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In this paper, Particle-in-Cell algorithms for the Vlasov-Poisson system are presented based on its Poisson bracket structure. The Poisson equation is solved by finite element methods, in which the appropriate finite element spaces are…
An efficient direct solver for volume integral equations with O(N) complexity for a broad range of problems is presented. The solver relies on hierarchical compression of the discretized integral operator, and exploits that off-diagonal…
We consider flux-corrected finite element discretizations of 3D convection-dominated transport problems and assess the computational efficiency of algorithms based on such approximations. The methods under investigation include…
The parallel linear equations solver capable of effectively using 1000+ processors becomes the bottleneck of large-scale implicit engineering simulations. In this paper, we present a new hierarchical parallel master-slave-structural…
This paper develops and analyzes a fully discrete finite element method for a class of semilinear stochastic partial differential equations (SPDEs) with multiplicative noise. The nonlinearity in the diffusion term of the SPDEs is assumed to…
Randomized linear system solvers have become popular as they have the potential to reduce floating point complexity while still achieving desirable convergence rates. One particularly promising class of methods, random sketching solvers,…
We explore how the analysis of the Carleman linearization can be extended to dynamical systems on infinite-dimensional Hilbert spaces with quadratic nonlinearities. We demonstrate the well-posedness and convergence of the truncated Carleman…
We are interested in the numerical solution of coupled nonlinear partial differential equations (PDEs) in two and three dimensions. Under certain assumptions on the domain, we take advantage of the Kronecker structure arising in standard…
The computation of the radiative transfer equation is expensive mainly due to two stiff terms: the transport term and the collision operator. The stiffness in the former comes from the fact that particles (such as photons) travels at the…
Coarse-grained modeling and efficient computer simulations are critical to the study of complex molecular processes with many degrees of freedom and multiple spatiotemporal scales. Variational implicit-solvent model (VISM) for biomolecular…
In this paper we propose a new class of iterative regularization methods for solving ill-posed linear operator equations. The prototype of these iterative regularization methods is in the form of second order evolution equation with a…
In this work, a systematic protocol is proposed to automatically parametrize implicit solvent models with polar and nonpolar components. The proposed protocol utilizes the classical Poisson model or the Kohn-Sham density functional theory…
The Poisson-Boltzmann equation (PBE) is a nonlinear elliptic PDE that arises in biomolecular modeling and is a fundamental tool for structural biology. It is used to calculate electrostatic potentials around an ensemble of fixed charges…
We present a deep learning-based iterative approach to solve the discrete heterogeneous Helmholtz equation for high wavenumbers. Combining classical iterative multigrid solvers and convolutional neural networks (CNNs) via preconditioning,…
This paper introduces a new robust interior point method analysis for semidefinite programming (SDP). This new robust analysis can be combined with either logarithmic barrier or hybrid barrier. Under this new framework, we can improve the…
Neural networks have shown promising potential in accelerating the numerical simulation of systems governed by partial differential equations (PDEs). Different from many existing neural network surrogates operating on high-dimensional…
A discretization scheme for variable coefficient elliptic PDEs in the plane is presented. The scheme is based on high-order Gaussian quadratures and is designed for problems with smooth solutions, such as scattering problems involving soft…
High-order implicit shock tracking (fitting) is a class of high-order numerical methods that use numerical optimization to simultaneously compute a high-order approximation to a conservation law solution and align elements of the…
This is a work in two parts in which we show how to solve a large class of Lindblad master equations for non-interacting particles on $L$ sites. In part I we concentrate on bosonic particles. We show how to reduce the problem to…
In this paper, a class of smoothing modulus-based iterative method was presented for solving implicit complementarity problems. The main idea was to transform the implicit complementarity problem into an equivalent implicit fixed-point…