Related papers: Basis adaptation and domain decomposition for stea…
Dynamical systems that are subject to continuous uncertain fluctuations can be modelled using Stochastic Differential Equations (SDEs). Controlling such system results in solving path constrained SDEs. Broadly, these problems fall under the…
High-dimensional partial differential equations (PDE) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment (CVA) models, or portfolio optimization models. The PDEs in…
Solving general high-dimensional partial differential equations (PDE) is a long-standing challenge in numerical mathematics. In this paper, we propose a novel approach to solve high-dimensional linear and nonlinear PDEs defined on arbitrary…
Numerical resolution of high-dimensional nonlinear PDEs remains a huge challenge due to the curse of dimensionality. Starting from the weak formulation of the Lawson-Euler scheme, this paper proposes a stochastic particle method (SPM) by…
Derivation of the probability density evolution provides invaluable insight into the behavior of many stochastic systems and their performance. However, for most real-time applica-tions, numerical determination of the probability density…
We introduce an adaptive element-based domain decomposition (DD) method for solving saddle point problems defined as a block two by two matrix. The algorithm does not require any knowledge of the constrained space. We assume that all sub…
Domain adaptation approaches have shown promising results in reducing the marginal distribution difference among visual domains. They allow to train reliable models that work over datasets of different nature (photos, paintings etc), but…
We consider parameter identification problems in parametrized partial differential equations (PDE). This leads to nonlinear ill-posed inverse problems. One way to solve them are iterative regularization methods, which typically require…
We develop innovative algorithms for solving the strong-constraint formulation of four-dimensional variational data assimilation in large-scale applications. We present a space-time decomposition approach that employs domain decomposition…
Stochastic partial differential equations (SPDEs) are ubiquitous in engineering and computational sciences. The stochasticity arises as a consequence of uncertainty in input parameters, constitutive relations, initial/boundary conditions,…
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…
Stochastic differential equations (SDEs) are established tools to model physical phenomena whose dynamics are affected by random noise. By estimating parameters of an SDE intrinsic randomness of a system around its drift can be identified…
Differential equations (DEs) are commonly used to describe dynamic systems evolving in one (ordinary differential equations or ODEs) or in more than one dimensions (partial differential equations or PDEs). In real data applications the…
This article is focused on two related topics within the study of partial differential equations (PDEs) that illustrate a beautiful connection between dynamics, topology, and analysis: stability and spatial dynamics. The first is a property…
In this paper, we set the mathematical foundations of the Dynamical Low-Rank Approximation (DLRA) method for stochastic differential equations (SDEs). DLRA aims at approximating the solution as a linear combination of a small number of…
We present a new method based on functional tensor decomposition and dynamic tensor approximation to compute the solution of a high-dimensional time-dependent nonlinear partial differential equation (PDE). The idea of dynamic approximation…
Recently, various evolutionary partial differential equations (PDEs) with a mixed derivative have been emerged and drawn much attention. Nonetheless, their PDE-theoretical and numerical studies are still in their early stage. In this paper,…
Foundation models for partial differential equations (PDEs) have emerged as powerful surrogates pre-trained on diverse physical systems, but adapting them to new downstream tasks remains challenging due to limited task-specific data and…
An acoustic wave propagation problem with a log normal random field approximation for wave speed is solved using a sampling-free intrusive stochastic Galerkin approach. The stochastic partial differential equation with the inputs and…
Solving analytically intractable partial differential equations (PDEs) that involve at least one variable defined on an unbounded domain arises in numerous physical applications. Accurately solving unbounded domain PDEs requires efficient…