Related papers: A Direct Matrix Method for Computing Analytical Ja…
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 introduce a deep neural network based method for solving a class of elliptic partial differential equations. We approximate the solution of the PDE with a deep neural network which is trained under the guidance of a probabilistic…
In this paper, we provide a structure-preserving one-sided cyclic Jacobi method for computing the singular value decomposition of a quaternion matrix. In this method, the columns of the quaternion matrix are orthogonalized in pairs by using…
We propose a novel numerical algorithm utilizing model reduction for computing solutions to stationary partial differential equations involving the spectral fractional Laplacian. Our approach utilizes a known characterization of the…
A new Jacobian approximation is developed for use in quasi-Newton methods for solving systems of nonlinear equations. The new hypersecant Jacobian approximation is intended for the special case where the evaluation of the functions whose…
This paper proposes a new methodology for deriving a point-based dimensionally homogeneous Jacobian, intended for performance evaluation and optimization of parallel manipulators with mixed degrees of freedom. Optimal manipulator often rely…
This contribution proposes a new formulation to efficiently compute directional derivatives of order one to fourth. The formulation is based on automatic differentiation implemented with dual numbers. Directional derivatives are particular…
In calculating integral or discrete transforms, use has been made of fast algorithms for multiplying vectors by matrices whose elements are specified as values of special (Chebyshev, Legendre, Laguerre, etc.) functions. The currently…
We present a discretization of the Jacobi last multiplier, with some applications to the computation of solutions of difference equations.
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…
In this work, we present an efficient method for computing in the generalized Jacobian of special singular curves, nodal curves. The efficiency of the operation is due to the representation of an element in the Jacobian group by a single…
We present a simple and easy to implement method for the numerical solution of a rather general class of Hamilton-Jacobi-Bellman (HJB) equations. In many cases, the considered problems have only a viscosity solution, to which, fortunately,…
The purpose of this paper is to bridge the gap between the Dbar method and the direct linearization approach for the lattice Korteweg-de Vries (KdV) type equations. We develop the Dbar method to study some discrete integrable equations in…
Neural tangent kernel (NTK) methods are computationally limited by the need to evaluate large Jacobians across many data points. Existing approaches reduce this cost primarily through projecting and sketching the Jacobian. We show that NTK…
We consider the application of implicit and linearly implicit (Rosenbrock-type) peer methods to matrix-valued ordinary differential equations. In particular the differential Riccati equation (DRE) is investigated. For the Rosenbrock-type…
The simple product formulae for derivatives of scalar functions raised to different powers are generalized for functions which take values in the set of symmetric positive definite matrices. These formulae are fundamental in derivation of…
Nonlinear model predictive control~(NMPC) generally requires the solution of a non-convex optimization problem at each sampling instant under strict timing constraints, based on a set of differential equations that can often be stiff and/or…
The complex-step derivative approximation is a numerical differentiation technique that can achieve analytical accuracy, to machine precision, with a single function evaluation. In this letter, the complex-step derivative approximation is…
We propose quantum methods for solving differential equations that are based on a gradual improvement of the solution via an iterative process, and are targeted at applications in fluid dynamics. First, we implement the Jacobi iteration on…
A high-order Newton multigrid method is proposed for steady-state shallow water flows in open channels with regular and irregular geometries. The method integrates a finite volume discretization with third-order weighted essentially…