Optimal stability polynomials for numerical integration of initial value problems

We consider the problem of finding optimally stable polynomial approximations to the exponential for application to one-step integration of initial value ordinary and partial differential equations. The objective is to find the largest stable step size and corresponding method for a given problem when the spectrum of the initial value problem is known. The problem is expressed in terms of a general least deviation feasibility problem. Its solution is obtained by a new fast, accurate, and robust algorithm based on convex optimization techniques. Global convergence of the algorithm is proven in the case that the order of approximation is one and in the case that the spectrum encloses a starlike region. Examples demonstrate the effectiveness of the proposed algorithm even when these conditions are not satisfied

Generalized Differential Quadrature and Its Application. G.U. Aero Report 9117

The technique of differential quadrature (DQ) for the solution of a partial differential equation is extended and generalized in this paper. The general formulation for determining the weighting coefficients of the first order derivative is obtained. A recurrence relationship for determining the weighting coefficients of the second and higher order partial derivatives is also obtained, and it is shown that generalized differential quadrature (GDQ) can be considered as a finite difference scheme of the highest order. Three typical formulas of weighting coefficients for the first order derivative are also given in the paper. The error estimations for the function and derivative approximation, and the eigenvalue structures of some basic GDQ spatial discretization matrices have been studied. The application of GDQ to model problems showed that accurate results can be obtained using a small number of grid points

Reduced Order and Surrogate Models for Gravitational Waves

We present an introduction to some of the state of the art in reduced order and surrogate modeling in gravitational wave (GW) science. Approaches that we cover include Principal Component Analysis, Proper Orthogonal Decomposition, the Reduced Basis approach, the Empirical Interpolation Method, Reduced Order Quadratures, and Compressed Likelihood evaluations. We divide the review into three parts: representation/compression of known data, predictive models, and data analysis. The targeted audience is that one of practitioners in GW science, a field in which building predictive models and data analysis tools that are both accurate and fast to evaluate, especially when dealing with large amounts of data and intensive computations, are necessary yet can be challenging. As such, practical presentations and, sometimes, heuristic approaches are here preferred over rigor when the latter is not available. This review aims to be self-contained, within reasonable page limits, with little previous knowledge (at the undergraduate level) requirements in mathematics, scientific computing, and other disciplines. Emphasis is placed on optimality, as well as the curse of dimensionality and approaches that might have the promise of beating it. We also review most of the state of the art of GW surrogates. Some numerical algorithms, conditioning details, scalability, parallelization and other practical points are discussed. The approaches presented are to large extent non-intrusive and data-driven and can therefore be applicable to other disciplines. We close with open challenges in high dimension surrogates, which are not unique to GW science.Comment: Invited article for Living Reviews in Relativity. 93 page

An Algebraic Framework for the Real-Time Solution of Inverse Problems on Embedded Systems

This article presents a new approach to the real-time solution of inverse problems on embedded systems. The class of problems addressed corresponds to ordinary differential equations (ODEs) with generalized linear constraints, whereby the data from an array of sensors forms the forcing function. The solution of the equation is formulated as a least squares (LS) problem with linear constraints. The LS approach makes the method suitable for the explicit solution of inverse problems where the forcing function is perturbed by noise. The algebraic computation is partitioned into a initial preparatory step, which precomputes the matrices required for the run-time computation; and the cyclic run-time computation, which is repeated with each acquisition of sensor data. The cyclic computation consists of a single matrix-vector multiplication, in this manner computation complexity is known a-priori, fulfilling the definition of a real-time computation. Numerical testing of the new method is presented on perturbed as well as unperturbed problems; the results are compared with known analytic solutions and solutions acquired from state-of-the-art implicit solvers. The solution is implemented with model based design and uses only fundamental linear algebra; consequently, this approach supports automatic code generation for deployment on embedded systems. The targeting concept was tested via software- and processor-in-the-loop verification on two systems with different processor architectures. Finally, the method was tested on a laboratory prototype with real measurement data for the monitoring of flexible structures. The problem solved is: the real-time overconstrained reconstruction of a curve from measured gradients. Such systems are commonly encountered in the monitoring of structures and/or ground subsidence.Comment: 24 pages, journal articl