Related papers: A Direct Matrix Method for Computing Analytical Ja…
In this paper, we introduce a quasi-Newton method optimized for efficiently solving quasi-linear elliptic equations and systems, with a specific focus on GPU-based computation. By approximating the Jacobian matrix with a combination of…
Many numerical methods for evaluating matrix functions can be naturally viewed as computational graphs. Rephrasing these methods as directed acyclic graphs (DAGs) is a particularly effective approach to study existing techniques, improve…
The objective of this study is to present a novel, efficient, and fast direct method for solving linear systems of equations whose coefficient matrix is a tridiagonal Quasi-Toeplitz matrix. Such matrices are frequently encountered in the…
Broyden's method is a general method commonly used for nonlinear systems of equations, when very little information is available about the problem. We develop an approach based on Broyden's method for nonlinear eigenvalue problems. Our…
Some general problems of Jacobian computations in non-full rank matrices are discussed in this work. In particular, the Jacobian of the Moore-Penrose inverse derived via matrix differential calculus is revisited. Then the Jacobian in the…
Jacobian-vector products (JVPs) form the backbone of many recent developments in Deep Networks (DNs), with applications including faster constrained optimization, regularization with generalization guarantees, and adversarial example…
We propose a gradient-based Jacobi algorithm for a class of maximization problems on the unitary group, with a focus on approximate diagonalization of complex matrices and tensors by unitary transformations. We provide weak convergence…
This paper proposes a new method for differentiating through optimal trajectories arising from non-convex, constrained discrete-time optimal control (COC) problems using the implicit function theorem (IFT). Previous works solve a…
We present a method for the numerical calculation of derivatives of functions of general complex matrices. The method can be used in combination with any algorithm that evaluates or approximates the desired matrix function, in particular…
The Newmark/Newton-Raphson (NNR) method is widely employed for solving nonlinear dynamic systems. However, the current NNR method exhibits limited applicability in complex nonlinear dynamic systems, as the acquisition of the Jacobian matrix…
An exact discretization method is being developed for solving linear systems of ordinary fractional-derivative differential equations with constant matrix coefficients (LSOFDDECMC). It is shown that the obtained linear discrete system in…
Jacobi's method is a well-known algorithm in linear algebra to diagonalize symmetric matrices by successive elementary rotations. We report about the generalization of these elementary rotations towards canonical transformations acting in…
Learning expressive probabilistic models correctly describing the data is a ubiquitous problem in machine learning. A popular approach for solving it is mapping the observations into a representation space with a simple joint distribution,…
Finding the optimal hyperparameters of a model can be cast as a bilevel optimization problem, typically solved using zero-order techniques. In this work we study first-order methods when the inner optimization problem is convex but…
We present a fast Jacobi-like algorithm for computing the eigenvalues, and optionally the eigenvectors, of a real normal matrix. The method gains a computational advantage by using Paardekooper's method for skew-symmetric matrices The…
The standard methodology handling nonlinear PDE's involves the two steps: numerical discretization to get a set of nonlinear algebraic equations, and then the application of the Newton iterative linearization or its variants to solve the…
We present a novel numerical method, called {\tt Jacobi-predictor-corrector approach}, for the numerical solution of fractional ordinary differential equations based on the polynomial interpolation and the Gauss-Lobatto quadrature w.r.t.…
This paper shows how to build a formal analytical solution for a differential equation of arbitrary order and with variable coefficients. It proofs that the most known approximated solutions for such a problem can be derived from the…
This article aims to demonstrate and discuss the applications of automatic differentiation (AD) for finding derivatives in PDE-constrained optimization problems and Jacobians in non-linear finite element analysis. The main idea is to…
In this paper, we study the equality constrained nonlinear least squares problem, where the Jacobian matrices of the objective function and constraints are unavailable or expensive to compute. We approximate the Jacobian matrices via…