this paper we extend previous work in mining recommendation spaces based on symbolic problem features to PDE problems with continuous-valued attributes. We identify the research issues in mining such spaces, present a dynamic programming algorithm from the data-mining literature, and describe how a priori domain metaknowledge can be used to control the complexity of induction. A visualization aid for continuous-valued recommendation spaces is also outlined. Two case studies are presented to illustrate our approach and tools: (i) a comparison of an iterative and a direct linear system solver on nearly singular problems, and (ii) a comparison of two iterative solvers on problems posed on nonrectangular domains. Both case studies involve continuously varying problem and method parameters which strongly influence the choice of best algorithm in particular cases. By mining the results from thousands of PDE solves, we can gain valuable insight into the relative performance of these methods on similar problems.

Abstract. A dual reordering strategy based on both threshold and graph reorderings is introduced to construct robust incomplete LU (ILU) factorization of indefinite matrices. The ILU matrix is constructed as a preconditioner for the original matrix to be used in a preconditioned iterative scheme. The matrix is first divided into two parts according to a threshold parameter to control diagonal dominance. The first part with large diagonal dominance is reordered using a graph-based strategy, followed by an ILU factorization. A partial ILU factorization is applied to the second part to yield an approximate Schur complement matrix. The whole process is repeated on the Schur complement matrix and continues for a few times to yield a multilevel ILU factorization. Analyses are conducted to show how the Schur complement approach removes small diagonal elements of indefinite matrices and how the stability of the LU factor affects the quality of the preconditioner. Numerical results are used to compare the new preconditioning strategy with two popular ILU preconditioning techniques and a multilevel block ILU threshold preconditioner.

A sparse approximate inverse technique is introduced to solve general sparse linear systems. The sparse approximate inverse is computed as a factored form and used as a preconditioner to work with some Krylov subspace methods. The new technique is derived from a matrix decomposition algorithm for inverting dense nonsymmetric matrices. Several strategies and special data structures are proposed to implement the algorithm efficiently. Sparsity patterns of the the factored inverse are exploited to reduce computational cost. The computation of the factored sparse approximate inverse is relatively cheaper than the techniques based on norm minimization techniques. The new preconditioner possesses much greater inherent parallelism than traditional preconditioners based on incomplete LU factorizations. Numerical experiments are used to show the effectiveness and efficiency of the new sparse approximate inverse preconditioner.

We investigate the use of sparse approximate inverse techniques in a multi-level block ILU preconditioner to design a robust and efficient parallelizable preconditioner for solving general sparse matrices. The resulting preconditioner retains robustness of the multi-level block ILU preconditioner (BILUM) and offers a new way to control the fill-in elements when large size blocks (subdomains) are used to form block independent set. Moreover, the new preconditioner affords maximum parallelism for operations within each level as well as for the coarsest level solution. Thus it has two advantages over the standard BILUM preconditioner: the ability to control sparsity and increased parallelism. Numerical experiments are used to show the effectiveness and efficiency of the new preconditioner. Key words: Sparse matrices, incomplete LU factorization, multi-level ILU preconditioner, sparse approximate inverse, Krylov subspace methods. AMS subject classifications: 65F10, 65N06. 1 Introduction ...

We discuss issues related to domain decomposition and multilevel preconditioning techniques which are often employed for solving large sparse linear systems in parallel computations. We introduce a class of parallel preconditioning techniques for general sparse linear systems based on a two level block ILU factorization strategy. We give some new data structures and strategies to construct local coefficient matrix and local Schur complement matrix in each processor. The preconditioner constructed is fast and robust for solving certain large sparse matrices. Numerical experiments show that our domain based two level block ILU preconditioners are more robust and more efficient than some published ILU preconditioners based on Schur complement techniques for parallel sparse matrix solutions.

A multi-level preconditioned iterative method based on a multi-level block ILU factorization preconditioning technique is introduced and is applied to the solution of unstructured sparse linear systems arising from the numerical simulation of coating problems. The coefficient matrices usually have several rows with zero diagonal values that may cause stability difficulty in standard ILU factorization techniques. The new preconditioning strategy employs a diagonal threshold tolerance and a local reordering of individual blocks to increase robustness of the multi-level block ILU factorization process. Keywords: sparse matrices, multi-level preconditioning, ILU factorization 1 Introduction In this paper, a multi-level block incomplete LU (ILU) preconditioning technique is designed for solving unstructured sparse linear systems from the numerical simulation of coating problems. Coating is a delicate process of putting a layer of one liquid material (film) over another solid material unifo...

We compare dierent preconditioning techniques in connection with Krylov methods for the solution of large dense complex symmetric nonHermitian systems arising in computational electromagnetics. Both implicit and explicit preconditioners are considered, and we emphasize sparse approximate inverse methods. We also investigate simple strategies suggested by the underlying problems, showing their eciency on this class of applications. Keywords : Krylov subspace methods, preconditioning techniques, sparse approximate inverse, electromagnetic scattering, boundary element method. 1

We design a grid based multilevel incomplete LU preconditioner (GILUM) for solving general sparse matrices. This preconditioner combines a high accuracy ILU factorization with an algebraic multilevel recursive reduction. The GILUM preconditioner is a compliment to the domain based multilevel block ILUT preconditioner. A major difference between these two preconditioners is the way that the coarse level nodes are chosen. In this sense the approach of GILUM is analogous to that of algebraic multigrid method. However, the GILUM construction is completely different from the algebraic multigrid construction. A partial ILUT factorization is applied to the reordered matrix and the coarse level system is obtained implicitly. The incomplete factorization process is repeated with the coarse level systems recursively. The GILUM approach avoids some controversial issues in algebraic multigrid method such as how to construct the interlevel transfer operators and how to compute the coarse level oper...

We investigate and compare a few parallel preconditioning techniques in the iterative solution of large sparse linear systems arising from solid Earth simulation with and without using contact information in the domain partitioning process. Previous studies are focused on using static or matrix pattern based incomplete LU (ILU) preconditioners in a localized preconditioner implementation. Our current studies are concerned about preconditioner performance for solving two different problem configurations with and without known contact information. For the cases with contact information, we use localized threshold value based incomplete LU (ILUT) preconditioner to improve efficiency. For the cases without contact information, we use a global sparse approximate inverse preconditioner with a static sparsity pattern to achieve robustness. Numerical results from simulating ground motion on a parallel supercomputer are given to compare the effectiveness of these parallel preconditioning techniques.

We introduce a general framework for constructing multilevel recursive incomplete LU preconditioning techniques (RILUM) for solving general sparse matrices. This technique is based on a recursive two by two block incomplete LU factorization on the coefficient matrix. The coarse level system is constructed as an (approximate) Schur complement. A dynamic preconditioner is obtained by solving the Schur complement matrix approximately. The novelty of the proposed techniques is to solve the Schur complement matrix by a preconditioned Krylov subspace method. The very preconditioner for this secondary iteration is constructed by considering the Schur complement matrix as a general sparse matrix and by applying to it the block ILU factorization process that was applied to the original matrix. This recursive procedure continues for a few times and results in a multilevel preconditioner. Different implementation strategies are discussed. We conduct numerical experiments with two particular RILUM...