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Among the family of fourth-order time integration schemes, the two-stage Gauss--Legendre method, which is an implicit Runge--Kutta method based on collocation, is the only superconvergent. The computational cost of this implicit scheme for…
Fully implicit timestepping methods have several potential advantages for atmosphere/ocean simulation. First, being unconditionally stable, they degrade more gracefully as the Courant number increases, typically requiring more solver…
The main objective of this series of papers is to explore the entire landscape of numerical methods for fast nonlinear Fourier transformation (NFT) within the class of integrators known as the exponential integrators. In this paper, we…
Cryptographic primitives have been used for various non-cryptographic objectives, such as eliminating or reducing randomness and interaction. We show how to use cryptography to improve the time complexity of solving computational problems.…
In this paper, two novel classes of implicit exponential Runge-Kutta (ERK) methods are studied for solving highly oscillatory systems. First of all, we analyze the symplectic conditions of two kinds of exponential integrators, and present a…
This paper presents an efficient reversible algorithm for linear regression, both with and without ridge regression. Our reversible algorithm matches the asymptotic time and space complexity of standard irreversible algorithms for this…
This paper develops a new class of linearly implicit time integration schemes called Linearly-Implicit Runge-Kutta-W (LIRK-W) methods. These schemes are based on an implicit-explicit approach which does not require a splitting of the right…
In many problems in Computational Physics and Chemistry, one finds a special kind of sparse matrices, termed "banded matrices". These matrices, which are defined as having non-zero entries only within a given distance from the main…
We address the general mathematical problem of computing the inverse $p$-th root of a given matrix in an efficient way. A new method to construct iteration functions that allow calculating arbitrary $p$-th roots and their inverses of…
Suitable discretizations through tensor product formulas of popular multidimensional operators (diffusion or diffusion--advection, for instance) lead to matrices with $d$-dimensional Kronecker sum structure. For evolutionary Partial…
We address a linear fractional differential equation and develop effective solution methods using algorithms for inversion of triangular Toeplitz matrices and the recently proposed QTT format. The inverses of such matrices can be computed…
An iterative algorithm is adopted to construct approximate representations of matrices describing the scattering properties of arbitrary objects. The method is based on the implicit evaluation of scattering responses from iteratively…
We present algorithms to compute the Smith Normal Form of matrices over two families of local rings. The algorithms use the \emph{black-box} model which is suitable for sparse and structured matrices. The algorithms depend on a number of…
We develop several efficient algorithms for the classical \emph{Matrix Scaling} problem, which is used in many diverse areas, from preconditioning linear systems to approximation of the permanent. On an input $n\times n$ matrix $A$, this…
When the inverse of an algorithm is well-defined -- that is, when its output can be deterministically transformed into the input producing it -- we say that the algorithm is invertible. While one can describe an invertible algorithm using a…
We formulate the problem of matrix completion with and without side information as a non-convex optimization problem. We design fastImpute based on non-convex gradient descent and show it converges to a global minimum that is guaranteed to…
In this paper we have considered a finite unitary matrix group with exact elements being unknown and only approximate elements available. Such a group becomes inconsistent with its own multiplication table. We found simple correction…
We present an algorithm to reduce the computational effort for the multiplication of a given matrix with an unknown column vector. The algorithm decomposes the given matrix into a product of matrices whose entries are either zero or integer…
The computation of the radiative transfer equation is expensive mainly due to two stiff terms: the transport term and the collision operator. The stiffness in the former comes from the fact that particles (such as photons) travels at the…
Stiff systems of ordinary differential equations (ODEs) arise in a wide range of scientific and engineering disciplines and are traditionally solved using implicit integration methods due to their stability and efficiency. However, these…