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Concrete examples of the rate of convergence of Chernoff approximations: numerical results for the heat semigroup and open questions on them (with appendix: full list of pictures and Python code)

K. A. Katalova, N. Nikbakht, I. D. Remizov

TL;DR

The paper empirically investigates the rate of convergence of Chernoff approximations to the heat semigroup by numerically solving the Cauchy problem for $u_t=u_{xx}$ with diverse initial data and two Chernoff functions. It demonstrates a robust power-law decay of the approximation error with the index $n$, with the exponent depending on both the smoothness of the initial data and the order of tangency (first vs second) of the Chernoff function. The study confirms that higher-order Chernoff tangency ($S$) generally yields faster convergence than first-order tangency ($G$), and that smoother initial data yield larger convergence exponents, in line with existing theory. It also reports several unexpected phenomena, such as oscillatory error patterns and instances where the regression fit deviates from a straight line, suggesting interesting directions for theory and further numerical exploration. Overall, the work provides a detailed data-driven view of convergence rates in Chernoff approximations and highlights open questions about superfast convergence and behavior outside the generator domain.

Abstract

The article is devoted to the construction of examples that illustrate (using computer calculations) the rate of convergence of Chernoff approximations to the solution of the Cauchy problem for the heat equation. We are interested in the Chernoff theorem in general and select the heat semigroup as a model case because this semigroup (and solutions of the heat equations) are known, so it is easy to measure the distance between the exact solution and its Chernoff approximations. Two Chernoff functions (of the first and second order of Chernoff tangency to the generator of the heat semigroup, i.e. to the operator of taking the second derivative) and several initial conditions of different smoothness are considered. From the numerically plotted graphs, visually, it is determined that the approximations are close to the solution. For each of the two Chernoff functions, for several initial conditions of different smoothness and for approximation numbers up to 11 inclusive, the error (i.e. the supremum of the absolute value of the difference between the exact solution and the approximating function) corresponding to each approximation was numerically found. As it turned out, in all the cases studied, the dependence of the error on the number of the approximation has an approximately power-law form (we call this power the order of convergence). This follows from the fact that, as we discovered, the dependence of the logarithm of the error on the logarithm of the approximation number is approximately linear. Using the considered family of initial conditions, an empirical dependence of the order of convergence on the smoothness class of the initial condition is found. The orders of convergence for all the initial conditions studied are collected in a table.

Concrete examples of the rate of convergence of Chernoff approximations: numerical results for the heat semigroup and open questions on them (with appendix: full list of pictures and Python code)

TL;DR

The paper empirically investigates the rate of convergence of Chernoff approximations to the heat semigroup by numerically solving the Cauchy problem for with diverse initial data and two Chernoff functions. It demonstrates a robust power-law decay of the approximation error with the index , with the exponent depending on both the smoothness of the initial data and the order of tangency (first vs second) of the Chernoff function. The study confirms that higher-order Chernoff tangency () generally yields faster convergence than first-order tangency (), and that smoother initial data yield larger convergence exponents, in line with existing theory. It also reports several unexpected phenomena, such as oscillatory error patterns and instances where the regression fit deviates from a straight line, suggesting interesting directions for theory and further numerical exploration. Overall, the work provides a detailed data-driven view of convergence rates in Chernoff approximations and highlights open questions about superfast convergence and behavior outside the generator domain.

Abstract

The article is devoted to the construction of examples that illustrate (using computer calculations) the rate of convergence of Chernoff approximations to the solution of the Cauchy problem for the heat equation. We are interested in the Chernoff theorem in general and select the heat semigroup as a model case because this semigroup (and solutions of the heat equations) are known, so it is easy to measure the distance between the exact solution and its Chernoff approximations. Two Chernoff functions (of the first and second order of Chernoff tangency to the generator of the heat semigroup, i.e. to the operator of taking the second derivative) and several initial conditions of different smoothness are considered. From the numerically plotted graphs, visually, it is determined that the approximations are close to the solution. For each of the two Chernoff functions, for several initial conditions of different smoothness and for approximation numbers up to 11 inclusive, the error (i.e. the supremum of the absolute value of the difference between the exact solution and the approximating function) corresponding to each approximation was numerically found. As it turned out, in all the cases studied, the dependence of the error on the number of the approximation has an approximately power-law form (we call this power the order of convergence). This follows from the fact that, as we discovered, the dependence of the logarithm of the error on the logarithm of the approximation number is approximately linear. Using the considered family of initial conditions, an empirical dependence of the order of convergence on the smoothness class of the initial condition is found. The orders of convergence for all the initial conditions studied are collected in a table.
Paper Structure (49 sections, 6 theorems, 23 equations)

This paper contains 49 sections, 6 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $C_0$-semigroups and their Chernoff approximations
  4. Theoretical bounds for the speed of convergence
  5. Open problem: do superfast converging approximations exist?
  6. $C_0$-semigroups and dynamical systems
  7. Numerical simulation results
  8. Problem setting
  9. Graphs of approximations for several trigonometric initial conditions
  10. Approximations for initial condition $u_0(x)=\sin(x)$ as the first model case
  11. Approximations for initial condition $u_0(x)=|\sin(x)|$
  12. Approximations for initial condition $\sqrt[4]{|\sin x|}$
  13. Approximations for initial condition $u_0(x)=|\sin(x)|^{3/2}$
  14. Approximations for initial condition $u_0(x)=|\sin(x)|^{13/2}$
  15. Graphs of approximations for several exponential initial conditions
  16. ...and 34 more sections

Key Result

Theorem 1

(P. R. Chernoff (1968), see EN1Chernoff). Let $\mathcal{F}$ be a Banach space, and $\mathscr{L}(\mathcal{F})$ be the space of all linear bounded operators in $\mathcal{F}$. Suppose that the operator $L\colon \mathcal{F}\supset D(L)\to \mathcal{F}$ is linear and closed, and function $C$ is defined on

Theorems & Definitions (23)

  • Definition 1
  • Definition 2
  • Remark 1
  • Theorem 1
  • Remark 2
  • Definition 3
  • Remark 3
  • Remark 4
  • Theorem 2
  • Remark 5
  • ...and 13 more