Topic
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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01 Jan 2012
TL;DR: The correlation of typical diffusion processes arising e.g. in financial modelling can be exploited—by means of asymptotic analysis of principal components—to make Feynman-Kac PDEs of high dimension computationally tractable.
Abstract: In this short article, we describe how the correlation of typical diffusion processes arising e.g. in financial modelling can be exploited—by means of asymptotic analysis of principal components—to make Feynman-Kac PDEs of high dimension computationally tractable. We explore the links to dimension adaptive sparse grids (Gerstner and Griebel, Computing 71:65–87, 2003), anchored ANOVA decompositions and dimension-wise integration (Griebel and Holtz, J Complexity 26:455–489, 2010), and the embedding in infinite-dimensional weighted spaces (Sloan and Woźniakowski, J Complexity 14:1–33, 1998). The approach is shown to give sufficient accuracy for the valuation of index options in practice. These numerical findings are backed up by a complexity analysis that explains the independence of the computational effort of the dimension in relevant parameter regimes.
6 citations
TL;DR: An adaptive approach of the stochastic collocation method to calculate the statistics of the solution is discussed, based on an adaptive approximation of the inverse diffusion parameter, which is an upper bound of the error on the solution.
6 citations
TL;DR: Two new parallel algorithms for the SGCT that supports the full distributed memory parallelization over the dimensions of the component grids, as well as across the grids as well, and the direct algorithm has generally better performance.
6 citations
TL;DR: This work investigates from the empirical point of view an alternative procedure, which involves an interpolation of the complex field derived from the sparse phase measurements, to bypass the PU step and obtain a regular-grid complex field from which a wrapped phase field can be extracted.
Abstract: Sparse phase measurements often need to be interpolated on regular grids, to extend the information to unsampled locations. Typical cases involve the removal of atmospheric phase screen information from Interferometric Synthetic Aperture Radar (InSAR) stacks, or the retrieval of displacement information over extended areas in Persistent Scatterers Interferometry (PSI) applications, when sufficient point densities are available. This operation is usually done after a phase unwrapping (PU) of the sparse measurements to remove the sharp phase discontinuities due to the wrap operation. PU is a difficult and errorprone operation, especially for sparse data. In this work, we investigate from the empirical point of view an alternative procedure, which involves an interpolation of the complex field derived from the sparse phase measurements. Unlike traditional approaches, our method allows to bypass the PU step and obtain a regular-grid complex field from which a wrapped phase field can be extracted. Under general conditions, this can be shown to be a good approximation of the original phase without noise. Moreover, the interpolated, wrapped phase field can be fed to state-of-the-art, regular-grid PU algorithms, to obtain an improved absolute phase field, compared to the canonical method consisting of first unwrapping the sparse-grid data. We evaluate the performance of the method in simulation, comparing it to the classical methodology described above, as well as to an alternative procedure, recently proposed, to reduce a sparse PU problem to a regular-grid one, through a nearest-neighbor interpolation step. Results confirm the increased robustness of the proposed method with respect to the effects of noise and undersampling.
6 citations
Posted Content•
TL;DR: This work considers the problem of approximating the statistics of a given Quantity of Interest that depends on the solution of a linear elliptic PDE defined over a random domain parameterized by $N$ random variables and significantly reduces the dimensionality of the stochastic problem.
Abstract: In this work we consider the problem of approximating the statistics of a given Quantity of Interest (QoI) that depends on the solution of a linear elliptic PDE defined over a random domain parameterized by $N$ random variables. The random domain is split into large and small variations contributions. The large variations are approximated by applying a sparse grid stochastic collocation method. The small variations are approximated with a stochastic collocation-perturbation method. Convergence rates for the variance of the QoI are derived and compared to those obtained in numerical experiments. Our approach significantly reduces the dimensionality of the stochastic problem. The computational cost of this method increases at most quadratically with respect to the number of dimensions of the small variations. Moreover, for the case that the small and large variations are independent the cost increases linearly.
6 citations