Showing papers on "Sparse grid published in 2007"

A sparse grid space-time discretization scheme for parabolic problems

Abstract: In this paper, we consider the discretization in space and time of parabolic differential equations where we use the so-called space-time sparse grid technique It employs the tensor product of a one-dimensional multilevel basis in time and a proper multilevel basis in space This way, the additional order of complexity of a direct space-time discretization can be avoided, provided that the solution fulfills a certain smoothness assumption in space-time, namely that its mixed space-time derivatives are bounded This holds in many applications due to the smoothing properties of the propagator of the parabolic PDE (heat kernel) In the more general case, the space-time sparse grid approach can be employed together with adaptive refinement in space and time and then leads to similar approximation rates as the non-adaptive method for smooth functions We analyze the properties of different space-time sparse grid discretizations for parabolic differential equations from both, the theoretical and practical point of view, discuss their implementational aspects and report on the results of numerical experiments

Sparse grids for the Schrödinger equation

Abstract: We present a sparse grid/hyperbolic cross discretization for many-particle problems. It involves the tensor product of a one-particle multilevel basis. Subsequent truncation of the associated series expansion then results in a sparse grid discretization. Here, depending on the norms involved, different variants of sparse grid techniques for many-particle spaces can be derived that, in the best case, result in complexities and error estimates which are independent of the number of particles. Furthermore we introduce an additional constraint which gives antisymmetric sparse grids which are suited to fermionic systems. We apply the antisymmetric sparse grid discretization to the electronic Schrodinger equation and compare costs, accuracy, convergence rates and scalability with respect to the number of electrons present in the system.

Fourier transform on sparse grids: Code design and the time dependent Schrödinger equation

Abstract: The pseudo-spectral method together with a Strang-splitting are well suited for the discretization of the time-dependent Schrodinger equation with smooth potential. The curse of dimensionality limits this approach to low dimensions, if we stick to full grids. Theoretically, sparse grids allow accurate computations in (moderately) higher dimensions, provided that we supply an efficient Fourier transform. Motivated by this application, the design of the Fourier transform on sparse grids in multiple dimensions is described in detail. The focus of this presentation is on issues of flexible implementation and numerical studies of the convergence.

The Smolyak agorithm, sampling on sparse grids and functions spaces of dominated 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.

A sparse grid based spectral stochastic collocation method for variations-aware capacitance extraction of interconnects under nanometer process technology

Abstract: In this paper, a Spectral Stochastic Collocation Method (SSCM) is proposed for the capacitance extraction of interconnects with stochastic geometric variations for nanometer process technology. The proposed SSCM has several advantages over the existing methods. Firstly, compared with the PFA (Principal Factor Analysis) modeling of geometric variations, the K-L (Karhunen-Loeve) expansion involved in SSCM can be independent of the discretization of conductors, thus significantly reduces the computation cost. Secondly, compared with the perturbation method, the stochastic spectral method based on Homogeneous Chaos expansion has optimal (exponential) convergence rate, which makes SSCM applicable to most geometric variation cases. Furthermore, Sparse Grid combined with a MST (Minimum Spanning Tree) representation is proposed to reduce the number of sampling points and the computation time for capacitance extraction at each sampling point. Numerical experiments have demonstrated that SSCM can achieve higher accuracy and faster convergence rate compared with the perturbation method.

Strang Splitting for the Time-Dependent Schrödinger Equation on Sparse Grids

Abstract: The time-dependent Schrodinger equation is discretized in space by a sparse grid pseudospectral method. The Strang splitting for the resulting evolutionary problem features first or second order convergence in time, depending on the smoothness of the potential and of the initial data. In contrast to the full grid case, where the frequency domain is the working place, the proof of the sufficient conditions for the convergence is done in the space realm.

Analysis of Linear Difference Schemes in the Sparse Grid Combination Technique

Abstract: Sparse grids are tailored to the approximation of smooth high-dimensional functions. On a $d$-dimensional tensor product space, the number of grid points is $N = \mathcal O(h^{-1} |\log h|^{d-1})$, where $h$ is a mesh parameter. The so-called combination technique, based on hierarchical decomposition and extrapolation, requires specific multivariate error expansions of the discretisation error on Cartesian grids to hold. We derive such error expansions for linear difference schemes through an error correction technique of semi-discretisations. We obtain overall error formulae of the type $\epsilon = \mathcal{O} (h^p |\log h|^{d-1})$ and analyse the convergence, with its dependence on dimension and smoothness, by examples of linear elliptic and parabolic problems, with numerical illustrations in up to eight dimensions.

