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In this work we introduce a theory of stochastic integration with respect to general cylindrical semimartingales defined on a locally convex space $\Phi$. Our construction of the stochastic integral is based on the theory of tensor products…
We study a numerical method to compute probability density functions of solutions of stochastic differential equations. The method is sometimes called the numerical path integration method and has been shown to be fast and accurate in…
Polynomial Chaos Expansions represent a powerful tool to simulate stochastic models of dynamical systems. Yet, deriving the expansion's coefficients for complex systems might require a significant and non-trivial manipulation of the model,…
Covariant stochastic partial (pseudo-)differential equations are studied in any dimension. In particular a large class of covariant interacting local quantum fields obeying the Morchio-Strocchi system of axioms for indefinite quantum field…
The existence and uniqueness of measure-valued solutions to stochastic nonlinear, non-local Fokker-Planck equations is proven. This type of stochastic PDE is shown to arise in the mean field limit of weakly interacting diffusions with…
The construction of stochastic solutions for nonlinear partial differential equations is a powerful method to obtain new exact results and to develop efficient numerical algorithms, in particular when domain decomposition techniques are…
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…
We derive novel algorithms for optimization problems constrained by partial differential equations describing multiscale particle dynamics, including non-local integral terms representing interactions between particles. In particular, we…
In this paper, we show that the concept of sigma-convergence associated to stochastic processes can tackle the homogenization of stochastic partial differential equations. In this regard, the homogenization problem for a stochastic…
An algorithm is proposed, analyzed, and tested experimentally for solving stochastic optimization problems in which the decision variables are constrained to satisfy equations defined by deterministic, smooth, and nonlinear functions. It is…
Physics Informed Neural Networks is a numerical method which uses neural networks to approximate solutions of partial differential equations. It has received a lot of attention and is currently used in numerous physical and engineering…
A comprehensive convergence and stability analysis of some probabilistic numerical methods designed to solve Cauchy-type inverse problems is performed in this study. Such inverse problems aim at solving an elliptic partial differential…
This paper develops a probabilistic numerical method for solution of partial differential equations (PDEs) and studies application of that method to PDE-constrained inverse problems. This approach enables the solution of challenging inverse…
The paper develops the method for construction of families of particular solutions to some classes of nonlinear Partial Differential Equations (PDE). Method is based on the specific link between algebraic matrix equations and PDE.…
We consider the problem of approximating the solution of variational problems subject to the constraint that the admissible functions must be convex. This problem is at the interface between convex analysis, convex optimization, variational…
Most inverse problems from physical sciences are formulated as PDE-constrained optimization problems. This involves identifying unknown parameters in equations by optimizing the model to generate PDE solutions that closely match measured…
Strong and weak approximation errors of a spatial finite element method are analyzed for stochastic partial differential equations(SPDEs) with one-sided Lipschitz coefficients, including the stochastic Allen--Cahn equation, driven by…
Some of recent developments, including recent results, ideas, techniques, and approaches, in the study of degenerate partial differential equations are surveyed and analyzed. Several examples of nonlinear degenerate, even mixed, partial…
We examine the challenges associated with numerical integration when applying Neural Networks to solve Partial Differential Equations (PDEs). We specifically investigate the Deep Ritz Method (DRM), chosen for its practical applicability and…
In this paper, we propose a numerical method to approximate the solution of partial differential equations in irregular domains with no-flux boundary conditions by means of spectral methods. The main features of this method are its…