Showing papers on "Sparse grid published in 2012"

On the optimal polynomial approximation of stochastic pdes by galerkin and collocation methods

Abstract: In this work we focus on the numerical approximation of the solution u of a linear elliptic PDE with stochastic coefficients. The problem is rewritten as a parametric PDE and the functional dependence of the solution on the parameters is approximated by multivariate polynomials. We first consider the stochastic Galerkin method, and rely on sharp estimates for the decay of the Fourier coefficients of the spectral expansion of u on an orthogonal polynomial basis to build a sequence of polynomial subspaces that features better convergence properties, in terms of error versus number of degrees of freedom, than standard choices such as Total Degree or Tensor Product subspaces. We consider then the Stochastic Collocation method, and use the previous estimates to introduce a new class of Sparse Grids, based on the idea of selecting a priori the most profitable hierarchical surpluses, that, again, features better convergence properties compared to standard Smolyak or tensor product grids. Numerical results show the effectiveness of the newly introduced polynomial spaces and sparse grids.

Topology optimization considering material and geometric uncertainties using stochastic collocation methods

Abstract: The aim of this paper is to introduce the stochastic collocation methods in topology optimization for mechanical systems with material and geometric uncertainties. The random variations are modeled by a memory-less transformation of spatially varying Gaussian random fields which ensures their physical admissibility. The stochastic collocation method combined with the proposed material and geometry uncertainty models provides robust designs by utilizing already developed deterministic solvers. The computational cost is discussed in details and solutions to decrease it, like sparse grids and discretization refinement are proposed and demonstrated as well. The method is utilized in the design of compliant mechanisms.

A compressed sensing approach for partial differential equations with random input data

Abstract: In this paper, a novel approach for quantifying the parametric uncertainty associated with a stochastic problem output is presented. As with Monte-Carlo and stochastic collocation methods, only point-wise evaluations of the stochastic output response surface are required allowing the use of legacy deterministic codes and precluding the need for any dedicated stochastic code to solve the uncertain problem of interest. The new approach differs from these standard methods in that it is based on ideas directly linked to the recently developed compressed sensing theory. The technique allows the retrieval of the modes that contribute most significantly to the approximation of the solution using a minimal amount of information. The generation of this information, via many solver calls, is almost always the bottle-neck of an uncertainty quantification procedure. If the stochastic model output has a reasonably compressible representation in the retained approximation basis, the proposed method makes the best use of the available information and retrieves the dominant modes. Uncertainty quantification of the solution of both a 2-D and 8-D stochastic Shallow Water problem is used to demonstrate the significant performance improvement of the new method, requiring up to several orders of magnitude fewer solver calls than the usual sparse grid-based Polynomial Chaos (Smolyak scheme) to achieve comparable approximation accuracy.

Sparse Grids in a Nutshell

Abstract: The technique of sparse grids allows to overcome the curse of dimensionality, which prevents the use of classical numerical discretization schemes in more than three or four dimensions, under suitable regularity assumptions. The approach is obtained from a multi-scale basis by a tensor product construction and subsequent truncation of the resulting multiresolution series expansion. This entry level article gives an introduction to sparse grids and the sparse grid combination technique.

Efficient Analysis of High Dimensional Data in Tensor Formats

Abstract: In this article we introduce new methods for the analysis of high dimensional data in tensor formats, where the underling data come from the stochastic elliptic boundary value problem. After discretisation of the deterministic operator as well as the presented random fields via KLE and PCE, the obtained high dimensional operator can be approximated via sums of elementary tensors. This tensors representation can be effectively used for computing different values of interest, such as maximum norm, level sets and cumulative distribution function. The basic concept of the data analysis in high dimensions is discussed on tensors represented in the canonical format, however the approach can be easily used in other tensor formats. As an intermediate step we describe efficient iterative algorithms for computing the characteristic and sign functions as well as pointwise inverse in the canonical tensor format. Since during majority of algebraic operations as well as during iteration steps the representation rank grows up, we use lower-rank approximation and inexact recursive iteration schemes.

