Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for this type of system. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.

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Numerical solution of saddle point problems

Solving unsymmetric sparse systems of linear equations with PARDISO

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Electronic Transport In Mesoscopic Systems

In the field of computational biomechanics, investigators have primarily used commercial software that is neither geared toward biological applications nor sufficiently flexible to follow the latest developments in the field This lack of a tailored software environment has hampered research progress, as well as dissemination of models and results To address these issues, we developed the FEBio software suite (http://mrlsciutahedu/software/febio), a nonlinear implicit finite element (FE) framework, designed specifically for analysis in computational solid biomechanics This paper provides an overview of the theoretical basis of FEBio and its main features FEBio offers modeling scenarios, constitutive models, and boundary conditions, which are relevant to numerous applications in biomechanics The open-source FEBio software is written in C++, with particular attention to scalar and parallel performance on modern computer architectures Software verification is a large part of the development and maintenance of FEBio, and to demonstrate the general approach, the description and results of several problems from the FEBio Verification Suite are presented and compared to analytical solutions or results from other established and verified FE codes An additional simulation is described that illustrates the application of FEBio to a research problem in biomechanics Together with the pre- and postprocessing software PREVIEW and POSTVIEW, FEBio provides a tailored solution for research and development in computational biomechanics

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FEBio: finite elements for biomechanics.

Derivatives, mostly in the form of gradients and Hessians, are ubiquitous in machine learning. Automatic differentiation (AD), also called algorithmic differentiation or simply "auto-diff", is a family of techniques similar to but more general than backpropagation for efficiently and accurately evaluating derivatives of numeric functions expressed as computer programs. AD is a small but established field with applications in areas including computational uid dynamics, atmospheric sciences, and engineering design optimization. Until very recently, the fields of machine learning and AD have largely been unaware of each other and, in some cases, have independently discovered each other's results. Despite its relevance, general-purpose AD has been missing from the machine learning toolbox, a situation slowly changing with its ongoing adoption under the names "dynamic computational graphs" and "differentiable programming". We survey the intersection of AD and machine learning, cover applications where AD has direct relevance, and address the main implementation techniques. By precisely defining the main differentiation techniques and their interrelationships, we aim to bring clarity to the usage of the terms "autodiff", "automatic differentiation", and "symbolic differentiation" as these are encountered more and more in machine learning settings.

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Automatic differentiation in machine learning: a survey

This paper discusses new pivoting factorization methods for solving sparse symmetric indenite sys- tems. As opposed to many existing pivoting methods, our SupernodenBunchnKaufman (SBK) pivoting method dy- namically selects and pivots and may be supplemented by pivot perturbation techniques. We demonstrate the effectiveness and the numerical accuracy of this algorithm and also show that a high performance implementa- tion is feasible. We will also show that symmetric maximum-weighted matching strategies add an additional level of reliability to SBK. These techniques can be seen as a complement to the alternative idea of using more complete pivoting techniques during the numerical factorization. Numerical experiments validate these conclusions. where is a diagonal matrix with and pivot blocks, is a sparse lower triangu- lar matrix, and is a symmetric indenite diagonal matrix that reects small half-machine precision perturbations, which might be necessary to tackle the problem of tiny pivots. is a reordering that is based on a symmetric weighted matching of the matrix , and tries to move the largest off-diagonal elements directly alongside the diagonal in order to form good initial or diagonal block pivots. is a ll reducing reordering which honors the structure of . We will present three new variations of a direct factorization scheme to tackle the is- sue of indeniteness in sparse symmetric linear systems. These methods restrict the pivoting search, to stay as long as possible within predened data structures for efcient Level-3 BLAS factorization and parallelization. On the other hand, the imposed pivoting restrictions can be reduced in several steps by taking the matching permutation into account. The rst al- gorithm uses SupernodenBunchnKaufman (SBK) pivoting and dynamically selects and pivots. It is supplemented by pivot perturbation techniques. It uses no more storage than a sparse Cholesky factorization of a positive denite matrix with the same sparsity structure due to restricting the pivoting to interchanges within the diagonal block associated to a single supernode. The coefcient matrix is perturbed whenever numerically acceptable and pivots cannot be found within the diagonal block. One or two steps of iterative re- nement may be required to correct the effect of the perturbations. We will demonstrate that this restricting notion of pivoting with iterative renement is effective for highly indenite symmetric systems. Furthermore the accuracy of this method is for a large set of matrices from different applications areas as accurate as a direct factorization method that uses com- plete sparse pivoting techniques. In addition, we will discuss two preprocessing algorithms to identify large entries in the coefcient matrix that, if permuted close to the diagonal, permit

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On fast factorization pivoting methods for sparse symmetric indefinite systems

Stencil calculations comprise an important class of kernels in many scientific computing applications ranging from simple PDE solvers to constituent kernels in multigrid methods as well as image processing applications. In such types of solvers, stencil kernels are often the dominant part of the computation, and an efficient parallel implementation of the kernel is therefore crucial in order to reduce the time to solution. However, in the current complex hardware micro architectures, meticulous architecture-specific tuning is required to elicit the machine's full compute power. We present a code generation and auto-tuning framework \textsc{Patus} for stencil computations targeted at multi- and many core processors, such as multicore CPUs and graphics processing units, which makes it possible to generate compute kernels from a specification of the stencil operation and a parallelization and optimization strategy, and leverages the auto tuning methodology to optimize strategy-dependent parameters for the given hardware architecture.

PATUS: A Code Generation and Autotuning Framework for Parallel Iterative Stencil Computations on Modern Microarchitectures

Supernode pivoting for unsymmetric matrices coupled with supernode partitioning and asynchronous computation can achieve high gigaflop rates for parallel sparse LU factorization on shared memory parallel computers. The progress in weighted graph matching algorithms helps to extend these concepts further and prepermutation of rows is used to place large matrix entries on the diagonal. Supernode pivoting allows dynamical interchanges of columns and rows during the factorization process. The BLAS-3 level efficiency is retained. An enhanced left-right looking scheduling scheme is uneffected and results in good speedup on SMP machines without increasing the operation count. These algorithms have been integrated into the recent unsymmetric version of the PARDISO solver. Experiments demonstrate that a wide set of unsymmetric linear systems can be solved and high performance is consistently achieved for large sparse unsymmetric matrices from real world applications.

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Solving Unsymmetric Sparse Systems of Linear Equations with PARDISO

An efficient sparse LU factorization algorithm on popular shared memory multi-processors is presented. Pipelining parallelism is essential to achieve higher parallel efficiency and it is exploited with a left-right looking algorithm. No global barrier is used and a completely asynchronous scheduling scheme is one central point of the implementation. The algorithm has been successfully tested on SUN Enterprise, DEC AlphaServer, SGI Origin 2000 and Cray T90 and J90 parallel computers, delivering up to 2.3 GFlop/s on an eight processor DEC AlphaServer for medium-size semiconductor device simulations and structural engineering problems.

Olaf Schenk

Papers

Solving unsymmetric sparse systems of linear equations with PARDISO

On fast factorization pivoting methods for sparse symmetric indefinite systems

PATUS: A Code Generation and Autotuning Framework for Parallel Iterative Stencil Computations on Modern Microarchitectures

Solving Unsymmetric Sparse Systems of Linear Equations with PARDISO

Efficient Sparse LU Factorization with Left-Right Looking Strategy on Shared Memory Multiprocessors