Tables of integrals

Numerical simulation of large-scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent large-scale nature of the models often leads to unmanageable demands on computational resources. Model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original large-scale system behavior. Model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books. However, parametric model reduction has emerged only more recently as an important and vibrant research area, with several recent advances making a survey paper timely. Thus, this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods. Parametric model reduction targets the broad class of problems for wh...

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A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

Most numerical integration techniques consist of approximating the integrand by a polynomial in a region or regions and then integrating the polynomial exactly. Often a complicated integrand can be factored into a non-negative ''weight'' function and another function better approximated by a polynomial, thus $\int_{a}^{b} g(t)dt = \int_{a}^{b} \omega (t)f(t)dt \approx \sum_{i=1}^{N} w_i f(t_i)$. Hopefully, the quadrature rule ${\{w_j, t_j\}}_{j=1}^{N}$ corresponding to the weight function $\omega$(t) is available in tabulated form, but more likely it is not. We present here two algorithms for generating the Gaussian quadrature rule defined by the weight function when: a) the three term recurrence relation is known for the orthogonal polynomials generated by $\omega$(t), and b) the moments of the weight function are known or can be calculated.

Calculation of Gauss quadrature rules.

Given the square matrices $A, B, D, E$ and the matrix $C$ of conforming dimensions, we consider the linear matrix equation $A{\mathbf X} E+D{\mathbf X} B = C$ in the unknown matrix ${\mathbf X}$. Our aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems, which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing.

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Computational Methods for Linear Matrix Equations

United States. Air Force Office of Scientific Research (Computational Mathematics Grant FA9550-12-1-0420)

We consider the computation of $u(t)=\exp(-tA)\varphi$ using rational Krylov subspace reduction for $0\le t<\infty$, where $u(t),\varphi\in\mathbf{R}^N$ and $0<A=A^*\in\mathbf{R}^{N\times N}$. The objective of this work is the optimization of the shifts for the rational Krylov subspace (RKS). We consider this problem in the frequency domain and reduce it to a classical Zolotaryov problem. The latter yields an asymtotically optimal solution with real shifts. We also construct an infinite sequence of shifts yielding a nested sequence of the RKSs with the same (optimal) Cauchy-Hadamard convergence rate. The effectiveness of the developed approach is demonstrated on an example of the three-dimensional diffusion problem for Maxwell's equation arising in geophysical exploration.

Solution of Large Scale Evolutionary Problems Using Rational Krylov Subspaces with Optimized Shifts

We compute $u(t)=\exp(-tA)\varphi$ using rational Krylov subspace reduction for $0\leq t<\infty$, where $u(t),\varphi\in\mathbf{R}^N$ and $0<A=A^*\in\mathbf{R}^{N\times N}$. A priori optimization of the rational Krylov subspaces for this problem was considered in [V. Druskin, L. Knizhnerman, and M. Zaslavsky, SIAM J. Sci. Comput., 31 (2009), pp. 3760-3780]. There was suggested an algorithm generating sequences of equidistributed shifts, which are asymptotically optimal for the cases with uniform spectral distributions. Here we develop a recursive greedy algorithm for choice of shifts taking into account nonuniformity of the spectrum. The algorithm is based on an explicit formula for the residual in the frequency domain allowing adaptive shift optimization at negligible cost. The effectiveness of the developed approach is demonstrated in an example of the three-dimensional diffusion problem for Maxwell's equation arising in geophysical exploration. We compare our approach with the one using the above-mentioned equidistributed sequences of shifts. Numerical examples show that our algorithm is able to adapt to the spectral density of operator $A$. For examples with near-uniform spectral distributions, both algorithms show the same convergence rates, but the new algorithm produces superior convergence for cases with nonuniform spectra.

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On Adaptive Choice of Shifts in Rational Krylov Subspace Reduction of Evolutionary Problems

We solve an electromagnetic frequency domain induction problem in $\mathbf{R}^3$ for a frequency interval using rational Krylov subspace (RKS) approximation. The RKS is constructed by spanning on the solutions for a certain a priori chosen set of frequencies. We reduce the problem of the optimal choice of these frequencies to the third Zolotaryov problem in the complex plane, having an approximate closed form solution, and determine the best Cauchy-Hadamard convergence rate. The theory is illustrated with numerical examples for Maxwell's equations arising in 3D magnetotelluric geophysical exploration.

On Optimal Convergence Rate of the Rational Krylov Subspace Reduction for Electromagnetic Problems in Unbounded Domains

Model reduction approaches have been shown to be powerful techniques in the numerical simulation of very large dynamical systems. The presence of multiple inputs and outputs (MIMO systems) makes the reduction process even more challenging. We consider projection-based approaches where the reduction of complexity is achieved by direct projection of the problem onto a rational Krylov subspace of significantly smaller dimension. We present an effective way to treat multiple inputs by dynamically choosing the next direction vectors to expand the space. We apply the new strategy to the approximation of the transfer matrix function and to the solution of the Lyapunov matrix equation. Numerical results confirm that the new approach is competitive with respect to state-of-the-art methods both in terms of CPU time and memory requirements.

Adaptive Tangential Interpolation in Rational Krylov Subspaces for MIMO Dynamical Systems

Time-domain problems for controlled-source electromagnetic exploration require accurate discretization of the solution for multiple spacial and temporal scales. Therefore, forward simulation using conventional computational methods becomes computationally expensive, even without accounting for induced-polarization (IP) effects. These effects create another complication caused by the presence of a convolution integral in the time-domain Maxwell system. We suggested a novel, fast, and robust algorithm to solve the 3D time-domain electromagnetic (EM) problems that can be considered as a generalization of the spectral Lanczos decomposition method. The new method also allowed us to incorporate the IP effects without significant cost increase. The discretized large-scale Maxwell system was projected onto a small subspace consisting of the Laplace-domain solutions (the so-called parameter-dependent Krylov subspace) for an optimally chosen set of Laplace parameters. The projected system preserved stabilit...

Mikhail Zaslavsky

Papers

Solution of Large Scale Evolutionary Problems Using Rational Krylov Subspaces with Optimized Shifts

On Adaptive Choice of Shifts in Rational Krylov Subspace Reduction of Evolutionary Problems

On Optimal Convergence Rate of the Rational Krylov Subspace Reduction for Electromagnetic Problems in Unbounded Domains

Adaptive Tangential Interpolation in Rational Krylov Subspaces for MIMO Dynamical Systems

Solution of 3D time-domain electromagnetic problems using optimal subspace projection