Abstract. In this paper we generalize the hierarchically semiseparable (HSS) representations and propose some fast algorithms for HSS matrices. We provide a new linear complexity U LV T factorization algorithm for symmetric positive definite HSS matrices with small off-diagonal ranks. The corresponding factors can be used to solve compact HSS systems also in linear complexity. Numerical examples demonstrate the efficiency of the solver. We also present fast algorithms including new HSS structure generation, HSS form Cholesky factorization, and model compression. These algorithms are useful for problems where off-diagonal blocks have small numerical ranks.

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Abstract. 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 T into a Cauchy-like matrix C, which is known to have low-numerical-rank offdiagonal blocks. Thus, we design a fast scheme for constructing a hierarchically semiseparable (HSS) matrix approximation to C, where the HSS generators have internal structures. Unlike classical HSS methods, our solver employs randomized sampling techniques together with fast Toeplitz matrixvector multiplications, and thus converts the direct compression of the off-diagonal blocks of C into the compression of much smaller blocks. A strong rank-revealing QR factorization method is used to generate/preserve certain special structures, and also to ensure stability. A fast ULV HSS factorization scheme is provided to take advantage of the special structures. We also propose a precomputation procedure for the HSS construction so as to further improve the efficiency. The complexity of these methods is significantly lower than some similar Toeplitz solvers for large matrix size n. Detailed flop counts are given, with the aid of a rank relaxation technique. The total cost of our methods includes O(n) flops for HSS operations and O(n log 2 n) flops for matrix multiplications via FFTs, where n is the order of T . Various numerical tests on classical examples, including ill-conditioned ones, demonstrate the efficiency, and also indicate that the methods are stable in practice. This work shows a practical way of using randomized sampling in the development of fast rank structured methods.

Rank structures provide an opportunity to develop new efficient numerical methods for practical problems, when the off-diagonal blocks of certain dense intermediate matrices have small (numerical) ranks. In this work, we present a framework of structured direct factorizations for general sparse matrices, including discretized PDEs on general meshes, based on the multifrontal method and hierarchically semiseparable (HSS) matrices. We prove the idea of replacing certain complex structured operations by fast simple ones performed on compact reduced matrix forms. Such forms result from the hierarchical factorization of a tree-structured HSS matrix in a ULV-type scheme, so that the tree structure is reduced into a single node, the root of the original tree. This idea is shown to be very useful in the partial ULV factorization of an HSS matrix (for quickly computing Schur complements) as well as the solution stage. These techniques are then built into the multifrontal method for sparse factorizations after nested dissection, so as to convert the intermediate dense factorizations into fast structured ones. This method keeps certain Schur complements dense so as to avoid complicated data assembly, and is much simpler and more general than some existing methods. In particular, if the matrix arises from the discretization of certain PDEs, the factorization costs roughly O(n) flops in two dimensions, and roughly O(n 4/3) flops or less in three dimensions. The solution cost and memory are nearly O(n) in both cases. These counts are obtained with an idea of rank relaxation, so that this method is more generally applicable to problems where the intermediate off-diagonal ranks are not small. We demonstrate the performance of the method with two-and three-dimensional discretized equations, as well as various examples from a sparse matrix collection. The ideas here are also useful in future developments of fast structured solvers.

Given a symmetric positive definite matrix A, we compute a structured approximate Cholesky factorization A ≈ R T R up to any desired accuracy, where R is an upper triangular hierarchically semiseparable (HSS) matrix. The factorization is stable, robust, and efficient. The method compresses off-diagonal blocks with rank-revealing orthogonal decompositions. In the meantime, positive semidefinite terms are automatically and implicitly added to Schur complements in the factorization so that the approximation R T R is guaranteed to exist and be positive definite. The approximate factorization can be used as a structured preconditioner which does not break down. No extra stabilization step is needed. When A has an off-diagonal low-rank property, or when the off-diagonal blocks of A have small numerical ranks, the preconditioner is data sparse and is especially efficient. Furthermore, the method has a good potential to give satisfactory preconditioning bounds even if this low-rank property is not obvious. Numerical experiments are used to demonstrate the performance of the method. The method can be used to provide effective structured preconditioners for large sparse problems when combined with some sparse matrix techniques. The hierarchical compression scheme in this work is also useful in the development of more HSS algorithms.

We propose randomized direct solvers for large sparse linear systems, which integrate randomization into rank structured multifrontal methods. The use of randomization highly simplifies various essential steps in structured solutions, where fast operations on skinny matrix-vector products replace traditional complex ones on dense or structured matrices. The new methods thus significantly enhance the flexibility and efficiency of using structured methods in sparse solutions. We also consider a variety of techniques, such as some graph methods, the inclusion of additional structures, the concept of reduced matrices, information reuse, and adaptive schemes. The methods are applicable to various sparse matrices with certain rank structures. Particularly, for discretized matrices whose factorizations yield dense fill-in with some off-diagonal rank patterns, the factorizations cost about O(n) flops in two dimensions (2D), and about O(n) to O(n 4/3) flops in three dimensions (3D). The solution costs and memory sizes are nearly O(n) in both 2D and 3D. These counts are obtained based on two optimization strategies and a sparse rank relaxation idea. The methods are especially useful for approximate solutions and preconditioning. Numerical tests on both discretized PDEs and more general problems are used to demonstrate the efficiency and accuracy. The ideas here also have the potential to be generalized to matrix-free sparse direct solvers based on matrix-vector multiplications in future developments.

Abstract. 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 solvers, 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.

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