Sparse Grids and Applications

Abstract: In the recent decade, there has been a growing interest in the numerical treatment of high-dimensional problems. It is well known that classical numerical discretization schemes fail in more than three or four dimensions due to the curse of dimensionality. The technique of sparse grids helps overcome this problem to some extent under suitable regularity assumptions. This discretization approach is obtained from a multi-scale basis by a tensor product construction and subsequent truncation of the resulting multiresolution series expansion. This volume of LNCSE is a collection of the papers from the proceedings of the workshop on sparse grids and its applications held in Bonn in May 2011. The selected articles present recent advances in the mathematical understanding and analysis of sparse grid discretization. Aspects arising from applications are given particular attention.

Efficient Spectral Sparse Grid Methods and Applications to High-Dimensional Elliptic Equations II. Unbounded Domains

Abstract: This is the second part in a series of papers on using spectral sparse grid methods for solving higher-dimensional PDEs. We extend the basic idea in the first part [J. Shen and H. Yu, SIAM J. Sci. Comp., 32 (2010), pp. 3228-3250] for solving PDEs in bounded higher-dimensional domains to unbounded higher-dimensional domains and apply the new method to solve the electronic Schrodinger equation. By using modified mapped Chebyshev functions as basis functions, we construct mapped Chebyshev sparse grid methods which enjoy the following properties: (i) the mapped Chebyshev approach enables us to build sparse grids with Smolyak's algorithms based on nested, spectrally accurate quadratures and allows us to build fast transforms between the values at the sparse grid points and the corresponding expansion coefficients; (ii) the mapped Chebyshev basis functions lead to identity mass matrices and very sparse stiffness matrices for problems with constant coefficients and allow us to construct a matrix-vector product algorithm with quasi-optimal computational cost even for problems with variable coefficients; and (iii) the resultant linear systems for elliptic equations with constant or variable coefficients can be solved efficiently by using a suitable iterative scheme. Ample numerical results are presented to demonstrate the efficiency and accuracy of the proposed algorithms.

Sparse tensor approximation of parametric eigenvalue problems

Abstract: We design and analyze algorithms for the efficient sensitivity computation of eigenpairs of parametric elliptic self-adjoint eigenvalue problems on high-dimensional parameter spaces. We quantify the analytic dependence of eigenpairs on the parameters. For the efficient approximate evaluation of parameter sensitivities of isolated eigenpairs on the entire parameter space we propose and analyze a sparse tensor spectral collocation method on an anisotropic sparse grid in the parameter domain. The stable numerical implementation of these methods is discussed and their error analysis is given. Applications to parametric elliptic eigenvalue problems with infinitely many parameters arising from elliptic differential operators with random coefficients are presented.

Spatially Adaptive Refinement

Abstract: While sparse grids allow one to tackle problems in higher dimensionalities than possible for standard grid-based discretizations, real-world applications often come along with requirements or restrictions which enforce problem-dependent adaptations of the standard sparse grid technique Consider, for example, interpolations where the function values at grid points are obtained via time-consuming numerical simulations Then, only very few grid points can be spent; classical convergence might be out of reach Another hurdle is that real-world problems often do not meet the smoothness requirements of the sparse grid method Thus, the standard approach has to be fine-tuned to the problem at hand, especially in higher-dimensional settings Therefore, a suitable choice of basis functions can be required, as well as criteria for problem-adapted refinement Fortunately, and in contrast to full grids, the hierarchical basis formulation of the direct sparse grid approach conveniently provides a reasonable criterion for spatially adaptive refinement practically for free This can serve as a starting point to develop suitable modifications We show several problems stemming from different fields of application and demonstrate modifications of the standard sparse grid approach They enable one to cope with the properties and requirements of the corresponding problem and can serve as examples for similar challenges

