Related papers: Error analysis of truncated expansion solutions to…
In this work, we investigate the numerical approximation of the second order non-autonomous semilnear parabolic partial differential equation (PDE) using the finite element method. To the best of our knowledge, only the linear case is…
Reynolds' lubrication approximation is used extensively to study flows between moving machine parts, in narrow channels, and in thin films. The solution of Reynolds' equation may be thought of as the zeroth order term in an expansion of the…
In this paper, a multidimensional system of parabolic partial differential equations arising in European option pricing under a regime-switching market model is studied in details. For solving that numerically, one must truncate the domain…
In this work, we determine the full expression for the global truncation error of hyperbolic partial differential equations (PDEs). In particular, we use theoretical analysis and symbolic algebra to find exact expressions for the…
In this paper, we introduce a multiscale framework based on adaptive edge basis functions to solve second-order linear elliptic PDEs with rough coefficients. One of the main results is that we prove the proposed multiscale method achieves…
We propose a general, very fast method to quickly approximate the solution of a parabolic Partial Differential Equation (PDEs) with explicit formulas. Our method also provides equaly fast approximations of the derivatives of the solution,…
In this paper we introduce a numerical method for nonlinear parabolic PDEs that combines operator splitting with deep learning. It divides the PDE approximation problem into a sequence of separate learning problems. Since the computational…
We show how a posteriori goal oriented error estimation can be used to efficiently solve the subproblems occurring in a Model Predictive Control (MPC) algorithm. In MPC, only an initial part of a computed solution is implemented as a…
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…
In uncertainty quantification for parametric partial differential equations (PDEs), it is common to model uncertain random field inputs using countably infinite sequences of independent and identically distributed random variables. The…
We investigate the first-order correction in the homogenization of linear parabolic equations with random coefficients. In dimension $3$ and higher and for coefficients having a finite range of dependence, we prove a pointwise version of…
We introduce a method for the fast numerical approximation of linear, second-order parabolic partial differential equations (PDEs for short) with time-independent coefficients based on model order reduction techniques and the Laplace…
The problem of increasing the accuracy of an approximate solution is considered for boundary value problems for parabolic equations. For ordinary differential equations (ODEs), nonstandard finite difference schemes are in common use for…
A new asymptotic expansion scheme for backward SDEs (BSDEs) is proposed.The perturbation parameter is introduced just to scale the forward stochastic variables within a BSDE. In contrast to the standard small-diffusion asymptotic expansion…
This paper develops validated computational methods for studying infinite dimensional stable manifolds at equilibrium solutions of parabolic PDEs, synthesizing disparate errors resulting from numerical approximation. To construct our…
In this paper, we introduce a higher-order multiscale method for time-dependent problems with highly oscillatory coefficients. Building on the localized orthogonal decomposition (LOD) framework, we construct enriched correction operators to…
We propose a finite-dimensional control-based method to approximate solution operators for evolutional partial differential equations (PDEs), particularly in high-dimensions. By employing a general reduced-order model, such as a deep neural…
Many high-dimensional uncertainty quantification problems are solved by polynomial dimensional decomposition (PDD), which represents Fourier-like series expansion in terms of random orthonormal polynomials with increasing dimensions. This…
When using a finite difference method to solve an initial--boundary--value problem, the truncation error is often of lower order at a few grid points near boundaries than in the interior. Normal mode analysis is a powerful tool to analyze…
This paper establishes the optimal $H^1$-norm error estimate for a nonstandard finite element method for approximating $H^2$ strong solutions of second order linear elliptic PDEs in non-divergence form with continuous coefficients. To…