1,032 research outputs found

    Calculation of aggregate loss distributions

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    Estimation of the operational risk capital under the Loss Distribution Approach requires evaluation of aggregate (compound) loss distributions which is one of the classic problems in risk theory. Closed-form solutions are not available for the distributions typically used in operational risk. However with modern computer processing power, these distributions can be calculated virtually exactly using numerical methods. This paper reviews numerical algorithms that can be successfully used to calculate the aggregate loss distributions. In particular Monte Carlo, Panjer recursion and Fourier transformation methods are presented and compared. Also, several closed-form approximations based on moment matching and asymptotic result for heavy-tailed distributions are reviewed

    Exploiting spatial symmetries for solving Poisson's equation

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    This paper presents a strategy to accelerate virtually any Poisson solver by taking advantage of s spatial reflection symmetries. More precisely, we have proved the existence of an inexpensive block diagonalisation that transforms the original Poisson equation into a set of 2s fully decoupled subsystems then solved concurrently. This block diagonalisation is identical regardless of the mesh connectivity (structured or unstructured) and the geometric complexity of the problem, therefore applying to a wide range of academic and industrial configurations. In fact, it simplifies the task of discretising complex geometries since it only requires meshing a portion of the domain that is then mirrored implicitly by the symmetries’ hyperplanes. Thus, the resulting meshes naturally inherit the exploited symmetries, and their memory footprint becomes 2s times smaller. Thanks to the subsystems’ better spectral properties, iterative solvers converge significantly faster. Additionally, imposing an adequate grid points’ ordering allows reducing the operators’ footprint and replacing the standard sparse matrix-vector products with the sparse matrixmatrix product, a higher arithmetic intensity kernel. As a result, matrix multiplications are accelerated, and massive simulations become more affordable. Finally, we include numerical experiments based on a turbulent flow simulation and making state-of-theart solvers exploit a varying number of symmetries. On the one hand, algebraic multigrid and preconditioned Krylov subspace methods require up to 23% and 72% fewer iterations, resulting in up to 1.7x and 5.6x overall speedups, respectively. On the other, sparse direct solvers’ memory footprint, setup and solution costs are reduced by up to 48%, 58% and 46%, respectively.This work has been financially supported by two competitive R+D projects: RETOtwin (PDC2021-120970-I00), given by MCIN/AEI/10.13039/501100011033 and European Union Next GenerationEU/PRTR, and FusionCAT (001-P-001722), given by Generalitat de Catalunya RIS3CAT-FEDER. Àdel Alsalti-Baldellou has also been supported by the predoctoral grants DIN2018-010061 and 2019-DI-90, given by MCIN/AEI/10.13039/501100011033 and the Catalan Agency for Management of University and Research Grants (AGAUR), respectively.Peer ReviewedPostprint (published version

