Jianlin Xia

Superfast Multifrontal Method for Large Structured Linear Systems of Equations

In this paper we develop a fast direct solver for large discretized linear systems using the supernodal multifrontal method together with low-rank approximations. For linear systems arising from certain partial differential equations such as elliptic equations, during the Gaussian elimination of the matrices with proper ordering, the fill-in has a low-rank property: all off-diagonal blocks have small numerical ranks with proper definition of off-diagonal blocks. Matrices with this low-rank property can be efficiently approximated with semiseparable structures called hierarchically semiseparable (HSS) representations. We reveal the above low-rank property by ordering the variables with nested dissection and eliminating them with the multifrontal method. All matrix operations in the multifrontal method are performed in HSS forms. We present efficient ways to organize the HSS structured operations along the elimination. Some fast HSS matrix operations using tree structures are proposed. This new structured multifrontal method has nearly linear complexity and a linear storage requirement. Thus, we call it a superfast multifrontal method. It is especially suitable for large sparse problems and also has natural adaptability to parallel computations and great potential to provide effective preconditioners. Numerical results demonstrate the efficiency.

Fast algorithms for hierarchically semiseparable matrices

Semiseparable matrices and many other rank-structured matrices have been widely used in developing new fast matrix algorithms. In this paper, we generalize the hierarchically semiseparable (HSS) matrix representations and propose some fast algorithms for HSS matrices. We represent HSS matrices in terms of general binary HSS trees and use simplified postordering notation for HSS forms. Fast HSS algorithms including new HSS structure generation and HSS form Cholesky factorization are developed. Moreover, we provide a new linear complexity explicit ULV factorization algorithm for symmetric positive definite HSS matrices with a low-rank property. The corresponding factors can be used to solve the HSS systems also in linear complexity. Numerical examples demonstrate the efficiency of the algorithms. All these algorithms have nice data locality. They are useful in developing fast-structured numerical methods for large discretized PDEs (such as elliptic equations), integral equations, eigenvalue problems, etc. Some applications are shown. Copyright q 2009 John Wiley & Sons, Ltd.

Order conditions for ARKN methods solving oscillatory systems

For the perturbed oscillators in one-dimensional case, J.M. Franco designed the so-called Adapted Runge–Kutta–Nystrom (ARKN) methods and derived the sufficient order conditions as well as the necessary and sufficient order conditions for ARKN methods based on the B-series theory [J.M. Franco, Runge–Kutta–Nystrom methods adapted to the numerical integration of perturbed oscillators, Comput. Phys. Comm. 147 (2002) 770–787]. These methods integrate exactly the unperturbed oscillators and are highly efficient when the perturbing function is small. Unfortunately, some critical mistakes have been made in the derivation of order conditions in that paper. On the basis of the results from that paper, Franco extended directly the ARKN methods and the corresponding order conditions to multidimensional case where the perturbed function f does not depend on the first derivative y′ [J.M. Franco, New methods for oscillatory systems based on ARKN methods, Appl. Numer. Math. 56 (2006) 1040–1053]. In this paper, we present the order conditions for the ARKN methods for the general multidimensional perturbed oscillators where the perturbed function f may depend on only y or on both y and y′.

A Superfast Algorithm for Toeplitz Systems of Linear Equations

In this paper we develop a new superfast solver for Toeplitz systems of linear equations. To solve Toeplitz systems many people use displacement equation methods. With displacement structures, Toeplitz matrices can be transformed into Cauchy-like matrices using the FFT or other trigonometric transformations. These Cauchy-like matrices have a special property, that is, their off-diagonal blocks have small numerical ranks. This low-rank property plays a central role in our superfast Toeplitz solver. It enables us to quickly approximate the Cauchy-like matrices by structured matrices called sequentially semiseparable (SSS) matrices. The major work of the constructions of these SSS forms can be done in precomputations (independent of the Toeplitz matrix entries). These SSS representations are compact because of the low-rank property. The SSS Cauchy-like systems can be solved in linear time with linear storage. Excluding precomputations the main operations are the FFT and SSS system solve, which are both very efficient. Our new Toeplitz solver is stable in practice. Numerical examples are presented to illustrate the efficiency and the practical stability.

A Superfast Structured Solver for Toeplitz Linear Systems via Randomized Sampling

We propose a superfast solver for Toeplitz linear systems based on rank structured matrix methods and randomized sampling. The solver uses displacement equations to transform a Toeplitz matrix

Fast Methods for Estimating the Distance to Uncontrollability

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Two low accuracy methods for stiff systems

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Superfast and Stable Structured Solvers for Toeplitz Least Squares via Randomized Sampling

The distance to uncontrollability for a linear control system is the distance (in the 2-norm) to the nearest uncontrollable system. We present an algorithm based on methods of Gu and Burke-Lewis-Overton that estimates the distance to uncontrollability to any prescribed accuracy. The new method requires O(n4) operations on average, which is an improvement over previous methods which have complexity O(n6), where n is the order of the system. Numerical experiments indicate that the new method is reliable in practice.

Randomized Sparse Direct Solvers

Two low accuracy explicit one-step methods for stiff ordinary differential equations are extended directly to solve systems of equations. Some defects of the component form of these methods are avoided. To perform these, a new set of vector computations are introduced. Some numerical experiments are presented to show the superiority of the new methods.

Efficient Structured Multifrontal Factorization for General Large Sparse Matrices

We present some superfast (