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Remarks on choosing a regularization parameter using the quasi-optimality and ratio criterion
181 knapsack problem and 3. TRISHIN V.N., An adaptive algorithm for solving the multidimensional the recognition of a monotonic Boolean function, Izv. Akad. Nauk SSSR, Tekhn. Kibernetika, No.4, 11-18, 1982. Algorithms on graphs generated by contradictory systems of conditions and 4. GAINANOV D.N., their application in quality control problems, Avtoref. Dis. na soiskanie uch. st. kand. tekhn. nauk Sverdlovsk, Ural'skii Politekhn. Inst., 1981. 5. ZHURAVLBV W.I., On an algebraic approach to the solution of recognition or classification (Probl. kibernetiki), No.33, 5-68, Nauka, Moscow, problems, in: Problems of cybernetics 1978. 6. MAZUROV VL.D. and TYAGUNOV L.I., The method of committees in pattern recognition, in: The method of committees in pattern recognition (Meted komitetov v pacposnavanii obrazovl, Inst. Matem. i Mekhan. Ural'sk. Nauchn. Tsentr. Akad. Nauk SSSR, Sverdlovsk, No.6, 10-40, 1974. Translated
U.S.S.R. Comput.Maths.Math.Phys., Printed in Great Britain
vo1.24,No.4,pp.181~182,1984 0
by E.L.S.
0041-5553/84 $10.00+0.00 1985 Pergamon Press Ltd.
SHORT COMMUNICATIONS REMARKS ON CHOOSING A REGULARIZATIONPARAMETER USING THE QUASI-OPTIMALITYANDRATIOCRITERION* A.B. BAKUSHINSKII It is shown that the methods mentioned in the general case do not generate regularizing algorithms in A.N. Tkhonov's sense.
1. A method of choosing a parameter from the rough data of the problem, termed quasioptimal, was suggested in /l/ for a variational A.N. Tikhonov scheme of a mapping approximation which is inverse to the linear bounded operator. This method has been fairly widely used in practice, and modifications /2/ andanalogues/3/ (proportioncriteria) of it were subsequently proposed. At the present time the method is formally valid (in the sense of Until the theory of regularization) only when solving linear algebraic equations /5/. recently, the question of whether these methods generate regularizing algorithms (r.a.'s) /4/ for the question of solving a linear operator equation in the general infinite dimensional In this It seems that, in general, the answer can only be negative. case remained open. note it is shown that neither the method of quasi-optimal selection, nor the ratio method, nor their modifications /2, 3/ generate r.a.' s in the general infinite dimensional case, even when they can be formally implemented. 2. We shall use the definition reference. Suppose G is a function defined Definition. The function in this subset and the mapping
of r.a.'s from /6/.
We cite it for convenience
in the metric space X with values
G is termed regularizable in some subset Xx(@-Y R(z,b)=Ra(.r). exists, acting from limA(Ra,6,z)=O VrsD, b-0
in the metric
of space
Y.
DzY. if it isdefined and such that
where A (8,.6,I)=
rrlp Pp (?d("),G(r))s
To be specific we shall consider the set (6) to be a positive semi-axis of real numbers. The explicit dependence of R&(X) on 6 is fundamental. The following simple theorem is the basis of the argument: Theorem. The mapping G is regularizable by the mapping R (,), not dependent explicitly on 6, when and only when G is extended to all X and this extension is continuous in D as a mapping defined on all X. We can take as R (.) the mapping i:- on extension of G on X continuous on D (in X) to prove the sufficiency. When proving necessity, the regularizing algorithm R (.).not dependis exactly the required extension of G to all x. ent on 6, Suppose c=.l-l. where A is a one-to-one linear mapping of the finite-dimensional Hilbert space H, in the space Hz, not having a bounded inverse mapping (e.g. the integral operator Since n-1 in L, I. We take as D DA-l - the area of definition of ..I-'. is not relatively continuous on DA-, in the general infinite dimensional case, then, obviously, the extension on D,,_, does not exist. According to the theorem, such a of A-1 t0 all llz, COntinUOUS mapping cannot be regularizable by the operator R, independent of 6. The following is a general scheme for constructing the approximate values A-0 in the methods indicated in the title, and in their modifications.
lZh.vychisl.l4at.mat.Fiz.,24,8,1258-1259,1984 "SSB24:4-L
182 A parametric is considered,
family
(2,)of solutions of the equations (aC2.I..l);,=n'i.il,f-/llst,C6. where A is a linear bounded operator from the Hilbert
space
Ii, into I!!. The
choice of a==; is then made by various means, by only the approximate element f is always The quantity 6 is not part of the constructions. used. This is specially emphasized in /5/, i.e. U=Z(~). 1; is then taken as an approximate solution. Thus, :.=H(f).
