Speed of Convergence of Chernoff Approximations to Solutions of Evolution Equations

Vedenin, A. V.; Voevodkin, V. S.; Galkin, V. D.; Karatetskaya, E. Yu.; Remizov, Ivan D.

doi:10.1134/s0001434620090151

Cited by 7 publications

(9 citation statements)

References 7 publications

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“…Now, let us check the convergence speed for u 0 = [x → sin(x)] and Chernoff function (8), the prototype of which was proposed in [20]. The main result of this paragraph is presented in the following Theorem 5.…”

Section: Analysis Of Convergence Speed Formentioning

confidence: 99%

“…For the further convenience of the interpretation of numerical studies we provide the definition of an approximation subspace proposed in [20].…”

Section: Conjectures On Convergence Speed For Chernoff Approximationsmentioning

confidence: 99%

“…We will examine later in this chapter that for the heat equation this is not the case already. Here, we will analyse the convergence speed of Chernoff approximations defined by ( 15), (16), and (17) for u 0 = [x → sin(x)] to the exact solution given by (20). The graphs of the initial condition and investigated solution for t = 1 are shown in Figure 15.…”

Section: Numerical Experiments On Model Examplesmentioning

confidence: 99%

“…Nevertheless, this area is not well studied at the present time in general. Starting from [20] members of I.D. Remizov's group examine the problem systematically.…”

Section: Introductionmentioning

confidence: 99%

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Speed of convergence of Chernoff approximations for two model examples: heat equation and transport equation

Prudnikov

2020

Preprint

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Paul Chernoff in 1968 proposed his approach to approximations of one-parameter operator semigroups while trying to give a rigorous mathematical meaning to Feynman's path integral formulation of quantum mechanics. In early 2000's Oleg Smolyanov noticed that Chernoff's theorem may be used to obtain approximations to solutions of initial-value problems for linear partial differential equations (LPDEs) of evolution type with variable coefficients, including parabolic equations, Schrödinger equation, and some other. Chernoff expressions are explicit formulas containing variable coefficients of LPDE and the initial condition, hence they can be used as a numerical method for solving LPDEs. However, the speed of convergence of such approximations at the present time is understudied which makes it risky to employ this class of numerical methods.In the present paper we take two equations with known solutions (heat equation and transport equation) and study both analytically and numerically the speed of decay of the norm of the difference between Chernoff approximations and exact solutions. We also provide graphical illustrations of convergence and its rate. These model examples, being relatively simple, allow to demonstrate general properties of Chernoff approximations. The observations obtained build a base for the future employment of the approach based on Chernoff's theorem to the problem of construction of new numerical methods for solving initial-value problem for parabolic LPDEs with variable coefficients.

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Section: Analysis Of Convergence Speed Formentioning

confidence: 99%

“…For the further convenience of the interpretation of numerical studies we provide the definition of an approximation subspace proposed in [20].…”

Section: Conjectures On Convergence Speed For Chernoff Approximationsmentioning

confidence: 99%

Section: Numerical Experiments On Model Examplesmentioning

confidence: 99%

“…Nevertheless, this area is not well studied at the present time in general. Starting from [20] members of I.D. Remizov's group examine the problem systematically.…”

Section: Introductionmentioning

confidence: 99%

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Speed of convergence of Chernoff approximations for two model examples: heat equation and transport equation

Prudnikov

2020

Preprint

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“…Let us first provide examples of arbitrary high and arbitrary slow convergence. Our first examples of such kind we proposed in [6], and now we develop them. There are also some examples and results obtained by other researchers, we will add references here in the final version of the text.…”

Section: Examples Of Arbitrary Slow and Arbitrary Fast Convergencementioning

confidence: 99%

Upper and lower estimates for rate of convergence in the Chernoff product formula for semigroups of operators

Galkin¹,

Remizov²

2021

Preprint

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This paper is devoted to the speed (rate) of convergence of Chernoff approximations to strongly continuous one-parameter semigroups. We provide simple natural examples for which this convergence: is arbitrary high; is arbitrary slow; holds in the strong operator topology but does not hold in the norm operator topology. We also prove general theorem that gives estimate from above for the speed of decay of the norm of the residual appearing in Chernoff approximations. We provide also supplementary theorems which makes it easier to check the conditions of the main theorem. This text is a first draft preprint.

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Chernoff approximations of Feller semigroups in Riemannian manifolds

Mazzucchi

Moretti

Remizov³

et al. 2022

Mathematische Nachrichten

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Chernoff approximations of Feller semigroups and the associated diffusion processes in Riemannian manifolds are studied. The manifolds are assumed to be of bounded geometry, thus including all compact manifolds and also a wide range of non-compact manifolds. Sufficient conditions are established for a class of second order elliptic operators to generate a Feller semigroup on a (generally non-compact) manifold of bounded geometry. A construction of Chernoff approximations is presented for these Feller semigroups in terms of shift operators. This provides approximations of solutions to initial value problems for parabolic equations with variable coefficients on the manifold. It also yields weak convergence of a sequence of random walks on the manifolds to the diffusion processes associated with the elliptic generator. For parallelizable manifolds this result is applied in particular to the representation of Brownian motion on the manifolds as limits of the corresponding random walks.

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Speed of Convergence of Chernoff Approximations to Solutions of Evolution Equations

Cited by 7 publications

References 7 publications

Speed of convergence of Chernoff approximations for two model examples: heat equation and transport equation

Speed of convergence of Chernoff approximations for two model examples: heat equation and transport equation

Upper and lower estimates for rate of convergence in the Chernoff product formula for semigroups of operators

Chernoff approximations of Feller semigroups in Riemannian manifolds

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