Related papers: Multiple Right-Hand Side Techniques in Semi-Explic…
In the present paper we consider the semiclassical magnetic Schr\"odinger equation, which describes the dynamics of charged particles under the influence of a electro-magnetic field. The solution of the time-dependent Schr\"odinger equation…
We explore the applicability of splitting methods involving complex coefficients to solve numerically the time-dependent Schr\"odinger equation. We prove that a particular class of integrators are conjugate to unitary methods for…
One of the open problems in scientific computing is the long-time integration of nonlinear stochastic partial differential equations (SPDEs). We address this problem by taking advantage of recent advances in scientific machine learning and…
Finite element representations of Maxwell's equations pose unusual challenges inherent to the variational representation of the `curl-curl' equation for the fields. We present a variational formulation based on classical field theory.…
Lasso problems arise in many areas, including signal processing, machine learning, and control, and are closely connected to sparse coding mechanisms observed in neuroscience. A continuous-time ordinary differential equation (ODE)…
We study solution techniques for an evolution equation involving second order derivative in time and the spectral fractional powers, of order $s \in (0,1)$, of symmetric, coercive, linear, elliptic, second-order operators in bounded domains…
We consider high order, implicit Runge-Kutta schemes to solve time-dependent stiff PDEs on dynamically adapted grids generated by multiresolution analysis for unsteady problems disclosing localized fronts. The multiresolution finite volume…
We extend the tensor-product direct solver from the Laplacian to the Schr\"odinger operator $-\Delta + V$. When the potential $V_1$ is separable, the operator $-\Delta + V_1$ is inverted or exponentiated at cost $O(N^{1+1/d})$ in $d$…
In this paper, a fast solver is studied for saddle point system arising from a second-order Crank-Nicolson discretization of an initial-valued parabolic PDE constrained optimal control problem, which is indefinite and ill-conditioned.…
Neural Ordinary Differential Equations (ODEs) represent a significant advancement at the intersection of machine learning and dynamical systems, offering a continuous-time analog to discrete neural networks. Despite their promise, deploying…
Modeling physical phenomena like heat transport and diffusion is crucially dependent on the numerical solution of partial differential equations (PDEs). A PDE solver finds the solution given coefficients and a boundary condition, whereas an…
This work aims to construct an efficient and highly accurate numerical method to address the time singularity at $t=0$ involved in a class of time-fractional parabolic integro-partial differential equations in one and two dimensions. The…
Numerical methods for solving the ideal magnetohydrodynamic (MHD) equations in more than one space dimension must confront the challenge of controlling errors in the discrete divergence of the magnetic field. One approach that has been…
This study focuses on the numerical discretization methods for the continuous-time discounted linear-quadratic optimal control problem (LQ-OCP) with time delays. By assuming piecewise constant inputs, we formulate the discrete system…
We develop new dynamically orthogonal tensor methods to approximate multivariate functions and the solution of high-dimensional time-dependent nonlinear partial differential equations (PDEs). The key idea relies on a hierarchical…
We study an inverse problem associated with an eddy current model. We first address the ill-posedness of the inverse problem by proving the compactness of the forward map with respect to the conductivity and the non-uniqueness of the…
A time domain electric field volume integral equation (TD-EFVIE) solver is proposed for analyzing electromagnetic scattering from dielectric objects with Kerr nonlinearity. The nonlinear constitutive relation that relates electric flux and…
Time-dependent wave equations represent an important class of partial differential equations (PDE) for describing wave propagation phenomena, which are often formulated over unbounded domains. Given a compactly supported initial condition,…
We discuss a new completely integrable case of the time-dependent Schroedinger equation in $R^n$ with variable coefficients for a modified oscillator, which is dual with respect to the time inversion to a model of the quantum oscillator…
Time delays are ubiquitous in industry, and they must be accounted for when designing control strategies. However, numerical optimal control (NOC) of delay differential equations (DDEs) is challenging because it requires specialized…