“
By virtue of the theorem and its corollaries, the methods indicated do not generate r.a.'s in the finite-dimensional case on D.,., (an generally on all the sets I)cD~-~.on which the operator .4-1 is not relatively continuous). These arguments can also be transferred to the more general case of a non-injective operator A, for which we must define A-' correspondingly. REFERENCES 1. TIKHONOV A.N. and GLASKO V.B., An approximate solution of Fredholm integral equations of the first kind. Zh. vychisl. Mat. mat. Fiz., Vo1.4, No.3,pp.S64-571, 1964. 2. LEONOV A.S., Criteria for choosing a regularization parameter when solving ill-posed problems. In: Metody reshenya nyekorrekhtnykh zadach i ikh pridlozheniya. Novosibirsk: VTS SO AN SSSR, pp.77-84, 1982. 3. ZAIKIN P.N. and MECHENOV A.S., Some problems of numerically solving integral equations of the first kind using the regularization method. Otchet VTS MGV. Moscow: No.144, Vo1.3. 1971. 4. LEONOV A.S., A validation of the choice of a regularization parameter using the criterion of quasi-optimality and ratio. Zh. vychisl. Mat. mat. Fir., ~01.18, No.6, pp.1363-1376, 1978. 5. TIKHONOV A.N. and ARSENIN V.YA., Methods of solving ill-posed problems. Moscow, Nauka, 1979. 6. VINOKUROV V-A., The concept of regularizability of discontinuous mappings. Zh. vychisl. Mat. mat. Fir., vo1.11, 1~0.5, pp.1097-1112, 1971. Translated
U.S.S.R.
Printed
Comput.Maths.Math.Phys.,Vo1.24,No.4,pp.182-185,1984 in Great Britain
by H.Z.
0041-5553/84 $10.00+0.00 0 1985 Pergamon Press Ltd.
EVALUATION OF MULTIDIMENSIONALINTEGRALS USING PEANO-CURVETYPE INVOLUTIONS* I.A. GLINKIN and A.G. SUKHAREV
A method is considered for approximately evaluating an integral over an n-dimensional cube, based on the reduction of the dimensions using a Peanocurve type involution. An estimate of the method's error is made, and the problems of its practical application are discussed_ The method is generalized to the case of integration over domains of a more complex form. Consider unit cube
the problem of evaluating D=(u=(a,,
an integral
of the function f over the n-dimensional
., K,,)~J/~~U,~‘.‘?, i=l, 2,....n).
Suppose o: [O.11-8 is a single-valued continuous mapping of the segment [O.I] on D. Then the evaluation of the integral { /(u)du can be accurately or approximately replaced by the b evaluation of the integral
for certain conditions on v. This idea, in a case when ~1 is a mapping of a Peano-curve type (see, e.g. /l/), is considered in /2/. A simiiar method of reducing the dimensions when solving extremal problems was used in /3/. 1n this note the mapping
The method of constructing an is different from that in /3/ only is as small as desired. in details. However, the structure of the mapping o_ is used essentially when proving the basic Theorem 1, and we will therefore reproduce some of the constructions /3/. *Zh.vychisl.Mat.mat.Fiz.
,24,8,1259-1263,1984
U.S.S.R. Comput.Maths.Math.Phys., Printed in Great Britain
vo1.24,No.4,pp.181~182,1984 0
by E.L.S.
0041-5553/84 $10.00+0.00 1985 Pergamon Press Ltd.
SHORT COMMUNICATIONS REMARKS ON CHOOSING A REGULARIZATIONPARAMETER USING THE QUASI-OPTIMALITYANDRATIOCRITERION* A.B. BAKUSHINSKII It is shown that the methods mentioned in the general case do not generate regularizing algorithms in A.N. Tkhonov's sense.
1. A method of choosing a parameter from the rough data of the problem, termed quasioptimal, was suggested in /l/ for a variational A.N. Tikhonov scheme of a mapping approximation which is inverse to the linear bounded operator. This method has been fairly widely used in practice, and modifications /2/ andanalogues/3/ (proportioncriteria) of it were subsequently proposed. At the present time the method is formally valid (in the sense of Until the theory of regularization) only when solving linear algebraic equations /5/. recently, the question of whether these methods generate regularizing algorithms (r.a.'s) /4/ for the question of solving a linear operator equation in the general infinite dimensional In this It seems that, in general, the answer can only be negative. case remained open. note it is shown that neither the method of quasi-optimal selection, nor the ratio method, nor their modifications /2, 3/ generate r.a.' s in the general infinite dimensional case, even when they can be formally implemented. 2. We shall use the definition reference. Suppose G is a function defined Definition. The function in this subset and the mapping
of r.a.'s from /6/.
We cite it for convenience
in the metric space X with values
G is termed regularizable in some subset Xx(@-Y R(z,b)=Ra(.r). exists, acting from limA(Ra,6,z)=O VrsD, b-0
in the metric
of space
Y.
