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In this work we apply the Deep Galerkin Method (DGM) described in Sirignano and Spiliopoulos (2018) to solve a number of partial differential equations that arise in quantitative finance applications including option pricing, optimal…
This paper discusses a Python interface for the recently published DUNE-FEM-DG module which provides highly efficient implementations of the Discontinuous Galerkin (DG) method for solving a wide range of non linear partial differential…
We are concerned with the arithmetic of solutions to ordinary or partial nonlinear differential equations which are algebraic in the indeterminates and their derivatives. We call these solutions D-algebraic functions, and their equations…
Linear algebraic primitives are at the core of many modern algorithms in engineering, science, and machine learning. Hence, accelerating these primitives with novel computing hardware would have tremendous economic impact. Quantum computing…
We consider series expansions in bases of classical orthogonal polynomials. When such a series solves a linear differential equation with polynomial coefficients, its coefficients satisfy a linear recurrence equation. We interpret this…
This paper is concerned with the problem of finding a quadratic common Lyapunov function for a family of stable linear systems. We present gradient iteration algorithms which give deterministic convergence for finite system families and…
Solving nonlinear partial differential equations (PDEs) with multiple solutions using neural networks has found widespread applications in various fields such as physics, biology, and engineering. However, classical neural network methods…
Numerical integration of the stiffness matrix in higher order finite element (FE) methods is recognized as one of the heaviest computational tasks in a FE solver. The problem becomes even more relevant when computing the Gram matrix in the…
We present simplified formulae for the analytic integration of the Newton potential of polynomials over boxes in two- and three-dimensional space. These are implemented in an easy-to-use C++ library that allows computations in arbitrary…
A fast algorithm (linear in the degrees of freedom) for the solution of linear variable-coefficient rational-order fractional integral and differential equations is described. The approach is related to the ultraspherical method for…
We are interested in the fast computation of the exact value of integrals of polynomial functions over convex polyhedra. We present speed ups and extensions of the algorithms presented in previous work. We present the new software…
Recent results by Harrow et. al. and by Ta-Shma, suggest that quantum computers may have an exponential advantage in solving a wealth of linear algebraic problems, over classical algorithms. Building on the quantum intuition of these…
In this paper, we tackle the parametric complete multiplicity problem for a univariate polynomial. Our approach to the parametric complete multiplicity problem has a significant difference from the classical method, which relies on repeated…
We present a library of efficient implementations of deep learning primitives. Deep learning workloads are computationally intensive, and optimizing their kernels is difficult and time-consuming. As parallel architectures evolve, kernels…
We present an efficient multi-accuracy algorithm for the computations of a set of special functions of a complex argument, z=x+iy. These functions include the complex probability function w(z), and closely related functions such as the…
Quantum computation represents a computational paradigm whose distinctive attributes confer the ability to devise algorithms with asymptotic performance levels significantly superior to those achievable via classical computation. Recent…
Zielonka's classic recursive algorithm for solving parity games is perhaps the simplest among the many existing parity game algorithms. However, its complexity is exponential, while currently the state-of-the-art algorithms have…
Recently, there has been a surge of interest for quantum computation for its ability to exponentially speed up algorithms, including machine learning algorithms. However, Tang suggested that the exponential speed up can also be done on a…
Quantum algorithm is an algorithm for solving mathematical problems using quantum systems encoded as information, which is found to outperform classical algorithms in some specific cases. The objective of this study is to develop a quantum…
Quantum computing is a new computational paradigm that promises applications in several fields, including machine learning. In the last decade, deep learning, and in particular Convolutional neural networks (CNN), have become essential for…