Adaptive Method for Non-Intrusive Spectral Projection—Application on a Stochastic Eddy Current NDT Problem

Abstract: The Non-Intrusive Spectral Projection (NISP) method is widely used for uncertainty quantification in stochastic models. The determination of the expansion of the solution on the polynomial chaos requires the computation of multidimensional integrals. An automatic adaptive algorithm based on nested sparse grids has been developed to evaluate those integrals. The adapted algorithm takes into account the weight of each random variable with respect to the output of the model. To achieve that it constructs anisotropic sparse grid of the mean, leading to a reduction of the number of numerical simulations. Furthermore, the spectral form of the solution is explicitly identified from the constructed quadrature scheme. Numerical results obtained on an industrial application in NDT demonstrate the efficiency of the proposed method.

A highly parallel Black–Scholes solver based on adaptive sparse grids

Abstract: In this paper, we present a highly efficient approach for numerically solving the Black–Scholes equation in order to price European and American basket options. Therefore, hardware features of contemporary high performance computer architectures such as non-uniform memory access and hardware-threading are exploited by a hybrid parallelization using MPI and OpenMP which is able to drastically reduce the computing time. In this way, we achieve very good speed-ups and are able to price baskets with up to six underlyings. Our approach is based on a sparse grid discretization with finite elements and makes use of a sophisticated adaption. The resulting linear system is solved by a conjugate gradient method that uses a parallel operator for applying the system matrix implicitly. Since we exploit all levels of the operator's parallelism, we are able to benefit from the compute power of more than 100 cores. Several numerical examples as well as an analysis of the performance for different computer architectures are provided.

The Combination Technique for the Initial Value Problem in Linear Gyrokinetics

Abstract: The simulation of hot fusion plasmas via the five-dimensional gyrokinetic equations is computationally intensive with one reason being the curse of dimensionality. Using the sparse grid combination technique could reduce the computational effort. For the computation of the full grid solutions, the plasma turbulence code GENE is used. It is shown that the combination technique is applicable to linear gyrokinetics by retrieving combination coefficients with a least squares approach. The retrieved sparse grid solution is actually close to the full grid one. Also, combination schemes were found which provided promising results with respect to the computational effort and accuracy.

Fastsg: A Fast Routines Library for Sparse Grids

Abstract: In a complex processor landscape dominated by multi-and many-core processors, simplifying programming plays a crucial role in enhancing developers’ productivity. One way is to use highly tuned library functions. In this paper we present fastsg, an optimized library for the sparse grid technique with support for dimensional truncation. With optimizations for best cache use and vectorization, we improve the performance on one processor core up to a factor of 10. Parallelization using OpenMP scales almost linearly on a 12-core system.

Adaptive Smolyak Pseudospectral Approximations

Abstract: Polynomial approximations of computationally intensive models are central to uncertainty quantification. This paper describes an adaptive method for non-intrusive pseudospectral approximation, based on Smolyak's algorithm with generalized sparse grids. We rigorously analyze and extend the non-adaptive method proposed in [6], and compare it to a common alternative approach for using sparse grids to construct polynomial approximations, direct quadrature. Analysis of direct quadrature shows that O(1) errors are an intrinsic property of some configurations of the method, as a consequence of internal aliasing. We provide precise conditions, based on the chosen polynomial basis and quadrature rules, under which this aliasing error occurs. We then establish theoretical results on the accuracy of Smolyak pseudospectral approximation, and show that the Smolyak approximation avoids internal aliasing and makes far more effective use of sparse function evaluations. These results are applicable to broad choices of quadrature rule and generalized sparse grids. Exploiting this flexibility, we introduce a greedy heuristic for adaptive refinement of the pseudospectral approximation. We numerically demonstrate convergence of the algorithm on the Genz test functions, and illustrate the accuracy and efficiency of the adaptive approach on a realistic chemical kinetics problem.