A dimension adaptive sparse grid combination technique for machine learning

Abstract: We introduce a dimension adaptive sparse grid combination technique for the machine learning problems of classification and regression. A function over a $d$-dimensional space, which assumedly describes the relationship between the features and the response variable, is reconstructed using a linear combination of partial functions that possibly depend only on a subset of all features. The partial functions are adaptively chosen during the computational procedure. This approach (approximately) identifies the \textsc{anova} decomposition of the underlying problem. Experiments on synthetic data, where the structure is known, show the advantages of a dimension adaptive combination technique in run time behaviour, approximation errors, and interpretability. References M. Griebel, M. Schneider, and C. Zenger. A combination technique for the solution of sparse grid problems. In P. de Groen and R. Beauwens, editors, Iterative Methods in Linear Algebra , pages 263--281. IMACS, Elsevier, North Holland, 1992. http://wissrech.ins.uni-bonn.de/research/pub/griebel/griesiam.ps.gz . M. Hegland. Adaptive sparse grids. In K. Burrage and Roger B. Sidje, editors, Proc. of 10th Computational Techniques and Applications Conference CTAC-2001 , volume 44 of ANZIAM J. , pages C335--C353, 2003. http://anziamj.austms.org.au/V44/CTAC2001/Hegl . M. Hegland, J. Garcke, and V. Challis. The combination technique and some generalisations. Linear Algebra and its Applications , 420(2--3):249--275, 2007. doi:10.1016/j.laa.2006.07.014 . Ian H. Sloan, Xiaoqun Wang, and Henryk Wozniakowski. Finite-order weights imply tractability of multivariate integration. Journal of Complexity , 20:46--74, 2004. doi:10.1016/j.jco.2003.11.003 . Jerome H. Friedman. Multivariate adaptive regression splines. Ann. Statist. , 19(1):1--141, 1991. http://projecteuclid.org/euclid.aos/1176347963 . J. Garcke. Regression with the optimised combination technique. In W. Cohen and A. Moore, editors, Proceedings of the 23rd ICML , pages 321--328, 2006. doi:10.1007/s006070170007 . J. Garcke, M. Griebel, and M. Thess. Data mining with sparse grids. Computing , 67(3):225--253, 2001. doi:10.1007/s006070170007 . T. Gerstner and M. Griebel. Dimension-Adaptive Tensor-Product Quadrature. Computing , 71(1):65--87, 2003. doi:10.1007/s00607-003-0015-5 .

Stochastic Sparse-grid Collocation Algorithm (SSCA) for Periodic Steady-State Analysis of Nonlinear System with Process Variations

Abstract: In this paper, stochastic collocation algorithm combined with sparse grid technique (SSCA) is proposed to deal with the periodic steady-state analysis for nonlinear systems with process variations. Compared to the existing approaches, SSCA has several considerable merits. Firstly, compared with the moment-matching parameterized model order reduction (PMOR), which equally treats the circuit response on process variables and frequency parameter by Taylor approximation, SSCA employs homogeneous chaos to capture the impact of process variations with exponential convergence rate and adopts Fourier series or wavelet bases to model the steady-state behavior in time domain. Secondly, contrary to stochastic Galerkin algorithm (SGA), which is efficient for stochastic linear system analysis, the complexity of SSCA is much smaller than that of SGA for nonlinear case. Thirdly, different from efficient collocation method, the heuristic approach which may results in "rank deficient problem" and "Runge phenomenon", sparse grid technique is developed to select the collocation points in SSCA in order to reduce the complexity while guaranteing the approximation accuracy. Furthermore, though SSCA is proposed for the stochastic nonlinear steady-state analysis, it can be applied for any other kinds of nonlinear system simulation with process variations, such as transient analysis, etc.

Asymmetric shuffle keyboard

Abstract: An asymmetric keyboard with a QWERTY style layout comprising a plurality of sparse grids and a plurality of dense grids is provided. A sparse grid is substantially large in size containing large keys with large labels, whereas a dense grid is substantially small in size containing small keys with small labels. All keys are functional but the larger keys in the sparse grid offer greater visibility and operability than the smaller keys in the dense grid. The user makes use of the sparse grid as the primary grid to input data. A swipe across a designated boundary interchanges the key labels between corresponding pairs of keys in the designated sparse and dense grids. On the software-based version, a swipe across another designated boundary compresses or decompresses a corresponding grid. On the hardware-based version, a bi-axial hinge allows the display and the keyboard to rotate around two axes.

