A new parallel solver suited for arbitrary semilinear parabolic partial differential equations based on generalized random trees

TL;DR

The paper expands the class of semilinear parabolic PDEs solvable by probabilistic means by introducing generalized random trees to represent nonlinear terms, removing the need for a constant potential and allowing arbitrary initial data. It develops two strategies (A and B) to obtain probabilistic representations, both leading to series that may diverge and thus requiring Pade approximants for summation. These representations are integrated into a generalized probabilistic domain decomposition (PDD) method, enabling pointwise solution evaluation and highly parallel, fault-tolerant computation on massive systems. Numerical experiments show that Strategy B yields linear-in-time computational growth and strong scalability up to 512 processors, with Pade-based summation providing accurate results in many cases and favorable comparisons to classical solvers.

Abstract

A probabilistic representation for initial value semilinear parabolic problems based on generalized random trees has been derived. Two different strategies have been proposed, both requiring generating suitable random trees combined with a Pade approximant for approximating accurately a given divergent series. Such series are obtained by summing the partial contribution to the solution coming from trees with arbitrary number of branches. The new representation greatly expands the class of problems amenable to be solved probabilistically, and was used successfully to develop a generalized probabilistic domain decomposition method. Such a method has been shown to be suited for massively parallel computers, enjoying full scalability and fault tolerance. Finally, a few numerical examples are given to illustrate the remarkable performance of the algorithm, comparing the results with those obtained with a classical method.

Figures (16)

• Figure 1: Two possible configuration diagrams. In (a) only one splitting event occurs, with $\alpha_1=3$. Rather in (b) two of such events occur, the first one with $\alpha_1=2$, and the second one with $\alpha_2=3$.
• Figure 2: Configuration diagram for the case of 4 branches and 3 splitting events. Here $S_i$ is a random time uniformly distributed between the previous generated time, and the final time, $T$. The corresponding labels $i$ of the random time $S_i$ are defined according to the rule explained in the text.
• Figure 3: Configuration diagrams for the case of 2, and 3 branches, illustrating the notation used in the probabilistic representation obtained for Strategy A in Eq. (\ref{['rep_strategyB']}). Here $\alpha_i$ denotes the power of the nonlinearity, chosen randomly between $2$ and $m$, and governing the number of branches created every time a splitting event $i$ takes place; $y_i$ denotes the position of the $i$th path of the stochastic process expiring at a given time ${\bar{S}}$. By ${\bar{S}}$ we denote the corresponding global time obtained by summing conveniently the random times $S_i$ according to the rules described in the text.
• Figure 4: Configuration diagram with $k$ branches. See Fig. 3 for a detailed explanation of the corresponding labels.
• Figure 5: Comparison between the probability function $P(k,m)$ obtained analytically and numerically simulating $10^6$ random trees for $m=2$ in (a), and $m=3$ in (b).