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

The last two decades have seen major developments in inter- polatory methods for model reduction of large-scale linear dynamical systems. Advances of note include the ability to produce (locally) opti- mal reduced models at modest cost; rened methods for deriving inter- polatory reduced models directly from input/output measurements; and extensions for the reduction of parametrized systems. This chapter oers a survey of interpolatory model reduction methods starting from basic principles and ranging up through recent developments that include weighted model reduction and structure-preserving methods based on generalized coprime representations. Our discussion is supported by an assortment of numerical examples. Mathematics Subject Classication (2010). 41A05, 93A15, 93C05, 37M99. Keywords. Rational interpolation, model reduction,H2 approximation, parametric systems, generalized coprime factorization, weighted model reduction, Loewner framework

Model Reduction by Rational Interpolation

Issues in Interpolatory Model Reduction: Inexact Solves, Second-order Systems and DAEs

Krylov subspace recycling is a process for accelerating the convergence of sequences of linear systems. Based on this technique, the recycling BiCG algorithm has been developed recently. Here, we now generalize and extend this recycling theory to BiCGSTAB. Recycling BiCG focuses on efficiently solving sequences of dual linear systems, while the focus here is on efficiently solving sequences of single linear systems (assuming nonsymmetric matrices for recycling BiCG and recycling BiCGSTAB). As compared with other methods for solving sequences of single linear systems with nonsymmetric matrices (e.g., recycling variants of GMRES), BiCG-based recycling algorithms, like recycling BiCGSTAB, have the advantage that they involve a short-term recurrence and hence do not suffer from storage issues and are also cheaper with respect to the orthogonalizations. We modify the BiCGSTAB algorithm to use a recycle space, which is built from left and right approximate invariant subspaces. Using our algorithm for a parametr...

Recycling BiCGSTAB with an Application to Parametric Model Order Reduction

We present an extension of the framework of Gaul et al. [SIAM J. Matrix Anal. Appl., 34 (2013), pp. 495--518] for deflated and augmented Krylov subspace methods satisfying a Galerkin condition to more general Petrov--Galerkin conditions. The main goal is to apply the framework to the biconjugate gradient method (BiCG) and some of its generalizations, including BiCGStab and IDR$(s)$. For such applications the assumptions of Gaul et al. were too restrictive. Our abstract approach does not depend on particular recurrences and thus simplifies the derivation of theoretical results. It easily leads to a variety of realizations by specific algorithms. We do not go into algorithmic details, but we show that for every method there are two different approaches for extending it by augmentation and deflation: one that explicitly takes care of the augmentation space in every step, and one that applies the appropriate Krylov subspace method to a projected problem but requires a correction step at the end. The latter ap...

Deflated and Augmented Krylov Subspace Methods: A Framework for Deflated BiCG and Related Solvers

Science and engineering problems frequently require solving a sequence of dual linear systems. Besides having to store only a few Lanczos vectors, using the biconjugate gradient method (BiCG) to solve dual linear systems has advantages for specific applications. For example, using BiCG to solve the dual linear systems arising in interpolatory model reduction provides a backward error formulation in the model reduction framework. Using BiCG to evaluate bilinear forms---for example, in quantum Monte Carlo (QMC) methods for electronic structure calculations---leads to a quadratic error bound. Since our focus is on sequences of dual linear systems, we introduce recycling BiCG, a BiCG method that recycles two Krylov subspaces from one pair of dual linear systems to the next pair. The derivation of recycling BiCG also builds the foundation for developing recycling variants of other bi-Lanczos based methods, such as CGS, BiCGSTAB, QMR, and TFQMR. We develop an augmented bi-Lanczos algorithm and a modified two-te...

Recycling BiCG with an Application to Model Reduction

Science and engineering problems frequently require solving a sequence of dual linear systems. Besides having to store only few Lanczos vectors, using the BiConjugate Gradient method (BiCG) to solve dual linear systems has advantages for specific applications. For example, using BiCG to solve the dual linear systems arising in interpolatory model reduction provides a backward error formulation in the model reduction framework. Using BiCG to evaluate bilinear forms -- for example, in quantum Monte Carlo (QMC) methods for electronic structure calculations -- leads to a quadratic error bound. Since our focus is on sequences of dual linear systems, we introduce recycling BiCG, a BiCG method that recycles two Krylov subspaces from one pair of dual linear systems to the next pair. The derivation of recycling BiCG also builds the foundation for developing recycling variants of other bi-Lanczos based methods, such as CGS, BiCGSTAB, QMR, and TFQMR. 
We develop an augmented bi-Lanczos algorithm and a modified two-term recurrence to include recycling in the iteration. The recycle spaces are approximate left and right invariant subspaces corresponding to the eigenvalues closest to the origin. These recycle spaces are found by solving a small generalized eigenvalue problem alongside the dual linear systems being solved in the sequence. 
We test our algorithm in two application areas. First, we solve a discretized partial differential equation (PDE) of convection-diffusion type. Such a problem provides well-known test cases that are easy to test and analyze further. Second, we use recycling BiCG in the Iterative Rational Krylov Algorithm (IRKA) for interpolatory model reduction. IRKA requires solving sequences of slowly changing dual linear systems. We show up to 70% savings in iterations, and also demonstrate that for a model reduction problem BiCG takes (about) 50% more time than recycling BiCG.

Science and engineering problems frequently require solving a sequence of dual linear systems. Two examples are the Iterative Rational Krylov Algorithm (IRKA) for model reduction and Quantum Monte Carlo (QMC) methods in electronic structure calculations. This paper introduces Recycling BiCG, a BiCG method that recycles two Krylov subspaces from one pair of linear systems to the next pair. We develop an augmented bi-Lanczos algorithm and a modified two-term recurrence to include recycling in the iteration. The recycle spaces are approximate left and right invariant subspaces corresponding to the eigenvalues close to the origin. These recycle spaces are found by solving a small generalized eigenvalue problem alongside the dual linear systems being solved in the sequence. We test our algorithm in two application areas. First, we solve a discretized partial differential equation of convection-diffusion type, because these are well-known model problems. Second, we use Recycling BiCG for the linear systems arising in IRKA for model reduction, which requires solving a sequence of slowly changing, dual linear systems. Our experiments with Recycling BiCG give good results.

Eun R. Chang

Papers

Recycling BiCG with an Application to Model Reduction

Recycling BiCG with an Application to Model Reduction

Recycling BiCG for Model Reduction