DzY. if it isdefined and such that
where A (8,.6,I)=
rrlp Pp (?d("),G(r))s
To be specific we shall consider the set (6) to be a positive semi-axis of real numbers. The explicit dependence of R&(X) on 6 is fundamental. The following simple theorem is the basis of the argument: Theorem. The mapping G is regularizable by the mapping R (,), not dependent explicitly on 6, when and only when G is extended to all X and this extension is continuous in D as a mapping defined on all X. We can take as R (.) the mapping i:- on extension of G on X continuous on D (in X) to prove the sufficiency. When proving necessity, the regularizing algorithm R (.).not dependis exactly the required extension of G to all x. ent on 6, Suppose c=.l-l. where A is a one-to-one linear mapping of the finite-dimensional Hilbert space H, in the space Hz, not having a bounded inverse mapping (e.g. the integral operator Since n-1 in L, I. We take as D DA-l - the area of definition of ..I-'. is not relatively continuous on DA-, in the general infinite dimensional case, then, obviously, the extension on D,,_, does not exist. According to the theorem, such a of A-1 t0 all llz, COntinUOUS mapping cannot be regularizable by the operator R, independent of 6. The following is a general scheme for constructing the approximate values A-0 in the methods indicated in the title, and in their modifications.
lZh.vychisl.l4at.mat.Fiz.,24,8,1258-1259,1984 "SSB24:4-L
182 A parametric is considered,
family
(2,)of solutions of the equations (aC2.I..l);,=n'i.il,f-/llst,C6. where A is a linear bounded operator from the Hilbert
space
Ii, into I!!. The
choice of a==; is then made by various means, by only the approximate element f is always The quantity 6 is not part of the constructions. used. This is specially emphasized in /5/, i.e. U=Z(~). 1; is then taken as an approximate solution. Thus, :.=H(f).
“
By virtue of the theorem and its corollaries, the methods indicated do not generate r.a.'s in the finite-dimensional case on D.,., (an generally on all the sets I)cD~-~.on which the operator .4-1 is not relatively continuous). These arguments can also be transferred to the more general case of a non-injective operator A, for which we must define A-' correspondingly. REFERENCES 1. TIKHONOV A.N. and GLASKO V.B., An approximate solution of Fredholm integral equations of the first kind. Zh. vychisl. Mat. mat. Fiz., Vo1.4, No.3,pp.S64-571, 1964. 2. LEONOV A.S., Criteria for choosing a regularization parameter when solving ill-posed problems. In: Metody reshenya nyekorrekhtnykh zadach i ikh pridlozheniya. Novosibirsk: VTS SO AN SSSR, pp.77-84, 1982. 3. ZAIKIN P.N. and MECHENOV A.S., Some problems of numerically solving integral equations of the first kind using the regularization method. Otchet VTS MGV. Moscow: No.144, Vo1.3. 1971. 4. LEONOV A.S., A validation of the choice of a regularization parameter using the criterion of quasi-optimality and ratio. Zh. vychisl. Mat. mat. Fir., ~01.18, No.6, pp.1363-1376, 1978. 5. TIKHONOV A.N. and ARSENIN V.YA., Methods of solving ill-posed problems. Moscow, Nauka, 1979. 6. VINOKUROV V-A., The concept of regularizability of discontinuous mappings. Zh. vychisl. Mat. mat. Fir., vo1.11, 1~0.5, pp.1097-1112, 1971. Translated
U.S.S.R.
Printed
Comput.Maths.Math.Phys.,Vo1.24,No.4,pp.182-185,1984 in Great Britain
by H.Z.
0041-5553/84 $10.00+0.00 0 1985 Pergamon Press Ltd.
EVALUATION OF MULTIDIMENSIONALINTEGRALS USING PEANO-CURVETYPE INVOLUTIONS* I.A. GLINKIN and A.G. SUKHAREV
A method is considered for approximately evaluating an integral over an n-dimensional cube, based on the reduction of the dimensions using a Peanocurve type involution. An estimate of the method's error is made, and the problems of its practical application are discussed_ The method is generalized to the case of integration over domains of a more complex form. Consider unit cube
the problem of evaluating D=(u=(a,,
an integral
of the function f over the n-dimensional
., K,,)~J/~~U,~‘.‘?, i=l, 2,....n).
Suppose o: [O.11-8 is a single-valued continuous mapping of the segment [O.I] on D. Then the evaluation of the integral { /(u)du can be accurately or approximately replaced by the b evaluation of the integral
for certain conditions on v. This idea, in a case when ~1 is a mapping of a Peano-curve type (see, e.g. /l/), is considered in /2/. A simiiar method of reducing the dimensions when solving extremal problems was used in /3/. 1n this note the mapping
The method of constructing an is different from that in /3/ only is as small as desired. in details. However, the structure of the mapping o_ is used essentially when proving the basic Theorem 1, and we will therefore reproduce some of the constructions /3/. *Zh.vychisl.Mat.mat.Fiz.
,24,8,1259-1263,1984