Showing papers on "Sparse grid published in 2006"

On coordinate transformation and grid stretching for sparse grid pricing of basket options

Abstract: We evaluate two coordinate transformation techniques in combination with a coordinate stretching for pricing basket options in a sparse grid setting. The sparse grid technique is a basic technique for solving a high-dimensional partial differential equation. By creating a small hypercube sub-grid in the 'composite' sparse grid we can also determine hedge parameters accurately. We evaluate these techniques for multi-asset examples with up to five underlying assets in the basket.

Smolyak’s Algorithm, Sampling on Sparse Grids and Function Spaces of Dominating Mixed Smoothness

Abstract: We investigate the rate of convergence in ‖ · |Lp‖, 1 ≤ p ≤ ∞, of the ddimensional Smolyak algorithm, associated to a sequence of sampling operators in the framework of periodic Sobolev and Besov spaces with dominating mixed smoothness.

Regression with the optimised combination technique

Abstract: We consider the sparse grid combination technique for regression, which we regard as a problem of function reconstruction in some given function space. We use a regularised least squares approach, discretised by sparse grids and solved using the so-called combination technique, where a certain sequence of conventional grids is employed. The sparse grid solution is then obtained by addition of the partial solutions with combination co-efficients dependent on the involved grids. This approach shows instabilities in certain situations and is not guaranteed to converge with higher discretisation levels. In this article we apply the recently introduced optimised combination technique, which repairs these instabilities. Now the combination coefficients also depend on the function to be reconstructed, resulting in a non-linear approximation method which achieves very competitive results. We show that the computational complexity of the improved method still scales only linear in regard to the number of data.

Space-Time Approximation with Sparse Grids

Abstract: In this article we introduce approximation spaces, especially suited for the approximation of solutions of parabolic problems, which are based on the tensor product construction of a multiscale basis in space and a multiscale basis in time. Proper truncation then leads to so-called space-time sparse grid spaces. For a uniform discretization of the spatial space of dimension d with O(Nd) degrees of freedom, these spaces involve for d > 1 also only O(Nd) degrees of freedom for the discretization of the whole space-time problem. But they provide the same approximation rate as classical space-time finite element spaces which need O(Nd+1) degrees of freedoms. This makes these approximation spaces well suited for conventional parabolic and time-dependent optimization problems. We analyze the approximation properties and the dimension of these sparse grid space-time spaces for general stable multiscale bases. We then restrict ourselves to an interpolatory multiscale basis, i.e., a hierarchical basis. Here, to be able to handle also complicated spatial domains Omega, we construct the hierarchical basis from a given spatial finite element basis as follows: First we determine coarse grid points recursively over the levels by the coarsening step of the algebraic multigrid method. Then, we derive interpolatory prolongation operators between the respective coarse and fine grid points by a least squares approach. This way we obtain an algebraic hierarchical basis for the spatial domain which we then use in our space-time sparse grid approach. We give numerical results on the convergence rate of the interpolation error of these spaces for various space-time problems with two spatial dimensions. Implementational issues, data structures, and questions of adaptivity also are addressed to some extent.

Estimation with Numerical Integration on Sparse Grids

Abstract: For the estimation of many econometric models, integrals without analytical solutions have to be evaluated. Examples include limited dependent variables and nonlinear panel data models. In the case of one-dimensional integrals, Gaussian quadrature is known to work efficiently for a large class of problems. In higher dimensions, similar approaches discussed in the literature are either very specific and hard to implement or suffer from exponentially rising computational costs in the number of dimensions - a problem known as the "curse of dimensionality" of numerical integration. We propose a strategy that shares the advantages of Gaussian quadrature methods, is very general and easily implemented, and does not suffer from the curse of dimensionality. Monte Carlo experiments for the random parameters logit model indicate the superior performance of the proposed method over simulation techniques.

A wavelet based sparse grid method for the electronic Schrödinger equation

Abstract: We present a direct discretization of the electronic Schrodinger equation. It is based on one-dimensional Meyer wavelets from which we build an anisotropic multiresolution analysis for general particle spaces by a tensor product construction. We restrict these spaces to the case of antisymmetric functions. To obtain finite-dimensional subspaces we first discuss semidiscretization with respect to the scale parameter by means of sparse grids which relies on mixed regularity and decay properties of the electronic wave functions. We then propose different techniques for a discretization with respect to the position parameter. Furthermore we present the results of our numerical experiments using this new generalized sparse grid methods for Schrodinger�s equation.