Dense Mapping for Range Sensors: Efficient Algorithms and Sparse Representations

Abstract: This paper focuses on efficient occupancy grid building based on wavelet occupancy grids, a new sparse grid representation and on a new update algorithm for range sensors. The update algorithm takes advantage of the natural multiscale properties of the wavelet expansion to update only parts of the environement that are modified by the sensor measurements and at the proper scale. The sparse wavelet representation coupled with an efficient algorithm presented in this paper provides efficient and fast updating of occupancy grids. It leads to realtime results especially in 2D grids and for the first time in 3D grids. Experiments and results are discussed for both real and simulated data.

Computing Stochastic Dynamic Economic Models with a Large Number of State Variables: A Description and Application of a Smolyak-Collocation Method

Abstract: We describe a sparse grid collocation algorithm to compute recursive solutions of dynamic economies with a sizable number of state variables We show how powerful this method may be in applications by computing the nonlinear recursive solution of an international real business cycle model with a substantial number of countries, complete insurance markets and frictions that impede frictionless international capital flows In this economy the aggregate state vector includes the distribution of world capital across different countries as well as the exogenous country-specific technology shocks We use the algorithm to efficiently solve models with 2, 4, and 6 countries (ie, up to 12 continuous state variables)

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

Interpolating Wavelets and Adaptive Finite Difference Schemes for Solving Maxwell's Equations: The Effects of Gridding

Abstract: In this paper, we discuss the use of the sparse point representation (SPR) methodology for adaptive finite-difference simulations in computational electromagnetics. The principle of the SPR method is to represent the solution only through those point values indicated by the significant wavelet coefficients, which are used as local regularity indicators. Recently, two kinds of SPR schemes have been considered for solving Maxwell's equations: 1) staggered grids in the time-space domain are used for the discretization of the magnetic and electrical fields, as in the finite-differences time-domain (FDTD) scheme and 2) nonstaggered grids are used in combination with Runge-Kutta ODE solvers. In both cases, 1-D simulations of the SPR method leads to sparse grids that adapt in space to the local smoothness of the fields, and, at the same time, track the evolution of the fields over time with substantial gain in memory and computational speed. However, in the latter case, we found spurious oscillations in the simulations. Therefore, before extending the implementation of the SPR method to higher dimensions, we wanted to evaluate which of these two SPR strategies is more convenient. After a careful theoretical analysis of stability and numerical dispersion comparing the schemes in staggered and nonstaggered grids, we conclude that schemes for staggered grids seem to be preferable from the dispersion viewpoint, especially for low-order schemes and coarse grids. However, by adapting the grid density and increasing the order, SPR schemes for nonstaggered grids also show good performance. In our experiments, no spurious oscillations were detected. We observed that, for a given accuracy, the adaptive scheme on a nonstaggered grid requires less computational effort. Since the use of nonstaggered grids increases the stability range and facilitates the implementation of adaptive strategies, we believe that the SPR method in nonstaggered grids has a very good potential for computational electromagnetics

An Optimised Sparse Grid Combination Technique for Eigenproblems

Abstract: Technische Universit¨at Berlin, Institut f ur Mathematik, MA 3-3¨Strase des 17. Juni 136, 10623 BerlinWe introduce the optimised sparse grid combination technique for the numerical solution of d-dimensional eigenproblems onsparse grids. We present numerical results for the stationary Schrodinger equation in the case of hydrogen.¨

Adaptive Sparse Grid Classification Using Grid Environments

Abstract: Common techniques tackling the task of classification in data mining employ ansatz functions associated to training data points to fit the data as well as possible. Instead, the feature space can be discretized and ansatz functions centered on grid points can be used. This allows for classification algorithms scaling only linearly in the number of training data points, enabling to learn from data sets with millions of data points. As the curse of dimensionality prohibits the use of standard grids, sparse grids have to be used. Adaptive sparse grids allow to get a trade-off between both worlds by refining in rough regions of the target function rather than in smooth ones. We present new results for some typical classification tasks and show first observations of dimension adaptivity. As the study of the critical parameters during development involves many computations for different parameter values, we used a grid environment which we present.

Path planning for formations using global optimization with sparse grids

Abstract: This paper proposes an original path planning method developed for leader following formations of car-like robots. In this approach a reference path calculated by the leader should be feasible for all following robots without changing a relative distance in the formation. This requirement can be satisfied using a solution which is composed of smoothly connected cubic splines and can be calculated in real time. Qualities of the result like the length and minimal radius of the resulting path as well as the distance to obstacles are merged into a discontinuous penalty function. The resulting global minimization problem is solved with a deterministic approach based on sparse grids. This algorithm has been tested for several applications in real robotics environments and compared with the results from a stochastic Particle Swarm Optimization method. Some of the determined solutions were then verified by simulations of formation movements.

