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This contribution combines a low-rank matrix approximation through Singular Value Decomposition (SVD) with second-order Krylov subspace-based Model Order Reduction (MOR), in order to efficiently propagate input uncertainties through a given vibroacoustic model. The vibroacoustic model consists of a plate coupled to a fluid into which the plate radiates sound due to a turbulent boundary layer excitation. This excitation is subject to uncertainties due to the stochastic nature of the turbulence and the computational cost of simulating the coupled problem with stochastic forcing is very high. The proposed method approximates the output uncertainties in an efficient way, by reducing the evaluation cost of the model in terms of DOFs and samples by using the factors of the SVD low-rank approximation directly as input for the MOR algorithm. Here, the covariance matrix of the vector of unknowns can efficiently be approximated with only a fraction of the original number of evaluations. Therefore, the approach is a promising step to further reducing the computational effort of large-scale vibroacoustic evaluations.

Predictive modeling involving simulation and sensor data at the same time, is a growing challenge in computational science. Even with large-scale finite element models, a mismatch to the sensor data often remains, which can be attributed to different sources of uncertainty. For such a scenario, the statistical finite element method (statFEM) can be used to condition a simulated field on given sensor data. This yields a posterior solution which resembles the data much better and additionally provides consistent estimates of uncertainty, including model misspecification. For frequency or parameter dependent problems, occurring, e.g. in acoustics or electromagnetism, solving the full order model at the frequency grid and conditioning it on data quickly results in a prohibitive computational cost. In this case, the introduction of a surrogate in form of a reduced order model yields much smaller systems of equations. In this paper, we propose a reduced order statFEM framework relying on Krylov-based moment matching. We introduce a data model which explicitly includes the bias induced by the reduced approximation, which is estimated by an inexpensive error indicator. The results of the new statistical reduced order method are compared to the standard statFEM procedure applied to a ROM prior, i.e. without explicitly accounting for the reduced order bias. The proposed method yields better accuracy and faster convergence throughout a given frequency range for different numerical examples., Comment: 32 pages, 12 figures, associated code available at https://github.com/herluc/statROM

Many application problems that lead to solving linear systems make use of preconditioned Krylov subspace solvers to compute their solution. Among the most popular preconditioning approaches are incomplete factorization methods either as single-level approaches or within a multilevel framework. We will present a block incomplete factorization that is based on skillfully blocking the system initially and throughout the factorization. This approach allows for the use of cache-optimized dense matrix kernels such as level-3 BLAS or LAPACK. We will demonstrate how this block approach outperforms the scalar method often by orders of magnitude on modern architectures, paving the way for its prospective use inside various multilevel incomplete factorization approaches or other applications where the core part relies on an incomplete factorization.

In this chapter we will give an insight into modern sparse elimination methods. These are driven by a preprocessing phase based on combinatorial algorithms which improve diagonal dominance, reduce fill-in, and improve concurrency to allow for parallel treatment. Moreover, these methods detect dense submatrices which can be handled by dense matrix kernels based on multithreaded level-3 BLAS. We will demonstrate for problems arising from circuit simulation, how the improvements in recent years have advanced direct solution methods significantly.

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 the largest 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 $LDL^T$ 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 accelerative the computation of a large-scale eigenvalue problem corresponding to the Anderson model of localization by several orders of magnitude., Comment: 9 .eps figures, submitted to SIAM J. Sci. Comp., SIAM LaTeX styles (included), high quality figures can be obtained from http://eprints.csc.warwick.ac.uk/archive/00000110/