    Parallel algorithms for computational fluid dynamics on unstructured meshes

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    La simulació numèrica directa (DNS) de fluxos complexes és actualment una utopia per la majoria d'aplicacions industrials ja que els requeriments computacionals son massa elevats. Donat un flux, la diferència entre els recursos computacionals necessaris i els disponibles és cobreix mitjançant la modelització/simplificació d'alguns termes de les equacions originals que regeixen el seu comportament. El creixement continuat dels recursos computacionals disponibles, principalment en forma de super-ordinadors, contribueix a reduir la part del flux que és necessari aproximar. De totes maneres, obtenir la eficiència esperada dels nous super-ordinadors no és una tasca senzilla i, per aquest motiu, part de la recerca en el camp de la Mecànica de Fluids Computacional es centra en aquest objectiu. En aquest sentit, algunes contribucions s'han presentat en el marc d'aquesta tesis. El primer objectiu va ser el desenvolupament d'un codi de CFD de propòsit general i paral·lel, basat en la metodologia de volums finits en malles no estructurades, per resoldre problemes de multi-física. Aquest codi, anomenat TermoFluids (TF), té un disseny orientat a objectes i pensat per ser usat de forma altament eficient en els super-ordinadors actuals. Amb el temps, ha esdevingut pel grup una eina fonamental en projectes tant de recerca bàsica com d'interès industrial. En el context d'aquesta tesis, el treball s'ha focalitzat en el desenvolupament de dos de les llibreries més bàsiques de TermoFluids: i) La Basics Objects Library (BOL), que es una plataforma de software sobre la qual estan programades la resta de llibreries del codi, i que conté els mètodes algebraics i geomètrics fonamentals per la implementació paral·lela dels algoritmes de discretització, ii) la Linear Solvers Library (LSL), que conté un gran nombre de mètodes per resoldre els sistemes d'equacions lineals derivats de les discretitzacions. El primer capítol d'aquesta tesi conté les principals idees subjacents al disseny i la implementació de la BOL i la LSL, juntament amb alguns exemples i algunes aplicacions industrials. En els capítols posteriors hi ha una explicació detallada de solvers específics per algunes aplicacions concretes. En el segon capítol, es presenta un solver paral·lel i directe per la resolució de l'equació de Poisson per casos en els quals una de les direccions del domini té condicions d'homogeneïtat. En la simulació de fluxos incompressibles, l'equació de Poisson es resol almenys una vegada en cada pas de temps, convertint-se en una de les parts més costoses i difícils de paral·lelitzar del codi. El mètode que proposem és una combinació d'una descomposició directa de Schur (DDS) i una diagonalització de Fourier. La darrera descompon el sistema original en un conjunt de sub-sistemes 2D independents que es resolen mitjançant l'algorisme DDS. Atès que no s'imposen restriccions a les direccions no periòdiques del domini, aquest mètode és aplicable a la resolució de problemes discretitzats mitjançat l'extrusió de malles 2D no estructurades. L'escalabilitat d'aquest mètode ha estat provada amb èxit amb un màxim de 8192 CPU per malles de fins a ~10⁹ volums de control. En el darrer capitol capítol, es presenta un mètode de resolució per l'equació de Transport de Boltzmann (BTE). La estratègia emprada es basa en el mètode d'Ordenades Discretes i pot ser aplicat en discretitzacions no estructurades. El flux per a cada ordenada angular es resol amb un mètode de substitució equivalent a la resolució d'un sistema lineal triangular. La naturalesa seqüencial d'aquest procés fa de la paral·lelització de l'algoritme el principal repte. Diversos algorismes de substitució han estat analitzats, esdevenint una de les heurístiques proposades la millor opció en totes les situacions analitzades, amb excel·lents resultats. Els testos d'eficiència paral·lela s'han realitzat usant fins a 2560 CPU.Direct Numerical Simulation (DNS) of complex flows is currently an utopia for most of industrial applications because computational requirements are too high. For a given flow, the gap between the required and the available computing resources is covered by modeling/simplifying of some terms of the original equations. On the other hand, the continuous growth of the computing power of modern supercomputers contributes to reduce this gap, reducing hence the unresolved physics that need to be attempted with approximated models. This growth, widely relies on parallel computing technologies. However, getting the expected performance from new complex computing systems is becoming more and more difficult, and therefore part of the CFD research is focused on this goal. Regarding to it, some contributions are presented in this thesis. The first objective was to contribute to the development of a general purpose multi-physics CFD code. referred to as TermoFluids (TF). TF is programmed following the object oriented paradigm and designed to run in modern parallel computing systems. It is also intensively involved in many different projects ranging from basic research to industry applications. Besides, one of the strengths of TF is its good parallel performance demonstrated in several supercomputers. In the context of this thesis, the work was focused on the development of two of the most basic libraries that compose TF: I) the Basic Objects Library (BOL), which is a parallel unstructured CFD application programming interface, on the top of which the rest of libraries that compose TF are written, ii) the Linear Solvers Library (LSL) containing many different algorithms to solve the linear systems arising from the discretization of the equations. The first chapter of this thesis contains the main ideas underlying the design and the implementation of the BOL and LSL libraries, together with some examples and some industrial applications. A detailed description of some application-specific linear solvers included in the LSL is carried out in the following chapters. In the second chapter, a parallel direct Poisson solver restricted to problems with one uniform periodic direction is presented. The Poisson equation is solved, at least, once per time-step when modeling incompressible flows, becoming one of the most time consuming and difficult to parallelize parts of the code. The solver here proposed is a combination of a direct Schur-complement based decomposition (DSD) and a Fourier diagonalization. The latter decomposes the original system into a set of mutually independent 2D sub-systems which are solved by means of the DSD algorithm. Since no restrictions are imposed in the non-periodic directions, the overall algorithm is well-suited for solving problems discretized on extruded 2D unstructured meshes. The scalability of the solver has been successfully tested using up to 8192 CPU cores for meshes with up to 10 9 grid points. In the last chapter, a solver for the Boltzmann Transport Equation (BTE) is presented. It can be used to solve radiation phenomena interacting with flows. The solver is based on the Discrete Ordinates Method and can be applied to unstructured discretizations. The flux for each angular ordinate is swept across the computational grid, within a source iteration loop that accounts for the coupling between the different ordinates. The sequential nature of the sweep process makes the parallelization of the overall algorithm the most challenging aspect. Several parallel sweep algorithms, which represent different options of interleaving communications and calculations, are analyzed. One of the heuristics proposed consistently stands out as the best option in all the situations analyzed. With this algorithm, good scalability results have been achieved regarding both weak and strong speedup tests with up to 2560 CPUs

