Related papers: Comments on "Bayesian Solution Uncertainty Quantif…
Mechanistic knowledge about the physical world is virtually always expressed via partial differential equations (PDEs). Recently, there has been a surge of interest in probabilistic PDE solvers -- Bayesian statistical models mostly based on…
Filtering-based probabilistic numerical solvers for ordinary differential equations (ODEs), also known as ODE filters, have been established as efficient methods for quantifying numerical uncertainty in the solution of ODEs. In practical…
In this article, we propose a wellposedness theory for a class of second order backward doubly stochastic differential equation (2BDSDE). We prove existence and uniqueness of the solution under a Lipschitz type assumption on the generator,…
Many problems in engineering and sciences require the solution of large scale optimization constrained by partial differential equations (PDEs). Though PDE-constrained optimization is itself challenging, most applications pose additional…
A brief comment on A variational Bayesian approach for inverse problems with skew-t error distributions (Guha et al., Journal of Computational Physics 301 (2015) 377-393) is given in this letter.
We discuss possible discretizations of complex analysis and some of their applications to probability and mathematical physics, following our recent work with Dmitry Chelkak, Hugo Duminil-Copin and Cl\'ement Hongler.
This paper develops meshless methods for probabilistically describing discretisation error in the numerical solution of partial differential equations. This construction enables the solution of Bayesian inverse problems while accounting for…
Partial differential equations (PDEs) are among the most universal and parsimonious descriptions of natural physical laws, capturing a rich variety of phenomenology and multi-scale physics in a compact and symbolic representation. This…
The paper concerns singular solutions of nonlinear elliptic equations.
We would like to congratulate the authors of "A Bayesian Conjugate Gradient Method" on their insightful paper, and welcome this publication which we firmly believe will become a fundamental contribution to the growing field of probabilistic…
Deep learning-based partial differential equation(PDE) solvers have received much attention in the past few years. Methods of this category can solve a wide range of PDEs with high accuracy, typically by transforming the problems into…
This paper is devoted to overview of the authors works for numerical solution of singular integral equations (SIE), polysingular integral equations and multi-dimensional singular integral equations of the second kind. The authors…
Comment on the Letter ``Polynomial-Time Simulation of Pairing Models on a Quantum Computer'', L. A. Wu, M. S. Byrd and D. A. Lidar, Phys. Rev. Lett. 89, 057904 (2002).
This paper is concerned with a class of uncertain backward stochastic differential equations (UBSDEs) driven by both an $m$-dimensional Brownian motion and a $d$-dimensional canonical process with uniform Lipschitzian coefficients. Such…
Inverse problems, particularly those governed by Partial Differential Equations (PDEs), are prevalent in various scientific and engineering applications, and uncertainty quantification (UQ) of solutions to these problems is essential for…
Statistical models can involve implicitly defined quantities, such as solutions to nonlinear ordinary differential equations (ODEs), that unavoidably need to be numerically approximated in order to evaluate the model. The approximation…
We study the problem of existence, uniqueness and regularity of probabilistic solutions of the Cauchy problem for nonlinear stochastic partial differential equations involving operators corresponding to regular (nonsymmetric) Dirichlet…
We introduce DDE-Solver, a Maple package designed for solving Discrete Differential Equations (DDEs). These equations are functional equations relating algebraically a formal power series F(t, u) with polynomial coefficients in a…
The paper by Bowen, Mancini, Fessatidis, and Murawski (2012 Phys. Scr. {\bf 85}, 065005) demonstrates in a dramatic fashion the serious difficulties that can arise when one rushes to perform numerical studies before understanding the…
This article reviews the 2008 quantum algorithm for linear systems of equations due to Harrow, Hassidim and Lloyd, as well as some of the followup and related work. It was submitted to the Springer Encyclopedia of Algorithms.