On the numerical recovery of a holomorphic mapping from a finite set of approximate values

Prieto, José María Marbán; Palencia, C.

doi:10.1007/s002110100363

Cited by 3 publications

(7 citation statements)

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“…Since typically P h = O(1) or P h = O(| ln h|) (see, e.g., [45]), (3.1) shows that essentially N = O(| ln ε|). Thus, the algorithm requires a moderate amount of memory and accurately solving the required least square problems (see [33]) is rather cheap. Therefore, the efficiency of the new method relies on that of the numerical integration of the underlying forward problem.…”

Section: The Numerical Algorithm For the Bcpmentioning

confidence: 99%

“…The coefficients in the expansion (2.6) will be denoted by To end this section we briefly describe the numerical algorithm in [33] for the recovery of holomorphic mappings.…”

Section: Notation and Preliminariesmentioning

confidence: 99%

“…The goal is to recoverf from knowledge of the approximate values w+δ w (see [33]). We also assume that |δ w n | ≤ ρ, 1 ≤ n ≤ N , and that |f (s)| ≤ H, s ∈ D, where ρ and H are a priori known.…”

Section: Notation and Preliminariesmentioning

confidence: 99%

“…Working with this basis, problem (2.9) reduces to a matrix format which can be efficiently solved by means of the SVD (see [22,33]). Finally, since…”

Section: Notation and Preliminariesmentioning

confidence: 99%

“…Our algorithm is based on the realistic assumption that (1.6) can be integrated forward in time, i.e., for t ≥ T , and it consists of two basic steps: (i) a standard forward integration in order to get fully discrete approximationsû n h (belonging to discrete spaces X h ) to the values u(t n ) of the solution at suitable future nodes t n ≥ T , 1 ≤ n ≤ N , and (ii) the numerical holomorphic recovery of u(t), 0 < t ≤ T , based on the previous approximations [33]. Let ε be the total uncertainty in the first step, i.e., the contributions of δ u T and the error of the forward integration procedure.…”

Section: Introductionmentioning

confidence: 99%

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A New Numerical Method for Backward Parabolic Problems in the Maximum-Norm Setting

Prieto

Palencia²

2002

SIAM J. Numer. Anal.

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A new method for solving numerically backward parabolic problems is proposed. As usual for this kind of ill posed problems, it is assumed that an a priori bound for the solution is available. The algorithm consists of two basic steps. First, a standard forward integration is performed, in order to approximate the solution at suitable future time levels. Second, a holomorphic recovery procedure is carried out, providing the required approximations for the preceding times. The analysis is valid in the maximum-norm setting, and rigorous estimates are derived. Among other advantages, the method can also be applied to nonlinear problems, and it produces a continuous output. Some numerical illustrations are presented.Key words. ill posed problems, backward parabolic problems, maximum-norm, holomorphic recovery, harmonic measure, Chebyshev nodes, least squares method AMS subject classifications. 65J10, 65J20, 65M15, 65M20, 65M30 PII. S00361429013864221. Introduction. The present paper is devoted to introducing and analyzing a new numerical method for backward parabolic problems. Our main estimate relies on an assumption (see (2.7) in section 2) that, for standard discretizations of classical parabolic problems, holds only in the maximum-norm setting. However, in order to outline the main difficulties associated with the kind of ill-posed problems we have in mind, let us start by adopting an abstract point of view. Thus, let X be a complex Banach space and let A : D(A) ⊂ X → X be the infinitesimal generator of a holomorphic semigroup S(t), t ≥ 0, of linear and bounded operators in X. We do not assume that A is densely defined so that S(t) might fail to be continuous at t = 0 (see, e.g., [48]). In particular, this allows us to consider diffusion problems in the context of X = L ∞ . It is well known that the forward Cauchy problem

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Section: The Numerical Algorithm For the Bcpmentioning

confidence: 99%

“…The coefficients in the expansion (2.6) will be denoted by To end this section we briefly describe the numerical algorithm in [33] for the recovery of holomorphic mappings.…”

Section: Notation and Preliminariesmentioning

confidence: 99%

Section: Notation and Preliminariesmentioning

confidence: 99%

“…Working with this basis, problem (2.9) reduces to a matrix format which can be efficiently solved by means of the SVD (see [22,33]). Finally, since…”

Section: Notation and Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A New Numerical Method for Backward Parabolic Problems in the Maximum-Norm Setting

Prieto

Palencia²

2002

SIAM J. Numer. Anal.

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F John's stability conditions versus A Carasso's SECB constraint for backward parabolic problems

Lee

Sheen

2009

Inverse Problems

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In order to solve backward parabolic problems F. John [Comm. Pure. Appl. Math. (1960)] introduced the two constraints " u(T ) ≤ M " and u(0) − g ≤ δ where u(t) satisfies the backward heat equation for t ∈ (0, T ) with the initial data u(0).The slow-evolution-from-the-continuation-boundary (SECB) constraint has been introduced by A. Carasso in [SIAM J. Numer. Anal. (1994)] to attain continuous dependence on data for backward parabolic problems even at the continuation boundary t = T . The additional "SECB constraint" guarantees a significant improvement in stability up to t = T. In this paper we prove that the same type of stability can be obtained by using only two constraints among the three. More precisely, we show that the a priori boundedness condition u(T ) ≤ M is redundant. This implies that the Carasso's SECB condition can be used to replace the a priori boundedness condition of F. John with an improved stability estimate. Also a new class of regularized solutions is introduced for backward parabolic problems with an SECB constraint. The new regularized solutions are optimally stable and we also provide a constructive scheme to compute. Finally numerical examples are provided.

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Laplace Transform Method for Parabolic Problems with Time-Dependent Coefficients

Lee¹,

Lee²,

Sheen³

2013

SIAM J. Numer. Anal.

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(Wednesday) Time : 3.30pm-4.30pm Venue: MAS Executive Classroom 1, MAS-03-06 School of Physical and Mathematical Sciences Laplace transformation method has proven to be very efficient to deal with parabolic problems whose coefficients are time-independent, and is easily parallelizable. Applications include solving integrodifferential equations backward parabolic problems, and option pricing. The numerical schemes proposed and analyzed in these papers are based fundamentally on the line of thoughts from the earlier work of D. Sheen, I. Sloan and V. Thomee (2000, 2003). However, the method has been doubtful to be applicable to any nonlinear or linear problems whose coefficients are time-dependent. The reason is that the Laplace transform of two time-dependent functions leads to a convolution of the Laplace transformed functions in the dual variable. In this paper, we propose a method of Laplace transformation to linear parabolic problems with time-dependent coefficients, which is as efficient as the method for parabolic problems time-independent coefficients. Several numerical results are provided, which support the efficiency of the proposed scheme. This is a joint work with Hyoseop Lee (Bell

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On the numerical recovery of a holomorphic mapping from a finite set of approximate values

Cited by 3 publications

References 21 publications

A New Numerical Method for Backward Parabolic Problems in the Maximum-Norm Setting

A New Numerical Method for Backward Parabolic Problems in the Maximum-Norm Setting

F John's stability conditions versus A Carasso's SECB constraint for backward parabolic problems

Laplace Transform Method for Parabolic Problems with Time-Dependent Coefficients

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