    The optimal P3M algorithm for computing electrostatic energies in periodic systems

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    We optimize Hockney and Eastwood's Particle-Particle Particle-Mesh (P3M) algorithm to achieve maximal accuracy in the electrostatic energies (instead of forces) in 3D periodic charged systems. To this end we construct an optimal influence function that minimizes the RMS errors in the energies. As a by-product we derive a new real-space cut-off correction term, give a transparent derivation of the systematic errors in terms of Madelung energies, and provide an accurate analytical estimate for the RMS error of the energies. This error estimate is a useful indicator of the accuracy of the computed energies, and allows an easy and precise determination of the optimal values of the various parameters in the algorithm (Ewald splitting parameter, mesh size and charge assignment order).Comment: 31 pages, 3 figure

    Simulation techniques for cosmological simulations

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    Modern cosmological observations allow us to study in great detail the evolution and history of the large scale structure hierarchy. The fundamental problem of accurate constraints on the cosmological parameters, within a given cosmological model, requires precise modelling of the observed structure. In this paper we briefly review the current most effective techniques of large scale structure simulations, emphasising both their advantages and shortcomings. Starting with basics of the direct N-body simulations appropriate to modelling cold dark matter evolution, we then discuss the direct-sum technique GRAPE, particle-mesh (PM) and hybrid methods, combining the PM and the tree algorithms. Simulations of baryonic matter in the Universe often use hydrodynamic codes based on both particle methods that discretise mass, and grid-based methods. We briefly describe Eulerian grid methods, and also some variants of Lagrangian smoothed particle hydrodynamics (SPH) methods.Comment: 42 pages, 16 figures, accepted for publication in Space Science Reviews, special issue "Clusters of galaxies: beyond the thermal view", Editor J.S. Kaastra, Chapter 12; work done by an international team at the International Space Science Institute (ISSI), Bern, organised by J.S. Kaastra, A.M. Bykov, S. Schindler & J.A.M. Bleeke

    Superfast Line Spectral Estimation

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    A number of recent works have proposed to solve the line spectral estimation problem by applying off-the-grid extensions of sparse estimation techniques. These methods are preferable over classical line spectral estimation algorithms because they inherently estimate the model order. However, they all have computation times which grow at least cubically in the problem size, thus limiting their practical applicability in cases with large dimensions. To alleviate this issue, we propose a low-complexity method for line spectral estimation, which also draws on ideas from sparse estimation. Our method is based on a Bayesian view of the problem. The signal covariance matrix is shown to have Toeplitz structure, allowing superfast Toeplitz inversion to be used. We demonstrate that our method achieves estimation accuracy at least as good as current methods and that it does so while being orders of magnitudes faster.Comment: 16 pages, 7 figures, accepted for IEEE Transactions on Signal Processin

    Diffeomorphic density registration

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    In this book chapter we study the Riemannian Geometry of the density registration problem: Given two densities (not necessarily probability densities) defined on a smooth finite dimensional manifold find a diffeomorphism which transforms one to the other. This problem is motivated by the medical imaging application of tracking organ motion due to respiration in Thoracic CT imaging where the fundamental physical property of conservation of mass naturally leads to modeling CT attenuation as a density. We will study the intimate link between the Riemannian metrics on the space of diffeomorphisms and those on the space of densities. We finally develop novel computationally efficient algorithms and demonstrate there applicability for registering RCCT thoracic imaging.Comment: 23 pages, 6 Figures, Chapter for a Book on Medical Image Analysi
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