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Anzengruber S.W.,TU Chemnitz | Hofmann B.,TU Chemnitz | Ramlau R.,Johannes Kepler University | Ramlau R.,Johann Radon Institute for Computational and Applied Mathematics
Inverse Problems | Year: 2013

The convergence rates results in ℓ1-regularization when the sparsity assumption is narrowly missed, presented by Burger et al (2013 Inverse Problems 29 025013), are based on a crucial condition which requires that all basis elements belong to the range of the adjoint of the forward operator. Partly it was conjectured that such a condition is very restrictive. In this context, we study sparsity-promoting varieties of Tikhonov regularization for linear ill-posed problems with respect to an orthonormal basis in a separable Hilbert space using ℓ1 and sublinear penalty terms. In particular, we show that the corresponding range condition is always satisfied for all basis elements if the problems are well-posed in a certain weaker topology and the basis elements are chosen appropriately related to an associated Gelfand triple. The Radon transform, Symm's integral equation and linear integral operators of Volterra type are examples for such behaviour, which allows us to apply convergence rates results for non-sparse solutions, and we further extend these results also to the case of non-convex ℓq-regularization with 0 < q < 1. © 2013 IOP Publishing Ltd.


Kugler P.,Johann Radon Institute for Computational and Applied Mathematics
PLoS ONE | Year: 2012

The inference of reaction rate parameters in biochemical network models from time series concentration data is a central task in computational systems biology. Under the assumption of well mixed conditions the network dynamics are typically described by the chemical master equation, the Fokker Planck equation, the linear noise approximation or the macroscopic rate equation. The inverse problem of estimating the parameters of the underlying network model can be approached in deterministic and stochastic ways, and available methods often compare individual or mean concentration traces obtained from experiments with theoretical model predictions when maximizing likelihoods, minimizing regularized least squares functionals, approximating posterior distributions or sequentially processing the data. In this article we assume that the biological reaction network can be observed at least partially and repeatedly over time such that sample moments of species molecule numbers for various time points can be calculated from the data. Based on the chemical master equation we furthermore derive closed systems of parameter dependent nonlinear ordinary differential equations that predict the time evolution of the statistical moments. For inferring the reaction rate parameters we suggest to not only compare the sample mean with the theoretical mean prediction but also to take the residual of higher order moments explicitly into account. Cost functions that involve residuals of higher order moments may form landscapes in the parameter space that have more pronounced curvatures at the minimizer and hence may weaken or even overcome parameter sloppiness and uncertainty. As a consequence both deterministic and stochastic parameter inference algorithms may be improved with respect to accuracy and efficiency. We demonstrate the potential of moment fitting for parameter inference by means of illustrative stochastic biological models from the literature and address topics for future research. © 2012 Philipp Kügler.


Kasumba H.,Johann Radon Institute for Computational and Applied Mathematics | Kunisch K.,University of Graz | Laurain A.,TU Berlin
Interfaces and Free Boundaries | Year: 2014

A bilevel shape optimization problem with the exterior Bernoulli free boundary problem as lower-level problem and the control of the free boundary as the upper-level problem is considered. Using the shape of the inner boundary as the control, we aim at reaching a specific shape for the free boundary. A rigorous sensitivity analysis of the bilevel shape optimization in the infinite-dimensional setting is performed. The numerical realization using two different cost functionals presented in this paper demonstrate the efficiency of the approach. © European Mathematical Society 2014


Shaheen R.,Johann Radon Institute for Computational and Applied Mathematics | Shaheen R.,Cairo University | Winterhof A.,Johann Radon Institute for Computational and Applied Mathematics
Designs, Codes, and Cryptography | Year: 2010

Let q be a prime power. For a divisor n of q - 1 we prove an asymptotic formula for the number of polynomials of the form f(X)= a-b/n (∑j=1n-1Xj(q-1)/nX+a+b(n-1)/n X ∈ Fq[X] such that the five (not necessarily different) polynomials f(X), f(X)±X and f(f(X))±X are all permutation polynomials over Fq . Such polynomials can be used to define check digit systems that detect the most frequent errors: single errors, adjacent transpositions, jump transpositions, twin errors and jump twin errors. © 2010 Springer Science+Business Media, LLC.


