Monte Carlo approximations of the Neumann problem

We introduce Monte Carlo methods to compute the solution of elliptic equations with pure Neumann boundary conditions. We first prove that the solution obtained by the stochastic representation has a zero mean value with respect to the invariant measure of the stochastic process associated to the equation. Pointwise approximations are computed by means of standard and new simulation schemes especially devised for local time approximation on the boundary of the domain. Global approximations are computed thanks to a stochastic spectral formulation taking into account the property of zero mean value of the solution. This stochastic formulation is asymptotically perfect in terms of conditioning. Numerical examples are given on the Laplace operator on a square domain with both pure Neumann and mixed Dirichlet-Neumann boundary conditions. A more general convection-diffusion equation is also numerically studied

Numerical implementation of some reweighted path integral methods

The reweighted random series techniques provide finite-dimensional approximations to the quantum density matrix of a physical system that have fast asymptotic convergence. We study two special reweighted techniques that are based upon the Levy-Ciesielski and Wiener-Fourier series, respectively. In agreement with the theoretical predictions, we demonstrate by numerical examples that the asymptotic convergence of the two reweighted methods is cubic for smooth enough potentials. For each reweighted technique, we propose some minimalist quadrature techniques for the computation of the path averages. These quadrature techniques are designed to preserve the asymptotic convergence of the original methods.Comment: 15 pages, 10 figures, submitted to JC

An overview on deep learning-based approximation methods for partial differential equations

It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). Recently, several deep learning-based approximation algorithms for attacking this problem have been proposed and tested numerically on a number of examples of high-dimensional PDEs. This has given rise to a lively field of research in which deep learning-based methods and related Monte Carlo methods are applied to the approximation of high-dimensional PDEs. In this article we offer an introduction to this field of research, we review some of the main ideas of deep learning-based approximation methods for PDEs, we revisit one of the central mathematical results for deep neural network approximations for PDEs, and we provide an overview of the recent literature in this area of research.Comment: 23 page

An explicit substructuring method for overlapping domain decomposition based on stochastic calculus

In a recent paper [{\em F. Bernal, J. Mor\'on-Vidal and J.A. Acebr\'on, Comp.$\&$ Math. App. 146:294-308 (2023)}] an hybrid supercomputing algorithm for elliptic equations has been put forward. The idea is that the interfacial nodal solutions solve a linear system, whose coefficients are expectations of functionals of stochastic differential equations confined within patches of about subdomain size. Compared to standard substructuring techniques such as the Schur complement method for the skeleton, the hybrid approach renders an explicit and sparse shrunken matrix -- hence suitable for being substructured again. The ultimate goal is to push strong scalability beyond the state of the art, by leveraging the scope for parallelisation of stochastic calculus. Here, we present a major revamping of that framework, based on the insight of embedding the domain in a cover of overlapping circles (in two dimensions). This allows for efficient Fourier interpolation along the interfaces (now circumferences) and -- crucially -- for the evaluation of most of the interfacial system entries as the solution of small boundary value problems on a circle. This is both extremely efficient (as they can be solved in parallel and by the pseudospectral method) and free of Monte Carlo error. Stochastic numerics are only needed on the relatively few circles intersecting the domain boundary. In sum, the new formulation is significantly faster, simpler and more accurate, while retaining all of the advantageous properties of PDDSparse. Numerical experiments are included for the purpose of illustration

The PDD method for solving linear, nonlinear, and fractional PDEs problems

We review the Probabilistic Domain Decomposition (PDD) method for the numerical solution of linear and nonlinear Partial Differential Equation (PDE) problems. This Domain Decomposition (DD) method is based on a suitable probabilistic representation of the solution given in the form of an expectation which, in turns, involves the solution of a Stochastic Differential Equation (SDE). While the structure of the SDE depends only upon the corresponding PDE, the expectation also depends upon the boundary data of the problem. The method consists of three stages: (i) only few values of the sought solution are solved by Monte Carlo or Quasi-Monte Carlo at some interfaces; (ii) a continuous approximation of the solution over these interfaces is obtained via interpolation; and (iii) prescribing the previous (partial) solutions as additional Dirichlet boundary conditions, a fully decoupled set of sub-problems is finally solved in parallel. For linear parabolic problems, this is based on the celebrated Feynman-Kac formula, while for semilinear parabolic equations requires a suitable generalization based on branching diffusion processes. In case of semilinear transport equations and the Vlasov-Poisson system, a generalization of the probabilistic representation was also obtained in terms of the Method of Characteristics (characteristic curves). Finally, we present the latest progress towards the extension of the PDD method for nonlocal fractional operators. The algorithm notably improves the scalability of classical algorithms and is suited to massively parallel implementation, enjoying arbitrary scalability and fault tolerance properties. Numerical examples conducted in 1D and 2D, including some for the KPP equation and Plasma Physics, are given.info:eu-repo/semantics/acceptedVersio

Copolymer Networks and Stars: Scaling Exponents

We explore and calculate the rich scaling behavior of copolymer networks in solution by renormalization group methods. We establish a field theoretic description in terms of composite operators. Our 3rd order resummation of the spectrum of scaling dimensions brings about remarkable features: The special convexity properties of the spectra allow for a multifractal interpretation while preserving stability of the theory. This behavior could not be found for power of field operators of usual $\phi^4$ field theory. The 2D limit of the mutually avoiding walk star apparently corresponds to results of a conformal Kac series. Such a classification seems not possible for the 2D limit of other copolymer stars. We furthermore provide a consistency check of two complementary renormalization schemes: epsilon expansion and renormalization at fixed dimension, calculating a large collection of independent exponents in both approaches.Comment: 30 pages, revtex, figures: 5 latex, 1 postscript. New Figs. Text improved. Citations adde