Sparse grid

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

Singular value decomposition versus sparse grids: refined complexity estimates

Abstract: We compare the cost complexities of two approximation schemes for functions which live on the product domain $\Omega_1\times\Omega_2$ of sufficiently smooth domains $\Omega_1\subset\mathbb{R}^{n_1}$ and $\Omega_2\subset\mathbb{R}^{n_2}$, namely the singular value / Karhunen-Loeve decomposition and the sparse grid representation. We assume that appropriate finite element methods with associated orders $r_1$ and $r_2$ of accuracy are given on the domains $\Omega_1$ and $\Omega_2$,respectively. This setting reflects practical needs, since often black-box solvers are used in numerical simulation which restrict the freedom in the choice of the underlying discretization. We compare the cost complexities of the associated singular value decomposition and the associated sparse grid approximation. It turns out that, in this situation, the approximation by the sparse grid is always equal or superior to the approximation by the singular value decomposition. The results in this article improve and generalize those from Griebel & Harbrecht (2014). We now especially consider the approximation of functions from generalized isotropic and anisotropic Sobolev spaces.

Comparison of approaches for random PDE optimization problems based on different matching functionals

Abstract: In this article, we consider an optimal control problem for an elliptic partial differential equation with random inputs. To determine an applicable deterministic control f(x), we consider the four cases which we compare for efficiency and feasibility. We prove the existence of optimal states, adjoint states and optimality conditions for each cases. We also derive the optimality systems for the four cases. The optimality system is then discretized by a standard finite element method and sparse grid collocation method for physical space and probability space, respectively. The numerical experiments are performed for their efficiency and feasibility.

Comparison of sparse-grid geometric and random sampling methods in nonlinear inverse solution uncertainty estimation

Abstract: A new uncertainty estimation method, which we recently introduced in the literature, allows for the comprehensive search of model posterior space while maintaining a high degree of computational efficiency The method starts with an optimal solution to an inverse problem, performs a parameter reduction step and then searches the resulting feasible model space using prior parameter bounds and sparse-grid polynomial interpolation methods After misfit rejection, the resulting model ensemble represents the equivalent model space and can be used to estimate inverse solution uncertainty While parameter reduction introduces a posterior bias, it also allows for scaling this method to higher dimensional problems The use of Smolyak sparse-grid interpolation also dramatically increases sampling efficiency for large stochastic dimensions Unlike Bayesian inference, which treats the posterior sampling problem as a random process, this geometric sampling method exploits the structure and smoothness in posterior distributions by solving a polynomial interpolation problem and then resampling from the resulting interpolant The two questions we address in this paper are 1) whether our results are generally compatible with established Bayesian inference methods and 2) how does our method compare in terms of posterior sampling efficiency We accomplish this by comparing our method for two electromagnetic problems from the literature with two commonly used Bayesian sampling schemes: Gibbs’ and Metropolis-Hastings While both the sparse-grid and Bayesian samplers produce compatible results, in both examples, the sparse-grid approach has a much higher sampling efficiency, requiring an order of magnitude fewer samples, suggesting that sparse-grid methods can significantly improve the tractability of inference solutions for problems in high dimensions or with more costly forward physics

The analysis of a sparse grid stochastic collocation method for partial differential equations with high-dimensional random input data.

Abstract: This work describes the convergence analysis of a Smolyak-type sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations with random coefficients and forcing terms (input data of the model) To compute solution statistics, the sparse grid stochastic collocation method uses approximate solutions, produced here by finite elements, corresponding to a deterministic set of points in the random input space This naturally requires solving uncoupled deterministic problems and, as such, the derived strong error estimates for the fully discrete solution are used to compare the computational efficiency of the proposed method with the Monte Carlo method Numerical examples illustrate the theoretical results and are used to compare this approach with several others, including the standard Monte Carlo

Application of sparse-grid technique to chance constrained optimal power flow

Abstract: The method of chance constrained optimisation (CCOPT) provides a suitable way to address operations planning of power transmission networks under uncertainty. However, since many uncertain variables as well as many output constraints have to be considered in optimal power flow (OPF), the computational demand to solve the CCOPT problem will be prohibitive. In this study, the authors employ the sparse-grid technique for computing the values and gradients of the objective function and probability constraints, with which the computation efficiency can be significantly enhanced. The authors consider optimal operations planning under uncertain demands at different nodes of networks. Based on the non-linear OPF model, the results of solving the CCOPT problem for the IEEE 57-bus and 118-bus systems demonstrate the efficiency of the proposed approach.