Emans M.,Johann Radon Institute for Computational and Applied Mathematics | Emans M.,IMCC GmbH | Liebmann M.,University of Graz
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) | Year: 2013

We explore a GPU implementation of a Krylov-accelerated algebraic multigrid (AMG) algorithm with flexible preconditioning. We demonstrate by means of two benchmarks from an industrial computational fluid dynamics (CFD) application that the acceleration with multiple graphics processing units (GPUs) speeds up the solution phase by a factor of up to 13. In order to achieve good performance for the whole AMG algorithm, we propose for the setup a substitution of the double-pairwise aggregation by a simpler aggregation scheme skipping the calculation of temporary grids and operators. The version with the revised setup reduces the total computing time on multiple GPUs by further 30% compared to the GPU implementation with the double-pairwise aggregation. We observe that the GPU implementation of the entire Krylov-accelerated AMG runs up to four times faster than the fastest central processing unit (CPU) implementation. © 2013 Springer-Verlag.


Emans M.,Johann Radon Institute for Computational and Applied Mathematics | Emans M.,MathConsult GmbH
Journal of Computational Science | Year: 2011

We present an agglomeration approach for the solution of the coarse-grid problems in algebraic multigrid for coupled systems. Our implementation relies on an appropriate reordering of the variables of the merged systems. A benchmark from fluid dynamics, representing the important class of mixed elliptic-hyperbolic problems, is used to demonstrate that the performance of the suggested agglomeration scheme comes much closer to the desired behaviour of the ideal multigrid than that of alternatives described in the literature. © 2011 Elsevier B.V.


Piotrowski Z.P.,National Water Research Institute | Matejczyk B.,Johann Radon Institute for Computational and Applied Mathematics | Marcinkowski L.,University of Warsaw | Smolarkiewicz P.K.,European Center for Medium Range Weather Forecasts
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) | Year: 2016

Effective preconditioning lies at the heart of multiscale flow simulation, including a broad range of geoscientific applications that rely on semi-implicit integrations of the governing PDEs. For such problems, conditioning of the resulting sparse linear operator directly responds to the squared ratio of largest and smallest spatial scales represented in the model. For thin-spherical-shell geometry of the Earth atmosphere the condition number is enormous, upon which implicit preconditioning is imperative to eliminate the stiffness resulting from relatively fine vertical resolution. Furthermore, the anisotropy due to the meridians convergence in standard latitude-longitude discretizations becomes equally detrimental as the horizontal resolution increases to capture nonhydrostatic dynamics. Herein, we discuss a class of effective preconditioners based on the parallel ADI approach. The approach has been implemented in the established high-performance all-scale model EULAG with flexible computational domain distribution, including a 3D processor array. The efficacy of the approach is demonstrated in the context of an archetypal simulation of global weather. © Springer International Publishing Switzerland 2016.


Su M.,Nankai University | Winterhof A.,Johann Radon Institute for Computational and Applied Mathematics
IEEE Transactions on Information Theory | Year: 2010

We combine the concepts of the p-periodic Legendre sequence, the (q-1)-periodic Sidelnikov sequence and the two-prime generator to introduce a new p(q-1)-periodic sequence called LegendreSidelnikov sequence. We show that this new sequence is balanced if p=q. For an arbitrary odd prime p and an arbitrary power q of an odd prime with gcd (p,q-1)=1 we determine the exact values of its (periodic) autocorrelation function and deduce an upper bound on its aperiodic autocorrelation function showing that it is small compared to its period. © 2006 IEEE.


PubMed | Johann Radon Institute for Computational and Applied Mathematics
Type: Journal Article | Journal: PloS one | Year: 2012

The inference of reaction rate parameters in biochemical network models from time series concentration data is a central task in computational systems biology. Under the assumption of well mixed conditions the network dynamics are typically described by the chemical master equation, the Fokker Planck equation, the linear noise approximation or the macroscopic rate equation. The inverse problem of estimating the parameters of the underlying network model can be approached in deterministic and stochastic ways, and available methods often compare individual or mean concentration traces obtained from experiments with theoretical model predictions when maximizing likelihoods, minimizing regularized least squares functionals, approximating posterior distributions or sequentially processing the data. In this article we assume that the biological reaction network can be observed at least partially and repeatedly over time such that sample moments of species molecule numbers for various time points can be calculated from the data. Based on the chemical master equation we furthermore derive closed systems of parameter dependent nonlinear ordinary differential equations that predict the time evolution of the statistical moments. For inferring the reaction rate parameters we suggest to not only compare the sample mean with the theoretical mean prediction but also to take the residual of higher order moments explicitly into account. Cost functions that involve residuals of higher order moments may form landscapes in the parameter space that have more pronounced curvatures at the minimizer and hence may weaken or even overcome parameter sloppiness and uncertainty. As a consequence both deterministic and stochastic parameter inference algorithms may be improved with respect to accuracy and efficiency. We demonstrate the potential of moment fitting for parameter inference by means of illustrative stochastic biological models from the literature and address topics for future research.

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