898 research outputs found
Enhanced multi-level block ILU preconditioning strategies for general sparse linear systems
AbstractThis paper introduces several strategies to deal with pivot blocks in multi-level block incomplete LU factorization (BILUM) preconditioning techniques. These techniques are aimed at increasing the robustness and controlling the amount of fill-ins of BILUM for solving large sparse linear systems when large-size blocks are used to form block-independent set. Techniques proposed in this paper include double-dropping strategies, approximate singular-value decomposition, variable size blocks and use of an arrowhead block submatrix. We point out the advantages and disadvantages of these strategies and discuss their efficient implementations. Numerical experiments are conducted to show the usefulness of the new techniques in dealing with hard-to-solve problems arising from computational fluid dynamics. In addition, we discuss the relation between multi-level ILU preconditioning methods and algebraic multi-level methods
LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS IN ARROWHEAD FORM
This paper deals with different approaches for solving linear systems of the first order differential equations with the system matrix in the symmetric arrowhead form.Some needed algebraic properties of the symmetric arrowhead matrix are proposed.We investigate the form of invariant factors of the arrowhead matrix.Also the entries of the adjugate matrix of the characteristic matrix of the arrowhead matrix are considered. Some reductions techniques for linear systems of differential equations with the system matrix in the arrowhead form are presented
A Massively Parallel Algorithm for the Approximate Calculation of Inverse p-th Roots of Large Sparse Matrices
We present the submatrix method, a highly parallelizable method for the
approximate calculation of inverse p-th roots of large sparse symmetric
matrices which are required in different scientific applications. We follow the
idea of Approximate Computing, allowing imprecision in the final result in
order to be able to utilize the sparsity of the input matrix and to allow
massively parallel execution. For an n x n matrix, the proposed algorithm
allows to distribute the calculations over n nodes with only little
communication overhead. The approximate result matrix exhibits the same
sparsity pattern as the input matrix, allowing for efficient reuse of allocated
data structures.
We evaluate the algorithm with respect to the error that it introduces into
calculated results, as well as its performance and scalability. We demonstrate
that the error is relatively limited for well-conditioned matrices and that
results are still valuable for error-resilient applications like
preconditioning even for ill-conditioned matrices. We discuss the execution
time and scaling of the algorithm on a theoretical level and present a
distributed implementation of the algorithm using MPI and OpenMP. We
demonstrate the scalability of this implementation by running it on a
high-performance compute cluster comprised of 1024 CPU cores, showing a speedup
of 665x compared to single-threaded execution
Explicit preconditioned domain decomposition schemes for solving nonlinear boundary value problems
AbstractA new class of inner-outer iterative procedures in conjunction with Picard-Newton methods based on explicit preconditioning iterative methods for solving nonlinear systems is presented. Explicit preconditioned iterative schemes, based on the explicit computation of a class of domain decomposition generalized approximate inverse matrix techniques are presented for the efficient solution of nonlinear boundary value problems on multiprocessor systems. Applications of the new composite scheme on characteristic nonlinear boundary value problems are discussed and numerical results are given
A General Framework for Fair Regression
Fairness, through its many forms and definitions, has become an important
issue facing the machine learning community. In this work, we consider how to
incorporate group fairness constraints in kernel regression methods, applicable
to Gaussian processes, support vector machines, neural network regression and
decision tree regression. Further, we focus on examining the effect of
incorporating these constraints in decision tree regression, with direct
applications to random forests and boosted trees amongst other widespread
popular inference techniques. We show that the order of complexity of memory
and computation is preserved for such models and tightly bound the expected
perturbations to the model in terms of the number of leaves of the trees.
Importantly, the approach works on trained models and hence can be easily
applied to models in current use and group labels are only required on training
data.Comment: 8 pages, 4 figures, 2 pages reference
Thick-restarted joint Lanczos bidiagonalization for the GSVD
The computation of the partial generalized singular value decomposition
(GSVD) of large-scale matrix pairs can be approached by means of iterative
methods based on expanding subspaces, particularly Krylov subspaces. We
consider the joint Lanczos bidiagonalization method, and analyze the
feasibility of adapting the thick restart technique that is being used
successfully in the context of other linear algebra problems. Numerical
experiments illustrate the effectiveness of the proposed method. We also
compare the new method with an alternative solution via equivalent eigenvalue
problems, considering accuracy as well as computational performance. The
analysis is done using a parallel implementation in the SLEPc library
- …