Related papers: Magnus integrators for linear and quasilinear dela…
Finding eigenvalue distributions for a number of sparse random matrix ensembles can be reduced to solving nonlinear integral equations of the Hammerstein type. While a systematic mathematical theory of such equations exists, it has not been…
A Lagrangian numerical scheme for solving nonlinear degenerate Fokker-Planck equations in space dimensions $d\ge2$ is presented. It applies to a large class of nonlinear diffusion equations, whose dynamics are driven by internal energies…
A classification of integrable two-component systems of non-evolutionary partial differential equations that are analogous to the Camassa-Holm equation is carried out via the perturbative symmetry approach. Independently, a classification…
We propose a method for transformating linear and nonlinear hypersingular integral equations into ordinary differential equations. Linear and nonlinear polyhypersingular integral equations are transformed into partial differential…
This paper focuses on discussing Newton's method and its hybrid with machine learning for the steady state Navier-Stokes Darcy model discretized by mixed element methods. First, a Newton iterative method is introduced for solving the…
A linearized numerical scheme is proposed to solve the nonlinear time fractional parabolic problems with time delay. The scheme is based on the standard Galerkin finite element method in the spatial direction, the fractional Crank-Nicolson…
This article aims to demonstrate and discuss the applications of automatic differentiation (AD) for finding derivatives in PDE-constrained optimization problems and Jacobians in non-linear finite element analysis. The main idea is to…
Sometimes it is necessary to obtain a numerical integration using only discretised data. In some cases, the data contains singularities which position is known but does not coincide with a discretisation point, and the jumps in the function…
Using exhaustion method and finite differences a new method to solve system of partial differential equations and is presented. This method allows design algorithm to solve linear and nonlinear systems in irregular domains. Applying this…
We consider the problem of robust matrix completion, which aims to recover a low rank matrix $L_*$ and a sparse matrix $S_*$ from incomplete observations of their sum $M=L_*+S_*\in\mathbb{R}^{m\times n}$. Algorithmically, the robust matrix…
We describe novel subgradient methods for a broad class of matrix optimization problems involving nuclear norm regularization. Unlike existing approaches, our method executes very cheap iterations by combining low-rank stochastic…
This work proposes a suite of numerical techniques to facilitate the design of structure-preserving integrators for nonlinear dynamics. The celebrated LaBudde-Greenspan integrator and various energy-momentum schemes adopt a difference…
We consider a linear algebra approach to establishing a discrete comparison principle for a nonmonotone class of quasilinear elliptic partial differential equations. In the absence of a lower order term, we require local conditions on the…
Discretizations of infinite-dimensional variational inequalities lead to linear and nonlinear complementarity problems with many degrees of freedom. To solve these problems in a parallel computing environment, we propose two active-set…
This paper is a summary of the theory of discrete embeddings introduced in [5]. A discrete embedding is an algebraic procedure associating a numerical scheme to a given ordinary differential equation. Lagrangian systems possess a…
The stability of integrators dealing with high order Differential Algebraic Equations (DAEs) is a major issue. The usual procedures give rise to instabilities that are not predicted by the usual linear analysis, rendering the common checks…
Some of recent developments, including recent results, ideas, techniques, and approaches, in the study of degenerate partial differential equations are surveyed and analyzed. Several examples of nonlinear degenerate, even mixed, partial…
Efficient numerical solvers for partial differential equations empower science and engineering. One of the commonly employed numerical solvers is the preconditioned conjugate gradient (PCG) algorithm which can solve large systems to a given…
This work is devoted to find the numerical solutions of several one dimensional second-order ordinary differential equations. In a heuristic way, in such equations the quadratic logistic maps regarded as a local function are inserted within…
Magnus expansion (ME) provides a general way to expand the real-time propagator of a time-dependent Hamiltonian within the exponential such that the unitarity is satisfied at any order. We use this property and explicit integration of…