Related papers: Algorithmic aspects of multigrid methods for optim…
This article introduces a general purpose framework and software to approximate partial differential equations (PDEs). The sparsity patterns of finite element discretized operators is identified automatically using the tools from…
This paper concerns models and convergence principles for dealing with stochasticity in a wide range of algorithms arising in nonlinear analysis and optimization in Hilbert spaces. It proposes a flexible geometric framework within which…
Matrix configurations coming from matrix models comprise many important aspects of modern physics. They represent special quantum spaces and are thus strongly related to noncommutative geometry. In order to establish a semiclassical limit…
The goal of this primer is to provide a relatively short exposition of the basics of multigrid methods, simplified by focusing on fundamental concepts in a variational setting. This is done by way of a quadratic energy minimization…
We study the convergence rate of Bregman gradient methods for convex optimization in the space of measures on a $d$-dimensional manifold. Under basic regularity assumptions, we show that the suboptimality gap at iteration $k$ is in…
We investigate a complex system involving multiple shapes to be optimized in a domain, taking into account geometric constraints on the shapes and uncertainty appearing in the physics. We connect the differential geometry of product shape…
The simulation of large nonlinear dynamical systems, including systems generated by discretization of hyperbolic partial differential equations, can be computationally demanding. Such systems are important in both fluid and kinetic…
In practice, optimization tasks have some structure that allows developing new algorithms for every problem with faster convergence rates. Using the structure of optimization tasks, we can propose algorithms with more optimistic convergence…
In recent years, various subspace algorithms have been developed to handle large-scale optimization problems. Although existing subspace Newton methods require fewer iterations to converge in practice, the matrix operations and full…
In this paper, we present an improved numerical algorithm for computing the intersection area of multiple circles and a complex polygon efficiently. This geometric problem is fundamental to applications such as wireless sensor networks and…
This paper puts forth a new formulation and algorithm for the elastic matching problem on unparametrized curves and surfaces. Our approach combines the frameworks of square root normal fields and varifold fidelity metrics into a novel…
Multigrid has become a popular method for solving some of the most challenging real-world computational problems, such as computational fluid dynamics (CFD). The reason for this is the very good scaling properties of multigrid, which is…
We have developed a set of techniques for performing large scale ab initio calculations using multigrid accelerations and a real-space grid as a basis. The multigrid methods permit efficient calculations on ill-conditioned systems with long…
We suggest a novel shape matching algorithm for three-dimensional surface meshes of disk or sphere topology. The method is based on the physical theory of nonlinear elasticity and can hence handle large rotations and deformations.…
We present evidence that multigrid (MG) works for wave equations in disordered systems, e.g. in the presence of gauge fields, no matter how strong the disorder. We introduce a "neural computations" point of view into large scale…
Quasi-one-dimensional quantum structures with spectra scaling faster than the square of the eigenmode number (superscaling) can generate intrinsic, off-resonant optical nonlinearities near the fundamental physical limits, independent of the…
The first order condition of the constrained minimization problem leads to a saddle point problem. A multigrid method using a multiplicative Schwarz smoother for saddle point problems can thus be interpreted as a successive subspace…
The article explores an encoding and structural information processing approach using sparse bit vectors and fixed-length linear vectors. The following are presented: a discrete method of speculative stochastic dimensionality reduction of…
We present a systematic computational framework for generating positive quadrature rules in multiple dimensions on general geometries. A direct moment-matching formulation that enforces exact integration on polynomial subspaces yields…
Visually sorted grid layouts provide an efficient method for organizing high-dimensional vectors in two-dimensional space by aligning spatial proximity with similarity relationships. This approach facilitates the effective sorting of…