Related papers: Matrix oriented reduction of space-time Petrov-Gal…
Because of the finiteness of the life span and boundedness of the physical space, the more reasonable or physical choice is the tempered power-law instead of pure power-law for the CTRW model in characterizing the waiting time and jump…
A novel compressed matrix format is proposed that combines an adaptive hierarchical partitioning of the matrix with low-rank approximation. One typical application is the approximation of discretized functions on rectangular domains; the…
Solving time-dependent Partial Differential Equations (PDEs) using a densely discretized spatial domain is a fundamental problem in various scientific and engineering disciplines, including modeling climate phenomena and fluid dynamics.…
Partial Differential Equations (PDEs) are the bedrock for modern computational sciences and engineering, and inherently computationally expensive. While PDE foundation models have shown much promise for simulating such complex…
We propose a procedure for the numerical approximation of invariance equations arising in the moment matching technique associated with reduced-order modeling of high-dimensional dynamical systems. The Galerkin residual method is employed…
We analyze Galerkin discretizations of a new well-posed mixed space-time variational formulation of parabolic PDEs. For suitable pairs of finite element trial spaces, the resulting Galerkin operators are shown to be uniformly stable. The…
We study linear parabolic initial-value problems in a space-time variational formulation based on fractional calculus. This formulation uses "time derivatives of order one half" on the bi-infinite time axis. We show that for linear,…
In this paper, we develop an ensemble-based time-stepping algorithm to efficiently find numerical solutions to a group of linear, second-order parabolic partial differential equations (PDEs). Particularly, the PDE models in the group could…
Partial Differential Equations (PDEs) are fundamental tools for modeling physical phenomena, yet most PDEs of practical interest cannot be solved analytically and require numerical approximations. The feasibility of such numerical methods,…
We propose a probabilistic way for reducing the cost of classical projection-based model order reduction methods for parameter-dependent linear equations. A reduced order model is here approximated from its random sketch, which is a set of…
We consider a sequence of elliptic partial differential equations (PDEs) with different but similar rapidly varying coefficients. Such sequences appear, for example, in splitting schemes for time-dependent problems (with one coefficient per…
Solving time-dependent partial differential equations (PDEs) that exhibit sharp gradients or local singularities is computationally demanding, as traditional physics-informed neural networks (PINNs) often suffer from inefficient point…
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…
Higher-order accurate solution to electromagnetic scattering problems are obtained at reduced computational cost in a {\it p}-variable finite volume time domain method. Spatial operators of lower, including first-order accuracy, are…
Distributed-order PDEs are tractable mathematical models for complex multiscaling anomalous transport, where derivative orders are distributed over a range of values. We develop a fast and stable Petrov-Galerkin spectral method for such…
We propose a new approach for solving systems of conservation laws that admit a variational formulation of the time-discretized form, and encompasses the p-system or the system of elastodynamics. The approach consists of using constrained…
Physics-informed neural networks (PINNs) have shown promising potential for solving partial differential equations (PDEs) using deep learning. However, PINNs face training difficulties for evolutionary PDEs, particularly for dynamical…
We propose a space-time scheme that combines an unfitted finite element method in space with a discontinuous Galerkin time discretisation for the accurate numerical approximation of parabolic problems with moving domains or interfaces. We…
An unsteady problem is considered for a space-fractional equation in a bounded domain. A first-order evolutionary equation involves a fractional power of an elliptic operator of second order. Finite element approximation in space is…
Partial differential equations can be used to model many problems in several fields of application including, e.g., fluid mechanics, heat and mass transfer, and electromagnetism. Accurate discretization methods (e.g., finite element or…