Related papers: Parallelisation of partial differential equations …
Optimization methods are at the core of many problems in signal/image processing, computer vision, and machine learning. For a long time, it has been recognized that looking at the dual of an optimization problem may drastically simplify…
We consider inverse problems estimating distributed parameters from indirect noisy observations through discretization of continuum models described by partial differential or integral equations. It is well understood that the errors…
This paper develops a probabilistic numerical method for solution of partial differential equations (PDEs) and studies application of that method to PDE-constrained inverse problems. This approach enables the solution of challenging inverse…
Modeling real-world problems with partial differential equations (PDEs) is a prominent topic in scientific machine learning. Classic solvers for this task continue to play a central role, e.g. to generate training data for deep learning…
Understanding how the dynamics of a given quantum system with many degrees of freedom is altered by the presence of a generic perturbation is a notoriously difficult question. Recent works predict that, in the overwhelming majority of…
Reciprocal transformations mix the role of the dependent and independent variables of (nonlinear partial) differential equations to achieve simpler versions or even linearized versions of them. These transformations help in the…
An overview is given of the methods for treating complicated problems without small parameters, when the standard perturbation theory based on the existence of small parameters becomes useless. Such complicated problems are typical of…
We study the convergence of the new family of mimetic finite difference schemes for linear diffusion problems recently proposed in [38]. In contrast to the conventional approach, the diffusion coefficient enters both the primary mimetic…
A remarkable number of different numerical algorithms can be understood and analyzed using the concepts of symmetric spaces and Lie triple systems, which are well known in differential geometry from the study of spaces of constant curvature…
We show that any second order linear ordinary diffrential equation with constant coefficients (including the damped and undumped harmonic oscillator equation) admits an exact discretization, i.e., there exists a difference equation whose…
Recent years have witnessed a growth in mathematics for deep learning--which seeks a deeper understanding of the concepts of deep learning with mathematics and explores how to make it more robust--and deep learning for mathematics, where…
We consider simulation of spatially one-dimensional space-time fractional diffusion. Whereas in an earlier paper of ours we have developed the basic theory of what we call parametric subordination via three-fold splitting applied to…
Systems of partial differential equations lie at the heart of physics. Despite this, the general theory of these systems has remained rather obscure in comparison to numerical approaches such as finite element models and various other…
The efficient solution of sparse, linear systems resulting from the discretization of partial differential equations is crucial to the performance of many physics-based simulations. The algorithmic optimality of multilevel approaches for…
The aim of this paper is to study symmetries of linearly singular differential equations, namely, equations that can not be written in normal form because the derivatives are multiplied by a singular linear operator. The concept of…
Computing many eigenpairs of the Schr{\"o}dinger operator presents a computational bottleneck in large-scale quantum simulations due to the global communication overhead of explicit orthogonalization. To address this issue, we propose a…
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…
In this paper we present an algebraic dimension-oblivious two-level domain decomposition solver for discretizations of elliptic partial differential equations. The proposed parallel solver is based on a space-filling curve partitioning…
Taking advantage of the flexibility of the variational method with coordinate transformations, we derive a self-consistent set of equations of motion from a discretized Lagrangian to study kinetic plasmas using a Fourier decomposed…
We present new convergence analyses for parallel subspace correction methods for unconstrained semicoercive and nearly semicoercive convex optimization problems, generalizing the theory of singular and nearly singular linear problems to a…