Mathematical Problems in Engineering

Volume 2015, Article ID 372109, 8 pages

http://dx.doi.org/10.1155/2015/372109

## Deflated BiCG with an Application to Model Reduction

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China

Received 21 July 2014; Revised 13 September 2014; Accepted 29 September 2014

Academic Editor: Jiuwen Cao

Copyright © 2015 Jing Meng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Most calculations in model reduction involve the solutions of a sequence of dual linear systems with multiple right-hand sides. To solve such systems efficiently, a new deflated BiCG method is explored in this paper. The proposed algorithm uses harmonic Ritz vectors to approximate left and right invariant subspaces inexpensively via small descenting direction vectors found by subsequent runs of deflated BiCG and then derives the deflated subspaces for the next pair of dual linear systems. This process leads to faster convergence for the next pair of systems. Numerical examples illustrate the effectiveness of the proposed method.

#### 1. Introduction

Large scale simulations play an important role in the study of a great variety of complex physical phenomena, leading often to overwhelming demands on computational resources [1–5]. Hence, the common approach is to produce a surrogate model of much smaller dimension which provides a high-fidelity approximation of the original model. For such problems, interpolatory model reduction method combines flexibility and scalability and has proven effectiveness. It transfers function interpolations in the frequency domain to meet various desirable approximation goals. During this process, it requires the solutions of dual linear systems with multiple right-hand sides (RHSs):where is a sparse, nonsymmetric matrix and RHSs are not available simultaneously.

For the solutions of primary linear systems , , deflated Krylov methods have been appearing. This is due to the fact that they take advantage of the fact that several systems share the same matrix. In addition, the convergence of Krylov subspace solvers for a linear system, to a great extent, depends on the spectrum of the matrix. If one could project the eigenvectors corresponding to the smallest eigenvalues out from the initial residual and then solve the deflated system it will converge much faster. The process is referred to as deflation. Variants of deflated Krylov solvers for the primary linear systems have been fully studied in the literature [2, 3, 6–14]. However, deflation for dual linear systems has not yet been fully investigated [15, 16].

In this paper we extend this idea to the BiCG algorithm for the dual case. The goal of this paper is to develop a new deflated BiCG method for solving dual linear systems with multiple RHSs. Likewise, for BiCG, it can be shown that if the primal Krylov subspace is deflated with right eigenvectors, the corresponding left eigenvectors are removed from the dual residual. Therefore, while solving a pair of systems, we select approximate left and right invariant subspaces of and then use those to accelerate the convergence of the next pair of systems. The proposed algorithm uses harmonic Ritz vectors to approximate left and right invariant subspaces inexpensively via small descenting direction vectors found by subsequent runs of deflated BiCG and then derives the deflated subspaces for the next pair of dual linear systems. Furthermore, we describe a cheap way to build the deflated subspaces.

In the next section, we describe the outline of the deflated Lanczos algorithm used in the derivation of the deflated BiCG method. In Section 3, we derive a deflated BiCG method using previously computed deflated subspace matrices. How to compute and update such deflated space matrices is given in Section 4. In Section 5, we investigate the deflated BiCG algorithm for solving the dual systems with multiple RHSs. The effectiveness of the proposed method is also demonstrated in Section 6. Finally, some conclusions are summarized in Section 7.

Throughout this paper, is referred to as the transpose conjugate operation of matrix , denotes the inner product, is defined as the zero matrix, and is biorthogonality.

#### 2. The Deflated Lanczos Algorithm

In this section, we describe a deflated Lanczos algorithm that builds two sequences , of vectors such that and , where and are two sets of linearly independent vectors.

We assume that the matrix is nonsingular. Let Since the matrices and are each other’s conjugate transpose, we can apply the standard Lanczos procedure to the above auxiliary matrices. Let unit vectors , be biorthogonal to , , respectively; then we compute the Lanczos vectors which satisfy where , , and are tridiagonal matrices, , are the last element of the last row of , and , respectively.

The Lanczos procedure guarantees that the vectors and . Since and , we have and for , via (5) and (6). Hence the sequences and satisfy the properties of (2).

To make these ideas more concrete, we give a deflated Lanczos algorithm by substituting the right-hand sides of (3) and (4).

#### 3. The Deflated BiCG Algorithm

In this section, we derive the deflated BiCG method from Algorithm 1 in exactly the same way as BiCG was derived from the Lanczos biorthogonalization procedure [17]. For convenience, we drop the superscript in (1) and refer toas the primary and the dual system, respectively.