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SDE-based methods such as denoising diffusion probabilistic models (DDPMs) have shown remarkable success in real-world sample generation tasks. Prior analyses of DDPMs have been focused on the exponential Euler discretization, showing…
We develop two new variants of alternating direction methods of multipliers (ADMM) and two parallel primal-dual decomposition algorithms to solve a wide range class of constrained convex optimization problems. Our approach relies on a novel…
This paper studies first-order algorithms for solving fully composite optimization problems over convex and compact sets. We leverage the structure of the objective by handling its differentiable and non-differentiable components…
The solution of a sparse system of linear equations is ubiquitous in scientific applications. Iterative methods, such as the Preconditioned Conjugate Gradient method (PCG), are normally chosen over direct methods due to memory and…
In this paper we provide nearly linear time algorithms for several problems closely associated with the classic Perron-Frobenius theorem, including computing Perron vectors, i.e. entrywise non-negative eigenvectors of non-negative matrices,…
This survey describes a class of methods known as "fast direct solvers". These algorithms address the problem of solving a system of linear equations $\boldsymbol{Ax}=\boldsymbol{b}$ arising from the discretization of either an elliptic PDE…
We consider the problem of efficiently solving large-scale linear least squares problems that have one or more linear constraints that must be satisfied exactly. Whilst some classical approaches are theoretically well founded, they can face…
Recent work has explored solver strategies for the linear system of equations arising from a spectral Galerkin approximation of the solution of PDEs with parameterized (or stochastic) inputs. We consider the related problem of a matrix…
This work is concerned with linear matrix equations that arise from the space-time discretization of time-dependent linear partial differential equations (PDEs). Such matrix equations have been considered, for example, in the context of…
In this work, we propose a preconditioned augmented Lagrangian method (ALM) for solving semidefinite programming (SDP) problems. The preconditioner is implemented via a weighted penalty function in the ALM subproblem, with the weight matrix…
We propose a class of very simple modifications of gradient descent and stochastic gradient descent. We show that when applied to a large variety of machine learning problems, ranging from logistic regression to deep neural nets, the…
In numerical approaches to solving differential equations on a lattice, a representation of the derivative operator that correctly matches the continuum behaviour of functions of momentum up to the band limit must be non-local. We present…
In this paper, we propose a low rank approximation method for efficiently solving stochastic partial differential equations. Specifically, our method utilizes a novel low rank approximation of the stiffness matrices, which can significantly…
Newton systems in quadratic programming (QP) methods are often solved using direct Cholesky or LDL factorizations. When the linear systems in successive iterations differ by a low-rank modification (as is common in active set and augmented…
A robust algorithm for non-negative matrix factorization (NMF) is presented in this paper with the purpose of dealing with large-scale data, where the separability assumption is satisfied. In particular, we modify the Linear Programming…
Low-rank matrix approximation is extremely useful in the analysis of data that arises in scientific computing, engineering applications, and data science. However, as data sizes grow, traditional low-rank matrix approximation methods, such…
We consider parametric families of partial differential equations--PDEs where the parameter $\kappa$ modifies only the (1,1) block of a saddle point matrix product of a discretization below. The main goal is to develop an algorithm that…
The factorization of skew-symmetric matrices is a critically understudied area of dense linear algebra, particularly in comparison to that of general and symmetric matrices. While some algorithms can be adapted from the symmetric case, the…
This paper introduces an algorithm for the nonnegative matrix factorization-and-completion problem, which aims to find nonnegative low-rank matrices X and Y so that the product XY approximates a nonnegative data matrix M whose elements are…
Persistent homology is a leading tool in topological data analysis (TDA). Many problems in TDA can be solved via homological -- and indeed, linear -- algebra. However, matrices in this domain are typically large, with rows and columns…