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We develop robust and scalable fully implicit nonlinear finite element solvers for the simulations of biological transportation networks driven by the gradient flow minimization of a non-convex energy cost functional. Our approach employs a…
We introduce a numerical solver for the steady-state Boltzmann equation based on the symmetric Gauss-Seidel (SGS) method. To solve the nonlinear system on each grid cell derived from the SGS method, a fixed-point iteration preconditioned…
We propose a new type of reduction for integrable systems of coupled matrix PDEs; this reduction equates one matrix variable with the transposition of another multiplied by an antisymmetric constant matrix. Via this reduction, we obtain a…
This paper introduces a discretization-accurate stopping criterion of symmetric iterative methods for solving systems of algebraic equations resulting from the finite element approximation. The stopping criterion consists of the evaluations…
The splitting method is a powerful method for solving partial differential equations. Various splitting methods have been designed to separate different physics, nonlinearities, and so on. Recently, a new splitting approach has been…
Time-implicit schemes are attractive since they allow numerical time steps that are much larger than those permitted by the Courant-Friedrich-Lewy criterion characterizing time-explicit methods. This advantage comes, however, with a cost:…
Automatic segmentation of an image to identify all meaningful parts is one of the most challenging as well as useful tasks in a number of application areas. This is widely studied. Selective segmentation, less studied, aims to use limited…
We introduce a simple, rigorous, and unified framework for solving nonlinear partial differential equations (PDEs), and for solving inverse problems (IPs) involving the identification of parameters in PDEs, using the framework of Gaussian…
In this paper, new implicit methods with reduced memory are developed for solving the time-dependent Boltzmann transport equation (BTE). One-group transport problems in 1D slab geometry are considered. The reduced-memory methods are…
In this paper, we consider effective discretization strategies and iterative solvers for nonlinear PDE-constrained optimization models for pattern evolution within biological processes. Upon a Sequential Quadratic Programming linearization…
In Compressed Sensing, a real-valued sparse vector has to be estimated from an underdetermined system of linear equations. In many applications, however, the elements of the sparse vector are drawn from a finite set. For the estimation of…
The explicit split-operator algorithm is often used for solving the linear and nonlinear time-dependent Schr\"{o}dinger equations. However, when applied to certain nonlinear time-dependent Schr\"{o}dinger equations, this algorithm loses…
Ion transport, often described by the Poisson--Nernst--Planck (PNP) equations, is ubiquitous in electrochemical devices and many biological processes of significance. In this work, we develop conservative, positivity-preserving, energy…
This manuscript proposes a probabilistic framework for algorithms that iteratively solve unconstrained linear problems $Bx = b$ with positive definite $B$ for $x$. The goal is to replace the point estimates returned by existing methods with…
This paper presents an iterative method suitable for inverting semilinear problems which are important kernels in many numerical applications. The primary idea is to employ a parametrization that is able to reduce semilinear problems into…
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 propose an optimization proxy in terms of iterative implicit gradient methods for solving constrained optimization problems with nonconvex loss functions. This framework can be applied to a broad range of machine learning settings,…
In this paper we introduce and analyze an iteratively re-weighted algorithm, that allows to approximate the weak solution of the $p$-Poisson problem for $1 < p \leq 2$ by iteratively solving a sequence of linear elliptic problems. The…
The solution of large sparse linear systems is often the most time-consuming part of many science and engineering applications. Computational fluid dynamics, circuit simulation, power network analysis, and material science are just a few…
In this work, we study the convergence and performance of nonlinear solvers for the Bidomain equations after decoupling the ordinary and partial differential equations of the cardiac system. Firstly, we provide a rigorous proof of the…