Fuzzy arithmetic based on dimension-adaptive sparse grids: a case study of a large-scale finite element model under uncertain parameters

Abstract: Fuzzy arithmetic provides a powerful tool to introduce uncertainty into mathematical models. With Zadeh's extension principle, one can obtain a fuzzy-valued extension of any real-valued objective function. An efficient and accurate approach to computing expensive multivariate functions of fuzzy numbers is given by fuzzy arithmetic based on sparse grids. In many cases, not all uncertain input parameters carry equal weight, or the objective model exhibits separable structure. These characteristics can be exploited by dimension-adaptive algorithms. As a result, the treatment of even higher-dimensional problems becomes possible. This is demonstrated in this paper by a case study involving two large-scale finite element models in vibration engineering that are subjected to fuzzy-valued input data.

Fast evaluation of trigonometric polynomials from hyperbolic crosses

Abstract: The discrete Fourier transform in d dimensions with equispaced knots in space and frequency domain can be computed by the fast Fourier transform (FFT) in $${\cal O}(N^d \log N)$$ arithmetic operations. In order to circumvent the ‘curse of dimensionality’ in multivariate approximation, interpolations on sparse grids were introduced. In particular, for frequencies chosen from an hyperbolic cross and spatial knots on a sparse grid fast Fourier transforms that need only $${\cal O}(N \log^d N)$$ arithmetic operations were developed. Recently, the FFT was generalised to nonequispaced spatial knots by the so-called NFFT. In this paper, we propose an algorithm for the fast Fourier transform on hyperbolic cross points for nonequispaced spatial knots in two and three dimensions. We call this algorithm sparse NFFT (SNFFT). Our new algorithm is based on the NFFT and an appropriate partitioning of the hyperbolic cross. Numerical examples confirm our theoretical results.

Parallelisation of sparse grids for large scale data analysis

Abstract: Sparse grids are the basis for efficient high dimensional approximation and have recently been applied successfully to predictive modelling. They are spanned by a collection of simpler function spaces represented by regular grids. The sparse grid combination technique prescribes how approximations on a collection of anisotropic grids can be combined to approximate high dimensional functions. In this paper we study the parallelisation of fitting data onto a sparse grid. The computation can be done entirely by fitting partial models on a collection of regular grids. This allows parallelism over the collection of grids. In addition, each of the partial grid fits can be parallelised as well, both in the assembly phase, where parallelism is done over the data, and in the solution stage using traditional parallel solvers for the resulting PDEs. Using a simple timing model we confirm that the most effective methods are obtained when both types of parallelism are used.

Sparse Grids, Adaptivity, and Symmetry

Abstract: Sparse grid methods represent a powerful and efficient technique for the representation and approximation of functions and particularly the solutions of partial differential equations in moderately high space dimensions. To extend the approach to truly high-dimensional problems as they arise in quantum chemistry, an additional property has to be brought into play, the symmetry or antisymmetry of the functions sought there.In the present article, an adaptive sparse grid refinement scheme is developed that takes full advantage of such symmetry properties and for which the amount of work and storage remains strictly proportional to the number of degrees of freedom. To overcome the problems with the approximation of the inherently complex antisymmetric functions, augmented sparse grid spaces are proposed.

Wavelet-Based Multiscale Methods for Electronic Structure Calculations

Abstract: In order to treat multiple energy- and length-scales in electronic structure calculations for extended systems, we have studied a wavelet based multiresolution analysis of electron correlations. Wavelets provide hierarchical basis sets that can be locally adapted to the length- and energy-scales of physical phenomena. The inherently high dimensional many-body problem can be kept tractable by using the sparse grid method for the construction of multivariate wavelets. These so called “hyperbolic” wavelets provide sparse representations for correlated wavefunctions and can be combined with diagrammatic techniques from quantum many-particle theory into a diagrammatic multiresolution analysis. Using sparsity features originating from the hierarchical structure and vanishing moments property of wavelet bases, this leads to many-particle methods with almost linear computational complexity for the treatment of electron correlations.

A Spectral Stochastic Collocation Method for Capacitance Extraction of Interconnects with Process Variations

Abstract: In this paper, a Spectral Stochastic Collocation Method (SSCM), is proposed for the capacitance extraction of interconnects with either on-chip process variations or off-chip rough surfaces. The proposed method is based on the stochastic spectral method combined with Sparse Grid technique, and has several advantages over the existing methods. Compared with the perturbation method, the stochastic spectral method based on Homogeneous Chaos expansion has exponential convergence rate, which makes it very promising for parasitic extraction with process variations. Furthermore, the Sparse Grid technique significantly reduces the amount of sampling points compared with Monte Carlo method, and greatly saves the computation time for capacitance extraction. Numerical experiments have demonstrated that SSCM can achieve higher accuracy while having the same efficiency compared with the existing methods.