A Non-static Data Layout Enhancing Parallelism and Vectorization in Sparse Grid Algorithms

Abstract: The name sparse grids denotes a highly space-efficient, grid-based numerical technique to approximate high-dimensional functions. Although employed in a broad spectrum of applications from different fields, there have only been few tries to use it in real time visualization (e.g. [1]), due to complex data structures and long algorithm runtime. In this work we present a novel approach inspired by principles of I/0-efficient algorithms. Locally applied coefficient permutations lead to improved cache performance and facilitate the use of vector registers for our sparse grid benchmark problem hierarchization. Based on the compact data structure proposed for regular sparse grids in [2], we developed a new algorithm that outperforms existing implementations on modern multi-core systems by a factor of 37 for a grid size of 127 million points. For larger problems the speedup is even increasing, and with execution times below 1 s, sparse grids are well-suited for visualization applications. Furthermore, we point out how a broad class of sparse grid algorithms can benefit from our approach.

A resourceful splitting technique with applications to deterministic and stochastic multiscale finite element methods

Abstract: In this paper we use a splitting technique to develop new multiscale basis functions for the multiscale finite element method (MsFEM). The multiscale basis functions are iteratively generated using a Green's kernel. The Green's kernel is based on the first differential operator of the splitting. The proposed MsFEM is applied to deterministic elliptic equations and stochastic elliptic equations, and we show that the proposed MsFEM can considerably reduce the dimension of the random parameter space for stochastic problems. By combining the method with sparse grid collocation methods, the need for a prohibitive number of deterministic solves is alleviated. We rigorously analyze the convergence of the proposed method for both the deterministic and stochastic elliptic equations. Computational complexity discussions are also offered to supplement the convergence analysis. A number of numerical results are presented to confirm the theoretical findings.

An Adaptive Sparse Grid Approach for Time Series Prediction

Abstract: A real valued, deterministic and stationary time series can be embedded in a—sometimes high-dimensional—real vector space This leads to a one-to-one relationship between the embedded, time dependent vectors in \({\mathbb{R}}^{d}\) and the states of the underlying, unknown dynamical system that determines the time series The embedded data points are located on an m-dimensional manifold (or even fractal) called attractor of the time series Takens’ theorem then states that an upper bound for the embedding dimension d can be given by d ≤ 2m + 1The task of predicting future values thus becomes, together with an estimate on the manifold dimension m, a scattered data regression problem in d dimensions In contrast to most of the common regression algorithms like support vector machines (SVMs) or neural networks, which follow a data-based approach, we employ in this paper a sparse grid-based discretization technique This allows us to efficiently handle huge amounts of training data in moderate dimensions Extensions of the basic method lead to space- and dimension-adaptive sparse grid algorithms They become useful if the attractor is only located in a small part of the embedding space or if its dimension was chosen too largeWe discuss the basic features of our sparse grid prediction method and give the results of numerical experiments for time series with both, synthetic data and real life data

Intraday Foreign Exchange Rate Forecasting Using Sparse Grids

Abstract: We present a machine learning approach using the sparse grid combination technique for the forecasting of intraday foreign exchange (fx) rates The aim is to learn the impact of trading rules used by technical analysts just from the empirical behaviour of the market To this end, the problem of analyzing a time series of transaction tick data is transformed by delay embedding into a D-dimensional regression problem using derived measurements from several different exchange rates Then, a grid-based approach is used to discretize the resulting high-dimensional feature space To cope with the curse of dimensionality we employ sparse grids in the form of the combination technique Here, the problem is discretized and solved for a collection of conventional grids The sparse grid solution is then obtained by linear combination of the solutions on these grids We give the results of this approach to fx forecasting using real historical exchange data of the Euro, the US dollar, the Japanese Yen, the Swiss Franc and the British Pound from 2001 to 2005

A Numerical Comparison Between Quasi-Monte Carlo and Sparse Grid Stochastic Collocation Methods

