Rate of Convergence of Chernoff Approximations     of Operator $$C_0$$-Semigroups

Galkin, Oleg Evgenjevich; Remizov, Ivan D.

doi:10.1134/s0001434622010345

Cited by 3 publications

(5 citation statements)

References 7 publications

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“…We see that on the initial condition with high smoothness (first line in the table), the first order of Chernoff tangency corresponds to a decreasing error rate of about 1/n, and the second order -a decreasing rate of about 1/n 2 . This is in accordance with the conjecture from [12] and theorem from [6].…”

Section: Discussionsupporting

confidence: 93%

“…As the smoothness is lost (second line in the table and below), theory from [6] stops working, and the experimental evidence is the following: the convergence speed gradually decreases and the advantages of the Chernoff function with the second order of Chernoff tangency gradually vanish. Let us present the results from the table graphically: fig.…”

Section: Discussionmentioning

confidence: 99%

“…. So one can expect that the bigger k is the better rate of convergence C(t/n) n f → e tL f as n → ∞ will be, if f belongs to the space D. This idea was proposed in [12], where two conjectures about the convergence speed were formulated explicitly, and one of them were recently proved in [7,6]. For initial conditions that are good enough and t fixed, Chernoff function with Chernoff tangency of order k by conjecture should provide u(t,…”

Section: Problem Settingmentioning

confidence: 99%

“…To our current knowledge all contributions to a very young "theory of rates of convergence in Chernoff's theorem" can be found in [8,19,20,7,6] and references therein. These papers provide estimates for the rate of convergence under some conditions but if these conditions are not satisfied then one can say nothing about the quality of Chernoff approximations.…”

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confidence: 99%

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Numerical Study of the Rate of Convergence of Chernoff Approximations to Solutions of the Heat Equation

Dragunova¹,

Garashenkova²,

Nikbakht³

et al. 2023

Preprint

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Chernoff approximations are a flexible and powerful tool of functional analysis, which can be used, in particular, to find numerically approximate solutions of some differential equations with variable coefficients. For many classes of equations such approximations have already been constructed, however, the speed of their convergence to the exact solution has not been properly studied. We developed a program in Python 3 that allows to model a wide class of Chernoff approximations to a wide class of evolution equations on the real line. After that we select the heat equation (with already known exact solutions) as a simple yet informative model example for the study of the rate of convergence of Chernoff approximations. Examples illustrating the rate of convergence of Chernoff approximations to the solution of the Cauchy problem for the heat conduction equation are constructed in the paper. Numerically we show that for initial conditions that are smooth enough the order of approximation is equal to the order of Chernoff tangency of the Chernoff function used. We also consider not smooth enough initial conditions and show how Hölder class of initial condition is related to the rate of convergence. This method of study can be applied to general second order parabolic equation with variable coefficients by a slight modification of our Python 3 code, the full text of it is provided in the appendix to the paper.

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Section: Discussionsupporting

confidence: 93%

Section: Discussionmentioning

confidence: 99%

Section: Problem Settingmentioning

confidence: 99%

mentioning

confidence: 99%

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Numerical Study of the Rate of Convergence of Chernoff Approximations to Solutions of the Heat Equation

Dragunova¹,

Garashenkova²,

Nikbakht³

et al. 2023

Preprint

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show abstract

“…Several papers on the topic showing the diversity of cases studied are [14,15,16,17,18,19,20,21], see also [12,22]. Speed of convergence of Chernoff approximations were studied in [23,13,8,9,7]. It is also interesting to compare an approach to Feynman-Kac formula without Chernoff approximations [6] with the way of obtaining Feynman-Kac formulas as a corollary of Feynman formulas which were proven with the use of the Chernoff theorem [2].…”

Section: Introductionmentioning

confidence: 99%

Chernoff approximations as a way of finding the resolvent operator with applications to finding the solution of linear ODE with variable coefficients

Remizov¹

2023

Preprint

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The method of Chernoff approximation is a powerful and flexible tool of functional analysis that in many cases allows expressing exp(tL) in terms of variable coefficients of linear differential operator L. In this paper we prove a theorem that allows us to apply this method to find the resolvent of operator L. We demonstrate this on the second order differential operator. As a corollary, we obtain a new representation of the solution of an inhomogeneous second order linear ordinary differential equation in terms of functions that are the coefficients of this equation playing the role of parameters for the problem.

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Fast converging Chernoff approximations to the solution of heat equation with variable coefficient of thermal conductivity

Vedenin

2022

Zhurnal SVMO

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This paper is devoted to a new method for constructing approximations to the solution of a parabolic partial differential equation. The Cauchy problem for the heat equation on a straight line with a variable heat conduction coefficient is considered. In this paper, a sequence of functions is constructed that converges to the solution of the Cauchy problem uniformly in the spatial variable and locally uniformly in time. The functions that make up the sequence are explicitly expressed in terms of the initial condition and the thermal conductivity coefficient, i.e. through functions that play the role of parameters. When constructing functions that converge to the solution, ideas and methods of functional analysis are used, namely, Chernoff's theorem on approximation of operator semigroups, which is why the constructed functions are called Chernoff approximations. In most previously published papers, the error (i. e., the norm of the difference between the exact solution and the Chernoff approximation with number n) does not exceed const/n. Therefore, approximations, when using which the error decreases to zero faster than const/n, we call fast convergent. This is exactly what the approximations constructed in this work are, as follows from the recently proved Galkin-Remizov theorem. Key formulas, explicit forms of constructed approximations, and proof schemes are given in the paper. The results obtained in this paper point the way to the construction of fast converging Chernoff approximations for a wider class of equations.

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Rate of Convergence of Chernoff Approximations of Operator $$C_0$$-Semigroups

Cited by 3 publications

References 7 publications

Numerical Study of the Rate of Convergence of Chernoff Approximations to Solutions of the Heat Equation

Numerical Study of the Rate of Convergence of Chernoff Approximations to Solutions of the Heat Equation

Chernoff approximations as a way of finding the resolvent operator with applications to finding the solution of linear ODE with variable coefficients

Fast converging Chernoff approximations to the solution of heat equation with variable coefficient of thermal conductivity

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