Related papers: A new parallel solver suited for arbitrary semilin…
Constraint problems can be trivially solved in parallel by exploring different branches of the search tree concurrently. Previous approaches have focused on implementing this functionality in the solver, more or less transparently to the…
We present a Bayesian methodology for infinite as well as finite dimensional parameter identification for partial differential equation models. The Bayesian framework provides a rigorous mathematical framework for incorporating prior…
In this paper, we introduce a tensor neural network based machine learning method for solving the elliptic partial differential equations with random coefficients in a bounded physical domain. With the help of tensor product structure, we…
We provide sufficient conditions for the existence of viscosity solutions of fractional semilinear elliptic PDEs of index $\alpha \in (1,2)$ with polynomial gradient nonlinearities on $d$-dimensional balls, $d\geq 2$. Our approach uses a…
Mathematical modeling is a standard approach to solve many real-world problems and {\em diversity} of solutions is an important issue, emerging in applying solutions obtained from mathematical models to real-world problems. Many studies…
Partial differential equations have a wide range of applications in modeling multiple physical, biological, or social phenomena. Therefore, we need to approximate the solutions of these equations in computationally feasible terms. Nowadays,…
This paper is concerned with model order reduction of parametric Partial Differential Equations (PDEs) using tree-based library approximations. Classical approaches are formulated for PDEs on Hilbert spaces and involve one single linear…
In this paper, we prove that there exists a unique solution to the Dirichlet boundary value problem for a general class of semilinear second order elliptic partial differential equations. Our approach is probabilistic. The theory of…
The numerical solution methods for partial differential equation (PDE) solution allow obtaining a discrete field that converges towards the solution if the method is applied to the correct problem. Nevertheless, the numerical methods…
Randomized algorithms provide solutions to two ubiquitous problems: (1) the distributed calculation of a principal component analysis or singular value decomposition of a highly rectangular matrix, and (2) the distributed calculation of a…
Machine learning based partial differential equations (PDEs) solvers have received great attention in recent years. Most progress in this area has been driven by deep neural networks such as physics-informed neural networks (PINNs) and…
In this paper, we investigate adaptive nonlinear regression and introduce tree based piecewise linear regression algorithms that are highly efficient and provide significantly improved performance with guaranteed upper bounds in an…
This paper presents an iterative method suitable for inverting semilinear problems which are important kernels in many numerical applications. The primary idea is to employ a parametrization that is able to reduce semilinear problems into…
In the theory and practice of inverse problems for partial differential equations (PDEs) much attention is paid to the problem of the identification of coefficients from some additional information. This work deals with the problem of…
Developing efficient numerical algorithms for the solution of high dimensional random Partial Differential Equations (PDEs) has been a challenging task due to the well-known curse of dimensionality. We present a new solution framework for…
Probability estimation is one of the fundamental tasks in statistics and machine learning. However, standard methods for probability estimation on discrete objects do not handle object structure in a satisfactory manner. In this paper, we…
Several recent works have developed a new, probabilistic interpretation for numerical algorithms solving linear systems in which the solution is inferred in a Bayesian framework, either directly or by inferring the unknown action of the…
We construct a space-time parallel method for solving parabolic partial differential equations by coupling the Parareal algorithm in time with overlapping domain decomposition in space. The goal is to obtain a discretization consisting of…
We propose a primal-dual parallel proximal splitting method for solving domain decomposition problems for partial differential equations. The problem is formulated via minimization of energy functions on the subdomains with coupling…
Parametric linear programming is a central operation for polyhedral computations, as well as in certain control applications.Here we propose a task-based scheme for parallelizing it, with quasi-linear speedup over large problems.This type…