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)

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.