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Sparse grid

About: Sparse grid is a research topic. Over the lifetime, 1013 publications have been published within this topic receiving 20664 citations.


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TL;DR: In this paper, the authors presented reduced models for pricing basket options with the Black-Scholes and the Heston model, which achieved speedups between 80 and 160 compared to the high-fidelity sparse grid model for 2-, 3-, and 4-asset options.
Abstract: This work presents reduced models for pricing basket options with the Black-Scholes and the Heston model. Basket options lead to multi-dimensional partial differential equations (PDEs) that quickly become computationally infeasible to discretize on full tensor grids. We therefore rely on sparse grid discretizations of the PDEs, which allow us to cope with the curse of dimensionality to some extent. We then derive reduced models with proper orthogonal decomposition. Our numerical results with the Black-Scholes model show that sufficiently accurate results are achieved while gaining speedups between 80 and 160 compared to the high-fidelity sparse grid model for 2-, 3-, and 4-asset options. For the Heston model, results are presented for a single-asset option that leads to a two-dimensional pricing problem, where we achieve significant speedups with our model reduction approach based on high-fidelity sparse grid models.

2 citations

Posted Content
TL;DR: This work develops a novel methodology that dramatically alleviates the curse of dimensionality, and demonstrates via extensive numerical experiments that the methodology can handle problems with a design space of hundreds of dimensions, improving both prediction accuracy and computational efficiency by orders of magnitude relative to typical alternative methods in practice.
Abstract: Stochastic kriging has been widely employed for simulation metamodeling to predict the response surface of a complex simulation model. However, its use is limited to cases where the design space is low-dimensional, because the number of design points required for stochastic kriging to produce accurate prediction, in general, grows exponentially in the dimension of the design space. The large sample size results in both a prohibitive sample cost for running the simulation model and a severe computational challenge due to the need of inverting large covariance matrices. Based on tensor Markov kernels and sparse grid experimental designs, we develop a novel methodology that dramatically alleviates the curse of dimensionality. We show that the sample complexity of the proposed methodology grows very mildly in the dimension, even under model misspecification. We also develop fast algorithms that compute stochastic kriging in its exact form without any approximation schemes. We demonstrate via extensive numerical experiments that our methodology can handle problems with a design space of more than 10,000 dimensions, improving both prediction accuracy and computational efficiency by orders of magnitude relative to typical alternative methods in practice.

2 citations

ReportDOI
20 Feb 2010
TL;DR: A sophisticated though efficient and accurate multiscale stochastic framework for uncertainty quantification is developed and a unique data-driven strategy to encode the limited information on initial texture and grain distribution in deformation processes and represent it in a finite-dimensional framework is developed.
Abstract: : The effect of diverse sources of uncertainties and the intrinsically multi-scale nature of physical systems poses a considerable challenge in their analysis. Such phenomena are particularly critical in material systems where the microstructural variability and randomness at different scales have a significant impact on the macroscopic behavior of the system. Toward this goal, during the period of this grant, we have developed a sophisticated though efficient and accurate multiscale stochastic framework for uncertainty quantification. A methodology is first developed to incorporate topological uncertainties in microstructures using a non-linear data-driven model reduction technique. This framework seamlessly allows for accessing the effects of microstructural variability on the reliability of macro-scale systems and provides an accurate stochastic input model into our stochastic system. Next, to solve the resulted stochastic partial different equations (SPDEs), an adaptive sparse grid collocation technique has been developed. In this framework, we construct the stochastic collocation points based on the function being represented, thus avoiding computational overhead. We further extended this framework to include the High Dimensional Model Representation (HDMR) technique in the stochastic space to represent the model output as a finite hierarchical correlated function expansion in terms of the stochastic inputs starting from lower-order to higher-order component functions. In this way, we can address the stochastic high dimensional problem for the first time in this area. We applied this framework for the design of general materials processes under uncertainty including the robust design of deformation processes of polycrystalline materials. We developed a unique data-driven strategy to encode the limited information on initial texture and grain distribution in deformation processes and represent it in a finite-dimensional framework.

2 citations

Posted Content
TL;DR: An elementary approach to the analysis and programming of sparse grid finite element methods, which can compute accurate solutions to partial differential equations, but using far fewer degrees of freedom than their classical counterparts.
Abstract: Our goal is to present an elementary approach to the analysis and programming of sparse grid finite element methods. This family of schemes can compute accurate solutions to partial differential equations, but using far fewer degrees of freedom than their classical counterparts. After a brief discussion of the classical Galerkin finite element method with bilinear elements, we give a short analysis of what is probably the simplest sparse grid method: the two-scale technique of Lin et al. (2001). We then demonstrate how to extend this to a multiscale sparse grid method which, up to choice of basis, is equivalent to the hierarchical approach, as described by, e.g., Bungartz and Griebel (2004). However, by presenting it as an extension of the two-scale method, we can give an elementary treatment of its analysis and implementation. For each method considered, we provide MATLAB code, and a comparison of accuracy and computational costs.

2 citations

Journal ArticleDOI
TL;DR: Lu et al. as mentioned in this paper extended the approach to higher order WENO simulations specifically the fifth-order WenO scheme, which achieved comparable third order accuracy in smooth regions of the solutions and nonlinear stability as that for computations on regular single grids.
Abstract: The weighted essentially non-oscillatory (WENO) schemes, especially the fifth order WENO schemes, are a popular class of high order accurate numerical methods for solving hyperbolic partial differential equations (PDEs). However when the spatial dimensions are high, the number of spatial grid points increases significantly. It leads to large amount of operations and computational costs in the numerical simulations by using nonlinear high order accuracy WENO schemes such as a fifth order WENO scheme. How to achieve fast simulations by high order WENO methods for high spatial dimension hyperbolic PDEs is a challenging and important question. In the literature, sparse-grid technique has been developed as a very efficient approximation tool for high dimensional problems. In a recent work [Lu, Chen and Zhang, Pure and Applied Mathematics Quarterly, 14 (2018) 57-86], a third order finite difference WENO method with sparse-grid combination technique was designed to solve multidimensional hyperbolic equations including both linear advection equations and nonlinear Burgers’ equations. Numerical experiments showed that WENO computations on sparse grids achieved comparable third order accuracy in smooth regions of the solutions and nonlinear stability as that for computations on regular single grids. In application problems, higher than third order WENO schemes are often preferred in order to efficiently resolve the complex solution structures. In this paper, we extend the approach to higher order WENO simulations specifically the fifth order WENO scheme. A fifth order WENO interpolation is applied in the prolongation part of the sparse-grid combination technique to deal with discontinuous solutions. Benchmark problems are first solved to show that significant CPU times are saved while both fifth order accuracy and stability of the WENO scheme are preserved for simulations on sparse grids. The fifth order sparse grid WENO method is then applied to kinetic problems modeled by high dimensional Vlasov based PDEs to further demonstrate large savings of computational costs by comparing with simulations on regular single grids. Several open problems are discussed at last.

2 citations

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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202314
202242
202157
202040
201960
201872