Efficient d-multigrid preconditioners for sparse-grid solution of high-dimensional partial differential equations

Abstract: Fast and efficient solution techniques are developed for high-dimensional parabolic partial differential equations (PDEs). In this paper we present a robust solver based on the Krylov subspace method Bi-CGSTAB combined with a powerful, and efficient, multigrid preconditioner. Instead of developing the perfect multigrid method, as a stand-alone solver for a single problem discretized on a certain grid, we aim for a method that converges well for a wide class of discrete problems arising from discretization on various anisotropic grids. This is exactly what we encounter during a sparse grid computation of a high-dimensional problem. Different multigrid components are discussed and presented with operator construction formulae. An option-pricing application is focused and presented with results computed with this method.

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.

Three-dimensional feet data measuring method to sparse grid based on curve subdivision

Abstract: A sparse lattice-oriented method based on the fine division of curved surface for measuring the data about 3D foot type includes such steps as reading the 3D foot model on sparse lattice, setting up the iterative parameters for fine division, finely dividing the 3D foot model on sparse lattice, smoothing the generated 3D foot model, examining the terminating conditions of said model, 2D projection of generated 3D foot model, choosing the characteristic dots on the 2D profile of 3D foot model, obtaining the peripheral length of foot type, and outputting the characteristic parameters about foot type.

Adaptive finite difference schemes based on interpolating wavelets for solving 2D Maxwell’s equations

Abstract: This paper describes a 2D application of interpolating wavelets and recursive interpolation schemes with thresholding, aiming the representation of the electric and magnetic fields in nonuniform, adaptive grids. Applied to Maxwell's equations, the method leads to sparse grids that adapt in space to the local smoothness of the fields, and at the same time track the evolution of the fields over time. In general, the number of points in the grid, Ns, is below the maximum number of points, N. It is possible to control Ns, by trading off representation accuracy and data compression, and therefore speed. A numerical example is presented showing the propagation of a Gaussian pulse within a 2D horn.

Multigrid preconditioners for Bi-CGSTAB for the sparse-grid solution of high-dimensional anisotropic diffusion equation

Abstract: Robust and efficient solution techniques are developed for high-dimensional parabolic partial differential equations (PDEs). Presented is a solver based on the Krylov subspace method Bi-CGSTAB preconditioned with d-multigrid. Developing the perfect multigrid method, as a stand-alone solver for a single problem discretized on a particular grid, often requires a lot of optimal tuning and expert insight; on the other hand Krylov-subspace based methods are robust but much less efficient unless used in combination with a very suitable preconditioner. The preconditioner that we employ is d-multigrid. We aim for a robust combination of the two so that it results in a solver that converges well for a wide class of discrete problems arising from discretization on various anisotropic grids. This is exactly what we encounter during the sparse grid solution of a high-dimensional problem. Different multigrid components are discussed and presented with operator construction formulae (in abstract d dimensions). We also present convergence diagrams for various multigrid (solvers and preconditioners) that we develop in this work, and explain their applicability.

High Dimensional Stochastic Simulation and Electric Ship Models

Abstract: Determination of statistical moments requires the numerical computation of integrals, but as the number of random dimensions increases, computational cost grows dramatically. There is great interest in the large scale complex systems encountered in the all-electric ship, and to deal with the large dimension, we present the sparse grid technique based on Smolyak's algorithm for evaluating mid-dimensional integrals. We compare its performance with collocation (full grid) and Monte Carlo methods, first focusing on a now standard package of six test integrals proposed by Genz. We then apply these methods to a validated electric ship model to compute mean and variance behaviors, given variation in physical parameters, both individually and in ranked sets.

Computing Stochastic Dynamic Economic Models with a Large Number of State Variables: A Description and Application of a Smolyak-Collocation Method

Abstract: We describe a sparse grid collocation algorithm to compute recursive solutions of dynamic economies with a sizable number of state variables. We show how powerful this method may be in applications by computing the nonlinear recursive solution of an international real business cycle model with a substantial number of countries, complete insurance markets and frictions that impede frictionless international capital flows. In this economy the aggregate state vector includes the distribution of world capital across different countries as well as the exogenous country-specific technology shocks. We use the algorithm to efficiently solve models with 2, 4, and 6 countries (i.e., up to 12 continuous state variables).