Sparse grid

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

Parallelizing a Black-Scholes solver based on finite elements and sparse grids

Abstract: We present the parallelization of a sparse grid finite element discretization of the Black-Scholes equation, which is commonly used for option pricing. Sparse grids allow to handle higher dimensional options than classical approaches on full grids and can be extended to a fully adaptive discretization method. We introduce the algorithmical structure of efficient algorithms operating on sparse grids and demonstrate how they can be used to derive an efficient parallelization with OpenMP of the Black-Scholes solver. We show results on different commodity hardware systems based on multi-core architectures with up to 24 cores and discuss the parallel performance using Intel and Advanced Micro Devices AMD CPUs. Copyright © 2012 John Wiley & Sons, Ltd.

Turbulence Simulation on Sparse Grids Using the Combination Method

Abstract: In this paper, we study the parallel numerical solution of the Navier-Stokes equations with the sparse grid combination method. This algorithmic concept is based on the independent solution of many problems with reduced size and their linear combination. We describe the algorithm for three-dimensional problems and we report on its application to turbulence simulation. Furthermore, statistical results on a pipe ow for Reynolds number Re cl = 6950 are presented and compared to results obtained from other numerical simulations and physical experiments. 1. Summary Sparse grid techniques are very promising for the solution of linear PDEs. There, for three-dimensional problems, only O(h ?1 m (log(h ?1 m)) 2) grid points are needed instead of O(h ?3 m) grid points like in the conventional full grid case. Here, h m = 2 ?m denotes the mesh size in the unit cube. However, the accuracy of the sparse grid solution is of the order O(h 2 m (log(h ?1 m)) 2) (with respect to the L 2 ? and L 1 ?norm) provided that the solution is suuciently smooth (i.e. j 6 u(x;y;z) x 2 y 2 z 2 jj 1). This is only slightly worse than the order O(h 2 m) obtained for the usual full grid solution. For further details on sparse grids, see 4], 9]. For the solution of problems arising from the sparse grid discretization approach, two diierent methods have been developed in the last years. First, multilevel-type solvers for the system that results from hierarchical basis like nite element//nite volume or nite diierence discretization (FE, FV, FD) and second, the so-called combination method. There, the solution is obtained on a sparse grid by a certain linear combination of discrete solutions on diierent meshes. The multilevel-type solvers need hierarchical data structures. Thus, specially designed solvers are necessary (see 3]). In the case of the combination method, however, only simple data structures are needed. This method merely handles three-dimensional arrays. Furthermore, it is possible to use any PDE solver as a "black-box" solution method within the combination approach to get a solution on the diierent meshes involved.

Efficient Optimization of Stimuli for Model-Based Design of Experiments to Resolve Dynamical Uncertainty

Abstract: This model-based design of experiments (MBDOE) method determines the input magnitudes of an experimental stimuli to apply and the associated measurements that should be taken to optimally constrain the uncertain dynamics of a biological system under study. The ideal global solution for this experiment design problem is generally computationally intractable because of parametric uncertainties in the mathematical model of the biological system. Others have addressed this issue by limiting the solution to a local estimate of the model parameters. Here we present an approach that is independent of the local parameter constraint. This approach is made computationally efficient and tractable by the use of: (1) sparse grid interpolation that approximates the biological system dynamics, (2) representative parameters that uniformly represent the data-consistent dynamical space, and (3) probability weights of the represented experimentally distinguishable dynamics. Our approach identifies data-consistent representative parameters using sparse grid interpolants, constructs the optimal input sequence from a greedy search, and defines the associated optimal measurements using a scenario tree. We explore the optimality of this MBDOE algorithm using a 3-dimensional Hes1 model and a 19-dimensional T-cell receptor model. The 19-dimensional T-cell model also demonstrates the MBDOE algorithm’s scalability to higher dimensions. In both cases, the dynamical uncertainty region that bounds the trajectories of the target system states were reduced by as much as 86% and 99% respectively after completing the designed experiments in silico. Our results suggest that for resolving dynamical uncertainty, the ability to design an input sequence paired with its associated measurements is particularly important when limited by the number of measurements.

Parallelized Dimensional Decomposition for Large-Scale Dynamic Stochastic Economic Models

Abstract: We introduce and deploy a generic, highly scalable computational method to solve high-dimensional dynamic stochastic economic models on high-performance computing platforms. Within an MPI---TBB parallel, nonlinear time iteration framework, we approximate economic policy functions using an adaptive sparse grid algorithm with d-linear basis functions that is combined with a dimensional decomposition scheme. Numerical experiments on "Piz Daint" (Cray XC30) at the Swiss National Supercomputing Centre show that our framework scales nicely to at least 1,000 compute nodes. As an economic application, we compute global solutions to international real business cycle models up to 200 continuous dimensions with significant speedup values over state-of-the-art techniques.