Inverse, forward and other dynamic computations computationally optimized with sparse matrix factorizations

We propose an algorithm to compute the dynamics of articulated rigid-bodies with different sensor distributions. Prior to the on-line computations, the proposed algorithm performs an off-line optimisation step to simplify the computational complexity of the underlying solution. This optimisation step consists in formulating the dynamic computations as a system of linear equations. The computational complexity of computing the associated solution is reduced by performing a permuted LU-factorisation with off-line optimised permutations. We apply our algorithm to solve classical dynamic problems: inverse and forward dynamics. The computational complexity of the proposed solution is compared to `gold standard' algorithms: recursive Newton-Euler and articulated body algorithm. It is shown that our algorithm reduces the number of floating point operations with respect to previous approaches. We also evaluate the numerical complexity of our algorithm by performing tests on dynamic computations for which no gold standard is available.Comment: 8 pages, 2 figure, conference RCAR 201

On large-scale diagonalization techniques for the Anderson model of localization

We propose efficient preconditioning algorithms for an eigenvalue problem arising in quantum physics, namely the computation of a few interior eigenvalues and their associated eigenvectors for large-scale sparse real and symmetric indefinite matrices of the Anderson model of localization. We compare the Lanczos algorithm in the 1987 implementation by Cullum and Willoughby with the shift-and-invert techniques in the implicitly restarted Lanczos method and in the Jacobi–Davidson method. Our preconditioning approaches for the shift-and-invert symmetric indefinite linear system are based on maximum weighted matchings and algebraic multilevel incomplete LDLT factorizations. These techniques can be seen as a complement to the alternative idea of using more complete pivoting techniques for the highly ill-conditioned symmetric indefinite Anderson matrices. We demonstrate the effectiveness and the numerical accuracy of these algorithms. Our numerical examples reveal that recent algebraic multilevel preconditioning solvers can accelerate the computation of a large-scale eigenvalue problem corresponding to the Anderson model of localization by several orders of magnitude

Proceedings of the second "international Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST'14)

The implicit objective of the biennial "international - Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST) is to foster collaboration between international scientific teams by disseminating ideas through both specific oral/poster presentations and free discussions. For its second edition, the iTWIST workshop took place in the medieval and picturesque town of Namur in Belgium, from Wednesday August 27th till Friday August 29th, 2014. The workshop was conveniently located in "The Arsenal" building within walking distance of both hotels and town center. iTWIST'14 has gathered about 70 international participants and has featured 9 invited talks, 10 oral presentations, and 14 posters on the following themes, all related to the theory, application and generalization of the "sparsity paradigm": Sparsity-driven data sensing and processing; Union of low dimensional subspaces; Beyond linear and convex inverse problem; Matrix/manifold/graph sensing/processing; Blind inverse problems and dictionary learning; Sparsity and computational neuroscience; Information theory, geometry and randomness; Complexity/accuracy tradeoffs in numerical methods; Sparsity? What's next?; Sparse machine learning and inference.Comment: 69 pages, 24 extended abstracts, iTWIST'14 website: http://sites.google.com/site/itwist1

Second order adjoints for solving PDE-constrained optimization problems

Inverse problems are of utmost importance in many fields of science and engineering. In the variational approach inverse problems are formulated as PDE-constrained optimization problems, where the optimal estimate of the uncertain parameters is the minimizer of a certain cost functional subject to the constraints posed by the model equations. The numerical solution of such optimization problems requires the computation of derivatives of the model output with respect to model parameters. The first order derivatives of a cost functional (defined on the model output) with respect to a large number of model parameters can be calculated efficiently through first order adjoint sensitivity analysis. Second order adjoint models give second derivative information in the form of matrix-vector products between the Hessian of the cost functional and user defined vectors. Traditionally, the construction of second order derivatives for large scale models has been considered too costly. Consequently, data assimilation applications employ optimization algorithms that use only first order derivative information, like nonlinear conjugate gradients and quasi-Newton methods. In this paper we discuss the mathematical foundations of second order adjoint sensitivity analysis and show that it provides an efficient approach to obtain Hessian-vector products. We study the benefits of using of second order information in the numerical optimization process for data assimilation applications. The numerical studies are performed in a twin experiment setting with a two-dimensional shallow water model. Different scenarios are considered with different discretization approaches, observation sets, and noise levels. Optimization algorithms that employ second order derivatives are tested against widely used methods that require only first order derivatives. Conclusions are drawn regarding the potential benefits and the limitations of using high-order information in large scale data assimilation problems

Performance Improvements of Common Sparse Numerical Linear Algebra Computations

Manufacturers of computer hardware are able to continuously sustain an unprecedented pace of progress in computing speed of their products, partially due to increased clock rates but also because of ever more complicated chip designs. With new processor families appearing every few years, it is increasingly harder to achieve high performance rates in sparse matrix computations. This research proposes new methods for sparse matrix factorizations and applies in an iterative code generalizations of known concepts from related disciplines. The proposed solutions and extensions are implemented in ways that tend to deliver efficiency while retaining ease of use of existing solutions. The implementations are thoroughly timed and analyzed using a commonly accepted set of test matrices. The tests were conducted on modern processors that seem to have gained an appreciable level of popularity and are fairly representative for a wider range of processor types that are available on the market now or in the near future. The new factorization technique formally introduced in the early chapters is later on proven to be quite competitive with state of the art software currently available. Although not totally superior in all cases (as probably no single approach could possibly be), the new factorization algorithm exhibits a few promising features. In addition, an all-embracing optimization effort is applied to an iterative algorithm that stands out for its robustness. This also gives satisfactory results on the tested computing platforms in terms of performance improvement. The same set of test matrices is used to enable an easy comparison between both investigated techniques, even though they are customarily treated separately in the literature. Possible extensions of the presented work are discussed. They range from easily conceivable merging with existing solutions to rather more evolved schemes dependent on hard to predict progress in theoretical and algorithmic research