673 research outputs found

    A Bramble-Pasciak conjugate gradient method for discrete Stokes equations with random viscosity

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    We study the iterative solution of linear systems of equations arising from stochastic Galerkin finite element discretizations of saddle point problems. We focus on the Stokes model with random data parametrized by uniformly distributed random variables and discuss well-posedness of the variational formulations. We introduce a Bramble-Pasciak conjugate gradient method as a linear solver. It builds on a non-standard inner product associated with a block triangular preconditioner. The block triangular structure enables more sophisticated preconditioners than the block diagonal structure usually applied in MINRES methods. We show how the existence requirements of a conjugate gradient method can be met in our setting. We analyze the performance of the solvers depending on relevant physical and numerical parameters by means of eigenvalue estimates. For this purpose, we derive bounds for the eigenvalues of the relevant preconditioned sub-matrices. We illustrate our findings using the flow in a driven cavity as a numerical test case, where the viscosity is given by a truncated Karhunen-Lo\`eve expansion of a random field. In this example, a Bramble-Pasciak conjugate gradient method with block triangular preconditioner outperforms a MINRES method with block diagonal preconditioner in terms of iteration numbers.Comment: 19 pages, 1 figure, submitted to SIAM JU

    A Stochastic Method for Solving Time-Fractional Differential Equations

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    We present a stochastic method for efficiently computing the solution of time-fractional partial differential equations (fPDEs) that model anomalous diffusion problems of the subdiffusive type. After discretizing the fPDE in space, the ensuing system of fractional linear equations is solved resorting to a Monte Carlo evaluation of the corresponding Mittag-Leffler matrix function. This is accomplished through the approximation of the expected value of a suitable multiplicative functional of a stochastic process, which consists of a Markov chain whose sojourn times in every state are Mittag-Leffler distributed. The resulting algorithm is able to calculate the solution at conveniently chosen points in the domain with high efficiency. In addition, we present how to generalize this algorithm in order to compute the complete solution. For several large-scale numerical problems, our method showed remarkable performance in both shared-memory and distributed-memory systems, achieving nearly perfect scalability up to 16,384 CPU cores.Comment: Submitted to the Journal of Computational Physic

    The parallel finite element system M++ with integrated multilevel preconditioning and multilevel Monte Carlo methods

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    We present a parallel data structure for the discretization of partial differential equations which is based on distributed point objects and which enables the flexible, transparent, and efficient realization of conforming, nonconforming, and mixed finite elements. This concepts is realized for elliptic, parabolic and hyperbolic model problems, and sample applications are provided by a tutorial complementing a lecture on scientific computing. The corresponding open-source software is based on this parallel data structure, and it supports multilevel methods on nested meshes and 2D and 3D as well as in space-time. Here, we present generic results on porous media applications including multilevel preconditioning and multilevel Monte Carlo methods for uncertainty quantification

    Forecasting Financial Volatility Using Nested Monte Carlo Expression Discovery

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    We are interested in discovering expressions for financial prediction using Nested Monte Carlo Search and Genetic Programming. Both methods are applied to learn from financial time series to generate non linear functions for market volatility prediction. The input data, that is a series of daily prices of European S&P500 index, is filtered and sampled in order to improve the training process. Using some assessment metrics, the best generated models given by both approaches for each training sub sample, are evaluated and compared. Results show that Nested Monte Carlo is able to generate better forecasting models than Genetic Programming for the majority of learning samples

    Solving forward-looking models of cross-country adjustment within the euro area

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    This paper generalizes the standard methods of solving rational expectations models to the case of time-varying nonstochastic parameters, recurring in a finite cycle. Such a specification occurs in a simple stylized New Keynesian model of the euro area when we combine the rotation in the ECB Governing Council (as constituted by the Treaty of Nice) and home bias in the interest rate decisions taken by its members. In small and mid-size economies, this combination slightly increases output and inflation volatility, as compared to a monetary policy setup without rotation. The method of Christiano (2002) has also been applied to solve the model when we assume a lagged perception of foreign macroeconomic shocks by domestic agents. When the cross-country synchronization of shocks is low or moderate and when these shocks are relatively persistent, the exclusion of contemporaneous foreign shocks from domestic agents' information sets may raise the volatility of output. There is also some tentative evidence that this effect could particularly affect mid-size economies.EMU, monetary policy, solving rational expectations models, generalized Schur decomposition, heterogeneity.

    Solving Forward-Looking Models of Cross-Country Adjustment within the Euro Area

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    This article introduces and applies two refinements to the algorithm of solving rational expectations models of a currency union. Firstly, building upon Klein (2000), it generalizes the standard methods of solving rational expectations models to the case of time-varying nonstochastic parameters, recurring in a finite cycle. Such a specification occurs in a simple stylized New Keynesian model of the euro area after a joint introduction of (i) rotation in the ECB Governing Council (as constituted by the Treaty of Nice) and (ii) home bias in the interest rate decisions preferred by its members. Secondly, we apply the method of Christiano (2002) to solve the model with heterogenous information sets. This is justified if we argue that the information set of domestic economic agents in a currency union is home-biased (i.e. foreign shocks enter only with a lag). Both methods of solution are illustrated with simulation results.solving rational expectations models, generalized Schur decomposition, heterogenous information sets, method of Christiano, divergences in EMU

    Detection of Topographic Contrast in the Scanning Electron Microscope at Low and Medium Resolution by Different Detectors and Detector Systems

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    Origins of topographic contrast in the scanning electron microscope (SEM) are different at different resolution levels. At low resolution, tilt contrast of large features dominates; at medium resolution, diffusion contrast of features smaller than an interaction volume of primary electrons dominates. The secondary electron (SE) signal, commonly used in the SEM, does not give a good tilt contrast; better contrast can be obtained with backscattered electron (BSE) signal of a converter and a sector-shaped ring detector. For obtaining topographic images from signals containing topographic and material contrast, signals of detector systems containing two or more detectors are mixed. Detector systems containing BSE detectors give more reproducible signals with a more uniform dependence on tilt angles than systems containing SE detectors. Tilt contrast of specimens coated with thin layers of heavy metals is similar to the contrast of uncoated specimens in the case of an SE detector, and better tilt contrast can be obtained with a sector-shaped ring BSE detector. Diffusion contrast dominates at medium resolution. Contrast obtained with three selected detectors: SE detector, sector-shaped ring BSE detector, annular top BSE detector, is also discussed. The contrast is lowest for the top BSE detector and highest in the case of SE detection, especially for materials of low density. In the case of coated specimens, the SE detector and the sector-shaped ring BSE detector give good contrast and both are suitable for medium resolution studies. The discussion in the paper concerns untilted or slightly tilted specimens

    A framework for the automation of generalised stability theory

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