Abstract: Quasi-Monte Carlo methods and stochastic collocation methods based on sparse grids have become popular with solving stochastic partial differential equations. These methods use deterministic points for multi-dimensional integration or interpolation without suffering from the curse of dimensionality. It is not evident which method is best, specially on random models of physical phenomena. We numerically study the error of quasi-Monte Carlo and sparse grid methods in the context of ground-water flow in heterogeneous media. In particular, we consider the dependence of the variance error on the stochastic dimension and the number of samples/collocation points for steady flow problems in which the hydraulic conductivity is a lognormal process. The suitability of each technique is identified in terms of computational cost and error tolerance.

Clustering based on density estimation with sparse grids

Abstract: We present a density-based clustering method The clusters are determined by splitting a similarity graph of the data into connected components The splitting is accomplished by removing vertices of the graph at which an estimated density function of the data evaluates to values below a threshold The density function is approximated on a sparse grid in order to make the method feasible in higher-dimensional settings and scalable in the number of data points With benchmark examples we show that our method is competitive with other modern clustering methods Furthermore, we consider a real-world example where we cluster nodes of a finite element model of a Chevrolet pick-up truck with respect to the displacements of the nodes during a frontal crash

Computational Method for Global Sensitivity Analysis of Reactor Neutronic Parameters

Abstract: The variance-based global sensitivity analysis technique is robust, has a wide range of applicability, and provides accurate sensitivity information for most models. However, it requires input variables to be statistically independent. A modification to this technique that allows one to deal with input variables that are blockwise correlated and normally distributed is presented. The focus of this study is the application of the modified global sensitivity analysis technique to calculations of reactor parameters that are dependent on groupwise neutron cross-sections. The main effort in this work is in establishing a method for a practical numerical calculation of the global sensitivity indices. The implementation of the method involves the calculation of multidimensional integrals, which can be prohibitively expensive to compute. Numerical techniques specifically suited to the evaluation of multidimensional integrals, namely, Monte Carlo and sparse grids methods, are used, and their efficiency is compared. The method is illustrated and tested on a two-group cross-section dependent problem. In all the cases considered, the results obtained with sparse grids achieved much better accuracy while using a significantly smaller number of samples. This aspect is addressed in a ministudy, and a preliminary explanation of the results obtained is given.

A Resourceful Splitting Technique with Applications to Deterministic and Stochastic Multiscale Finite Element Methods

Abstract: In this paper we use a splitting technique to develop new multiscale basis functions for the multiscale finite element method (MsFEM). The multiscale basis functions are iteratively generated using a Green's kernel. The Green's kernel is based on the first differential operator of the splitting. The proposed MsFEM is applied to deterministic elliptic equations and stochastic elliptic equations, and we show that the proposed MsFEM can considerably reduce the dimension of the random parameter space for stochastic problems. By combining the method with sparse grid collocation methods, the need for a prohibitive number of deterministic solves is alleviated. We rigorously analyze the convergence of the proposed method for both the deterministic and stochastic elliptic equations. Computational complexity discussions are also offered to supplement the convergence analysis. A number of numerical results are presented to confirm the theoretical findings.

An adaptive wavelet stochastic collocation method for irregular solutions of stochastic partial differential equations

Abstract: Accurate predictive simulations of complex real world applications require numerical approximations to first, oppose the curse of dimensionality and second, converge quickly in the presence of steep gradients, sharp transitions, bifurcations or finite discontinuities in high-dimensional parameter spaces. In this paper we present a novel multi-dimensional multi-resolution adaptive (MdMrA) sparse grid stochastic collocation method, that utilizes hierarchical multiscale piecewise Riesz basis functions constructed from interpolating wavelets. The basis for our non-intrusive method forms a stable multiscale splitting and thus, optimal adaptation is achieved. Error estimates and numerical examples will used to compare the efficiency of the method with several other techniques.