Sparse grid

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

A dynamically adaptive sparse grids method for quasi-optimal interpolation of multidimensional functions

Abstract: In this work we develop a dynamically adaptive sparse grids (SG) method for quasi-optimal interpolation of multidimensional analytic functions defined over a product of one dimensional bounded domains. The goal of such approach is to construct an interpolant in space that corresponds to the "best M -terms" based on sharp a priori estimate of polynomial coefficients. In the past, SG methods have been successful in achieving this, with a traditional construction that relies on the solution to a Knapsack problem: only the most profitable hierarchical surpluses are added to the SG. However, this approach requires additional sharp estimates related to the size of the analytic region and the norm of the interpolation operator, i.e., the Lebesgue constant. Instead, we present an iterative SG procedure that adaptively refines an estimate of the region and accounts for the effects of the Lebesgue constant. Our approach does not require any a priori knowledge of the analyticity or operator norm, is easily generalized to both affine and non-affine analytic functions, and can be applied to sparse grids built from one dimensional rules with arbitrary growth of the number of nodes. In several numerical examples, we utilize our dynamically adaptive SG to interpolate quantities of interest related to the solutions of parametrized elliptic and hyperbolic PDEs, and compare the performance of our quasi-optimal interpolant to several alternative SG schemes.

Using Adaptive Sparse Grids to Solve High‐Dimensional Dynamic Models

Abstract: We present a exible and scalable method for computing global solutions of high-dimensional stochastic dynamic models. Within a time iteration or value function iteration setup, we interpolate functions using an adaptive sparse grid algorithm. With increasing dimensions, sparse grids grow much more slowly than standard tensor product grids. Moreover, adaptivity adds a second layer of sparsity, as grid points are added only where they are most needed, for instance in regions with steep gradients or at non-differentiabilities. To further speed up the solution process, our implementation is fully hybrid parallel, combining distributed and shared memory parallelization paradigms, and thus permits an efficient use of high-performance computing architectures. To demonstrate the broad applicability of our method, we solve two very different types of dynamic models: first, high-dimensional international real business cycle models with capital adjustment costs and irreversible investment; second, multiproduct menu-cost models with temporary sales and economies of scope in price setting.

On additive Schwarz preconditioners for sparse grid discretizations

Abstract: Based on the framework of subspace splitting and the additive Schwarz scheme, we give bounds for the condition number of multilevel preconditioners for sparse grid discretizations of elliptic model problems For a BXP-like preconditioner we derive an estimate of the optimal orderO(1) and for a HB-like variant we obtain an estimate of the orderO(k 2 ·2 k/2 ), wherek denotes the number of levels employed Furthermore, we confirm these results by numerically computed condition numbers

Multi-asset option pricing using a parallel Fourier-based technique

Abstract: In this paper we present and evaluate a Fourier-based sparse grid method for pricing multi-asset options. This involves computing multidimensional integrals efficiently and we do it by the Fast Fourier Transform. We also propose and evaluate ways to deal with the curse of dimensionality by means of parallel partitioning of the Fourier transform and by incorporating a parallel sparse grids method. Finally, we test the presented method by solving pricing equations for options dependent on up to seven underlying assets.