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No full textOnder het door Annie Romein-Verschoor geïnspireerde motto 'Omzien in Verwondering', kijkt prof.dr. Odo Diekmann terug op het toeval dat soms richtingbepalend was. Ontwikkelingen in de populatiedynamica en de epidemiologie van infectieziekten, daarentegen, hebben een hoog 'hoe had het anders kunnen zijn' gehalte. Tot besluit komt aan bod hoe een evolutionaire invalshoek leidt tot inzichten in het spel en de spelers
Parallel, groen en snel
No full textHet is parallel, groen en snel. Dit klinkt als een raadsel en zo is het ook bedoeld. In de komende drie kwartier zal ik mijn best doen dit raadsel op te helderen en u te laten zien hoe deze woorden mijn vakgebied, scientific computing, karakteriseren. U rijdt misschien wel eens over de parallelweg, evenwijdig aan de snelweg. Of u droomt van een parallel universum, of wat bescheidener van een parallelle wereld, evenwijdig aan de echte, maar net iets anders. Zonder dat u het beseft gebruikt u waarschijnlijk al een parallelle computer, en dat is de betekenis van het woord ‘parallel ’ waar het vandaag om gaat. Als ´ık droom over iets parallels dan gaat het om parallelle algoritmen, rekenrecepten om parallelle computers te gebruiken
Persistence of noncompact normally hyperbolic invariant manifolds in bounded geometry = Persistance des variétés invariantes normalement hyperboliques non-compactes dans la
No full textWe prove a persistence result for noncompact normally hyperbolic invariant manifolds in Riemannian manifolds of bounded geometry. The bounded geometry of the ambient manifold is a crucial assumption in order to control the uniformity of all estimates throughout the proof. -------------------------------------------------------------------------------
McKay correspondence for Landau-Ginzburg models
Full text linkIn this paper we prove an analogue of the McKay correspondence for Landau-Ginzburg models. Our proof is based on the ideas introduced by T. Bridgeland, A. King and M. Reid, which reformulate and generalize the McKay correspondence in the language of derived categories, along with the techniques introduced by J.-C. Chen
Redheffer representations and relaxed commutant lifting
Full text linkIt is well known that the solutions of a (relaxed) commutant lifting problem can be described via a linear fractional representation of the Redheffer type. The coefficients of such Redheffer representations are analytic operator-valued functions defined on the unit disc D of the complex plane. In this paper we consider the converse question. Given a Redheffer representation, necessary and sufficient conditions on the coefficients are obtained guaranteeing the representation to appear in the description of the solutions to some relaxed commutant lifting problem. In addition, a result concerning a form of non-uniqueness appearing in the Redheffer representations under consideration and an harmonic maximal principle, generalizing a result of A. Biswas, are proved. The latter two results can be stated both on the relaxed commutant lifting as well as on the Redheffer representation level
Non-degeneracy conditions in KAM theory
No full textPersistence of invariant tori in a perturbed dynamical system requires two kinds of conditions to be met. A strong non-resonance condition ensures a dense quasi-periodic orbit on both the unperturbed and the perturbed torus. A non-degeneracy condition enforces a sufficiently large subset of the unperturbed tori to be non-resonant and thus yields persistence. In the past 60 years various such conditions have been formulated and a number of them are reviewed here
Two-dimensional cache-oblivious sparse matrix–vector multiplication
No full textIn earlier work, we presented a one-dimensional cache-oblivious sparse matrix–vector (SpMV) multiplication scheme which has its roots in one-dimensional sparse matrix partitioning. Partitioning is often used in distributed-memory parallel computing for the SpMV multiplication, an important kernel in many applications. A logical extension is to move towards using a two-dimensional partitioning. In this paper, we present our research in this direction, extending the one-dimensional method for cache-oblivious SpMV multiplication to two dimensions, while still allowing only row and column permutations on the sparse input matrix. This extension requires a generalisation of the compressed row storage data structure to a block-based data structure, for which several variants are investigated. Experiments performed on three different architectures show further improvements of the two-dimensional method compared to the one-dimensional method, especially in those cases where the one-dimensional method already provided significant gains. The largest gain obtained by our new reordering is over a factor of 3 in SpMV speed, compared to the natural matrix ordering
M.A. Lewis, M.A.J. Chaplain, J.P. Keener, P.K. Maini (eds.): “Mathematical Biology”
No full textMathematical Biolog
Bi-CGSTAB as an induced dimension reduction method
No full textThe Induced Dimension Reduction method [P. Wesseling, P. Sonneveld, Numerical experiments with a multiple grid- and a preconditioned Lanczos type method, in: Lecture Notes in Mathematics, vol. 771, Springer-Verlag, Berlin, 1980, pp. 543–562] was proposed in 1980 as an iterative method for solving large nonsymmetric linear systems of equations. IDR can be considered as the predecessor of methods like CGS (Conjugate Gradient Squared) [P. Sonneveld, CGS, a fast Lanczos-type solver for nonsymmetric linear systems, SIAM J. Sci. Statist. Comput. 10 (1989) 36–52] and Bi-CGSTAB (Bi-Conjugate Gradients STABilized [H.A. van der Vorst, Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Statist. Comput. 13 (2) (1992) 631–644]). All three methods are based on efficient short recurrences. An important similarity between the methods is that they use orthogonalizations with respect to a fixed ‘shadow residual’. Of the three methods, Bi-CGSTAB has gained the most popularity, and is probably still the most widely used short recurrence method for solving nonsymmetric systems. Recently, Sonneveld and van Gijzen revived the interest for IDR. In [P. Sonneveld, M. van Gijzen, IDR(s): a family of simple and fast algorithms for solving large nonsymmetric systems of linear equations, Preprint, Delft Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands, March 2007], they demonstrate that a higher dimensional shadow space, defined by the n×s matrix , can easily be incorporated into IDR, yielding a highly effective method. The original IDR method is closely related to Bi-CGSTAB. It is therefore natural to ask whether Bi-CGSTAB can be extended in a way similar to IDR. To answer this question we explore the relation between IDR and Bi-CGSTAB and use our findings to derive a variant of Bi-CGSTAB that uses a higher dimensional shadow spac
An object-oriented bulk synchronous parallel library for multicore programming
No full textWe show that the bulk synchronous parallel (BSP) model, originally designed for distributed-memory systems, is also applicable for shared-memory multicore systems and, furthermore, that BSP libraries are useful in scientific computing on these systems. A proof-of-concept MulticoreBSP library has been implemented in Java, and is used to show that BSP algorithms can attain proper speedups on multicore architectures. This library is based on the BSPlib implementation, adapted to an object-oriented setting. In comparison, the number of function primitives is reduced, while the overall design simplicity is improved. We detail applying the BSP model and library on the sparse matrix–vector (SpMV) multiplication problem, and show by performing numerical experiments that the resulting BSP SpMV algorithm attains speedups, in one case reaching a speedup of 3.5 for 4 threads. Whereas not described in detail in this paper, algorithms for the fast Fourier transform and the dense LU decomposition are also investigated; in one case, attaining superlinear speedups of 5 for 4 threads. The predictability of BSP algorithms in the case of the SpMV is also investigated
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