Related papers: A Fast Approach to Creative Telescoping
Random field models have been widely employed to develop a predictor of an expensive function based on observations from an experiment. The traditional framework for developing a predictor with random field models can fail due to the…
In this paper, we introduce the new construction of fractional derivatives and integrals with respect to a function, based on a matrix approach. We believe that this is a powerful tool in both analytical and numerical calculations. We begin…
We expose (without proofs) a unified computational approach to integrable structures (including recursion, Hamiltonian, and symplectic operators) based on geometrical theory of partial differential equations. We adopt a coordinate based…
The dynamic iteration method with a restricted additive Schwarz splitting is investigated to co-simulate linear differential algebraic equations system coming from RLC electrical circuit with linear components. We show the pure linear…
Tight binding (TB) models are one approach to the quantum mechanical many particle problem. An important role in TB models is played by hopping and overlap matrix elements between the orbitals on two atoms, which of course depend on the…
The aim of these notes is to provide a reasonably short and "hands-on" introduction to the differential calculus on associative algebras over a field of characteristic zero. Following a suggestion of Ginzburg's we call the resulting theory…
A characteristic feature of differential-algebraic equations is that one needs to find derivatives of some of their equations with respect to time, as part of so called index reduction or regularisation, to prepare them for numerical…
Elementary techniques from operational calculus, differential algebra, and noncommutative algebra lead to a new approach for change-point detection, which is an important field of investigation in various areas of applied sciences and…
This article presents a natural extension of the tensor algebra. In addition to "left multiplications" by vectors, we can consider "derivations" by covectors as basic operators on this extended algebra. These two types of operators satisfy…
It is well known that using high-order numerical algorithms to solve fractional differential equations leads to almost the same computational cost with low-order ones but the accuracy (or convergence order) is greatly improved, due to the…
We review some aspects of the current state of data-intensive astronomy, its methods, and some outstanding data analysis challenges. Astronomy is at the forefront of "big data" science, with exponentially growing data volumes and data…
We introduce a new approach for applying sampling-based sketches to two and three mode tensors. We illustrate our technique to construct sketches for the classical problems of $\ell_0$ sampling and producing $\ell_1$ embeddings. In both…
Dijkstra observed that verifying correctness of a program is difficult and conjectured that derivation of a program hand-in-hand with its proof of correctness was the answer. We illustrate this goal-oriented approach by applying it to the…
We consider a framework for the construction of iterative schemes for operator equations that combine low-rank approximation in tensor formats and adaptive approximation in a basis. Under fairly general assumptions, we obtain a rigorous…
The article proposes formulating and codifying a set of applied numerical methods, coined as Deep Learning Discrete Calculus (DLDC), that uses the knowledge from discrete numerical methods to interpret the deep learning algorithms through…
We propose an approach to image processing related to algebraic operators acting in the space of images. In view of the interest in the applications in optics and computer science, mathematical aspects of the paper have been simplified as…
In this paper, we introduce a novel semi-analytical method for solving a broad class of initial value problems involving differential, integro-differential, and delay equations, including those with fractional and variable-order…
We derive computationally tractable methods to select a small subset of experiment settings from a large pool of given design points. The primary focus is on linear regression models, while the technique extends to generalized linear models…
Many numerical methods for multiscale differential equations require a scale separation between the larger and the smaller scales to achieve accuracy and computational efficiency. In the area of multiscale dynamical systems, so-called,…
In difference algebra, summability arises as a basic problem upon which rests the effective solution of other more elaborate problems, such as creative telescoping problems and the computation of Galois groups of difference equations. In…