Sparse grid

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

Fitting Multidimensional Data Using Gradient Penalties and Combination Techniques

Abstract: Sparse grids, combined with gradient penalties provide an attractive tool for regularised least squares fitting. It has earlier been found that the combination technique, which allows the approximation of the sparse grid fit with a linear combination of fits on partial grids, is here not as effective as it is in the case of elliptic partial differential equations. We argue that this is due to the irregular and random data distribution, as well as the proportion of the number of data to the grid resolution. These effects are investigated both in theory and experiments. The application of modified “optimal” combination coefficients provides an advantage over the ones used originally for the numerical solution of PDEs, who in this case simply amplify the sampling noise. As part of this investigation we also show how overfitting arises when the mesh size goes to zero.

Efficient Pseudorecursive Evaluation Schemes for Non-adaptive Sparse Grids

Abstract: In this work we propose novel algorithms for storing and evaluating sparse grid functions, operating on regular (not spatially adaptive), yet potentially dimensionally adaptive grid types. Besides regular sparse grids our approach includes truncated grids, both with and without boundary grid points. Similar to the implicit data structures proposed in Feuersanger (Dunngitterverfahren fur hochdimensionale elliptische partielle Differntialgleichungen. Diploma Thesis, Institut fur Numerische Simulation, Universitat Bonn, 2005) and Murarasu et al. (Proceedings of the 16th ACM Symposium on Principles and Practice of Parallel Programming. Cambridge University Press, New York, 2011, pp. 25–34) we also define a bijective mapping from the multi-dimensional space of grid points to a contiguous index, such that the grid data can be stored in a simple array without overhead. Our approach is especially well-suited to exploit all levels of current commodity hardware, including cache-levels and vector extensions. Furthermore, this kind of data structure is extremely attractive for today’s real-time applications, as it gives direct access to the hierarchical structure of the grids, while outperforming other common sparse grid structures (hash maps, etc.) which do not match with modern compute platforms that well. For dimensionality d ≤ 10 we achieve good speedups on a 12 core Intel Westmere-EP NUMA platform compared to the results presented in Murarasu et al. (Proceedings of the International Conference on Computational Science—ICCS 2012. Procedia Computer Science, 2012). As we show, this also holds for the results obtained on Nvidia Fermi GPUs, for which we observe speedups over our own CPU implementation of up to 4.5 when dealing with moderate dimensionality. In high-dimensional settings, in the order of tens to hundreds of dimensions, our sparse grid evaluation kernels on the CPU outperform any other known implementation.

Robust Topology Optimization Under Load and Geometry Uncertainties by Using New Sparse Grid Collocation Method

Abstract: In this paper, a robust topology optimization method presents that insensitive to the uncertainty in geometry and applied load. Geometric uncertainty can be introduced in the manufacturing variability. Applied load uncertainty is occurring in magnitude and angle of force. These uncertainties can be modeled as a random field. A memory-less transformation of random fields used to random variation modeling. The Adaptive Sparse Grid Collocation (ASGC) method combined with the uncertainty models provides robust designs by utilizing already developed deterministic solvers. The proposed algorithm provides a computationally cheap alternative to previously introduced stochastic optimization methods based on Monte Carlo sampling by using the adaptive sparse grid method. Numerical examples, such as a 2D simply supported beam and cantilever beam as benchmark problems, are used to show the effectiveness and superiority of the ASGC method.

Calculation of default probability (PD) solving Merton Model PDEs on sparse grids

Abstract: Actual developements of the sub-prime crisis of 2008 have put a strong focus on the importance of credit default models. The Merton Model is one of these models, using partial differential equations to calculate the probability of default (PD) for a correlated credit portfolio. The resulting equations are discretized on structured sparse grids through the method of Finite-Differences and numerically solved using the software package SG2. Parallel Computing is used to speed up the calculations.

Importance analysis for models with dependent input variables by sparse grids

Abstract: To get a better understanding on the output uncertainty contributed by an individual variable as well as the correlated variables of models with dependent inputs, a method for decomposing Sobol’s first-order effect indices into uncorrelated variations and correlated variations is investigated. Instead of using Monte Carlo simulation or full tensor product-based numerical integration approaches, a new sparse grid numerical integration method is proposed for estimating Sobol’s main effect indices as well as the two decomposed sensitivity measures. Before conducting the sparse grid numerical integration-based algorithm, an orthogonal transformation is used to transform the dependent input variables and model performance function into independent space as the joint probability density function of the correlated variables cannot be written as the product of univariate density functions. An obvious advantage of the sparse grid numerical integration-based method is that it can decrease the computational cost of ...