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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TL;DR: Numerical results demonstrate that the proposed method can efficiently estimate the posterior statistics of input parameters and provide accurate results for history matching and prediction of the observed data with a moderate number of parameters.
Abstract:
Bayesian inference provides a convenient framework for history matching and prediction. In this framework, prior knowledge, system nonlinearity, and measurement errors can be directly incorporated into the posterior distribution of the parameters. The Markov-chain Monte Carlo (MCMC) method is a powerful tool to generate samples from the posterior distribution. However, the MCMC method usually requires a large number of forward simulations. Hence, it can be a computationally intensive task, particularly when dealing with large-scale flow and transport models.
To address this issue, we construct a surrogate system for the model outputs in the form of polynomials using the stochastic collocation method (SCM). In addition, we use interpolation with the nested sparse grids and adaptively take into account the different importance of parameters for high-dimensional problems. Furthermore, we introduce an additional transform process to improve the accuracy of the surrogate model in case of strong nonlinearities, such as a discontinuous or unsmooth relation between the input parameters and the output responses. Once the surrogate system is built, we can evaluate the likelihood with little computational cost. Numerical results demonstrate that the proposed method can efficiently estimate the posterior statistics of input parameters and provide accurate results for history matching and prediction of the observed data with a moderate number of parameters.
17 citations
21 Dec 2010
TL;DR: This chapter suggests an algorithm that is based on GHK but uses an adaptive version of sparse-grids integration (SGI) instead of simulation, which generalizes Gaussian quadrature in a way such that the computational costs do not grow exponentially with the number of dimensions.
Abstract: In empirical research, panel (and multinomial) probit models are leading examples for the use of maximum simulated likelihood estimators. The Geweke–Hajivassiliou–Keane (GHK) simulator is the most widely used technique for this type of problem. This chapter suggests an algorithm that is based on GHK but uses an adaptive version of sparse-grids integration (SGI) instead of simulation. It is adaptive in the sense that it uses an automated change-of-variables to make the integration problem numerically better behaved along the lines of efficient importance sampling (EIS) and adaptive univariate quadrature. The resulting integral is approximated using SGI that generalizes Gaussian quadrature in a way such that the computational costs do not grow exponentially with the number of dimensions. Monte Carlo experiments show an impressive performance compared to the original GHK algorithm, especially in difficult cases such as models with high intertemporal correlations.
17 citations
TL;DR: A multilevel algorithm is presented that solves general second order elliptic partial differential equations on adaptive sparse grids using a suitable discretization to provide that the discrete equation system can be solved in an efficient way.
Abstract: A multilevel algorithm is presented that solves general second order elliptic partial differential equations on adaptive sparse grids The multilevel algorithm consists of several V-cycles in \(x\)- and \(y\)-direction A suitable discretization provide that the discrete equation system can be solved in an efficient way Numerical experiments show a convergence rate of order \(O(1)\) for the multilevel algorithm
17 citations
TL;DR: The integrated approach inherits the advantages from both of the SGSCM and the MsFEM, in which each low-dimensional stochastic problem is decoupled each other at small number of Stochastic collocation points and is solved on a coarse spatial mesh.
17 citations
12 May 1998
TL;DR: A new approach to local nonlinear image restoration, based on approximating functions using a regular grid of points in a many-dimensional space, is described, which requires only a single presentation of the training samples and are a superset of order statistic filters.
Abstract: We describe a new approach to local nonlinear image restoration, based on approximating functions using a regular grid of points in a many-dimensional space. Symmetry reductions and compression of the sparse grid make it feasible to work with eight-dimensional grids as large as 14/sup 8/. Unlike polynomials and neural networks whose filtering complexity per pixel is linear in the number of filter coefficients, grid filters have O(1) complexity per pixel. Grid filters require only a single presentation of the training samples, are numerically stable, leave unusual image features unchanged, and are a superset of order statistic filters. Results are presented for blurring and additive noise.
17 citations