Sparse grid-oriented three-dimensional foot type fast acquiring method based on standard foot deformation

Abstract: A method for obtaining 3D foot shape quickly based on standard foot variant by utilizing sparse grid includes setting up, standard foot shape library, setting up 3D coordinate for structuring foot length vector, rotating sparse grid foot model to let said vector be parallel to Y axle, selecting foot length and width information from said library, adjusting posture and position of sparse grid foot model to let it distribute uniformly around standard foot model, finding out corresponding points of two models, revising control top point of sparse grid foot model and using revised point to rebuild sparse grid foot model for obtaining 3D foot shape data

A multilevel adaptive solver based on second-generation wavelet thresholding techniques

Abstract: In this manuscript, we introduce a second-generation wavelet thresholding technique used to construct a numerically stable non-dyadic sparse grid representation. The resulting second-generation wavelet projectors, when coupled to a multigrid solver, provide an elegant method for integrating the numerical solution. The combined method is then utilized in the solution of a singular perturbation problem that arises when modelling an n-MOS gate exhibiting quantum tunnelling. The resulting solution is compared with the full Schrodinger–Poisson system, and the two solutions are shown to be in good agreement. Copyright © 2006 John Wiley & Sons, Ltd.

On applicability of the sparse grid method in the worst case setting

Abstract: Let $T_k:W^{r_k}_p\to W^{s_k}_q$ be bounded linear operators. We provide several sufficient conditions for the validity of the inequality $\|\otimes_kT_k\|\le\prod_k\|T_k\|$ . These results can be applied to error and cost estimates for the sparse grid method.

Constructing dimension-adaptive sparse grid interpolants using parallel function evaluations

Abstract: Dimension-adaptive sparse grid interpolation is a powerful tool to obtain surrogate functions of smooth, medium to high-dimensional objective models. In case of expensive models, the efficiency of the sparse grid algorithm is governed by the time required for the function evaluations. In this paper, we first briefly analyze the inherent parallelism of the standard dimension-adaptive algorithm. Then, we present an enhanced version of the standard algorithm that permits, in each step of the algorithm, a specified number (equal to the number of desired processes) of function evaluations to be executed in parallel, thereby increasing the parallel efficiency.

Realistic smoke simulation using a frustum aligned grid

Abstract: Realistic simulation of smoke is used in the special effects industry to produce smoke in both feature films and video games. Traditional simulations utilize uniformly spaced rectangular computational grids to perform the smoke simulation. Various changes had been proposed to improve different aspects of the simulation, including level of details, memory usage and simulation speed. In this thesis, I propose a novel computational grid that improves upon the level of details as well as memory usage. I propose a frustum aligned grid that takes advantage of the viewing camera because details are most important in the area close to the camera. A frustum aligned grid reduces the amount of grid points necessary to cover the whole domain by placing a high concentration of grid points near the camera while having sparse grid points away from the camera. By using a larger number of grid lines in the direction parallel to the camera and fewer grid lines in the direction perpendicular to the camera, high level of details using a smaller amount of memory can be achieved. The grid is logically rectangular and a perspective transformation can map the grid into a spatially rectangular one. These properties enable the use of existing simulation tools with some modifications, thus maintaining the level of speed. Experimental results and comparison with a standard uniform grid demonstrate the practicality and effectiveness of the proposed method.

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.

Numerical analysis for anisotropic multivariate Levy processes

Abstract: Arbitrage-free prices u of financial derivatives on d 2 assets are considered where the underlyings are modeled by Markov processes of Levy type. They satisfy a high dimensional parabolic partial integrodierentia l equation (PIDE) @tu+Au = 0 on (0,1) d . Numerical pricing of these contracts by sparse Finite El- ement Methods requires the ecient discretization of the infinitesimal generator A of X. For a wide class of operators we present a new sparse grid based wavelet com- pression scheme for anisotropic tensor product wavelets that (asymptotically) reduces the matrix complexity from originally O(h 2d ) to O(h 1 ). Numerical results from joint work with C. Winter are presented for d = 2.

Modelling Gene Regulatory Networks Using Galerkin Techniques Based on State Space Aggregation and Sparse Grids.

Abstract: An important driver of the dynamics of gene regulatory networks is noise generated by transcription and translation processes involving genes and their products. As relatively small numbers of copies of each substrate are involved, such systems are best described by stochastic models. With these models, the stochastic master equations, one can follow the time development of the probability distributions for the states defined by the vectors of copy numbers of each substance. Challenges are posed by the large discrete state spaces, and are mainly due to high dimensionality.