than one xed densities. For example, the same error estimate should be valid for computing multidimensional absolutely continuous invariant measures [6], and it is possible to estimate the distance of the approximate xed density fn to the eigenspace of P corresponding to eigenvalue 1. References [1] A. Boyarsky, P. Gora, Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension, Birkhauser, Basel, 1997. [2] F. Chatelin, Spectral Approximation of Linear Operators, Academic Press, New York, 1983. [3] C. Chiu, Q. Du, T.-Y. Li, Error estimates of the Markov nite approximation of the Frobenius–Perron operator, Nonlinear Anal. 19 (5) (1992) 291–308. [4] J. Ding, T.-Y. Li, Markov nite approximations of Frobenius–Perron operator equations, Nonlinear Anal. 17 (8) (1991) 759–772. [5] J. Ding, T.-Y. Li, A convergence rate analysis for Markov nite approximations of a class of Frobenius– Perron operators, Nonlinear Anal. 31 (5=6) (1998) 765–777. [6] J. Ding, A. Zhou, Piecewise linear Markov approximations of Frobenius–Perron operators associated with multi-dimensional transformations, Nonlinear Anal. 25 (5) (1995) 399– 408. [7] N. Dunford, J. Schwartz, Linear Operators, Part I, General Theory, Interscience, New York, 1958. [8] F. Hunt, Approximating the invariant measures of randomly perturbed dissipative maps, J. Math. Anal. Appl. 198 (1996) 534–551. [9] F. Hunt, W. Miller, On the approximation of invariant measures, J. Statist. Phys. 66 (1992) 535–548. [10] G. Keller, Stochastic stability of some chaotic dynamical systems, Mh. Math. 94 (1982) 313–333. [11] A. Lasota, M. Mackey, Chaos, Fractals, and Noise, 2nd ed., Springer, Berlin, 1994. [12] A. Lasota, J.A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc. 186 (1973) 481– 488. [13] T.-Y. Li, Finite approximation for the Frobenius–Perron operator, a solution to Ulam’s conjecture, J. Approx. Theory 17 (1976) 177–186. [14] S.M. Ulam, A Collection of Mathematical Problems, Interscience, New York, 1960.