Results 1  10
of
74
An anisotropic sparse grid stochastic collocation method for elliptic partial differential equations with random input data
, 2007
"... ..."
Sparse grids and related approximation schemes for higher dimensional problems
"... The efficient numerical treatment of highdimensional problems is hampered by the curse of dimensionality. We review approximation techniques which overcome this problem to some extent. Here, we focus on methods stemming from Kolmogorovâ€™s theorem, the ANOVA decomposition and the sparse grid approach ..."
Abstract

Cited by 46 (12 self)
 Add to MetaCart
The efficient numerical treatment of highdimensional problems is hampered by the curse of dimensionality. We review approximation techniques which overcome this problem to some extent. Here, we focus on methods stemming from Kolmogorovâ€™s theorem, the ANOVA decomposition and the sparse grid approach and discuss their prerequisites and properties. Moreover, we present energynorm based sparse grids and demonstrate that, for functions with bounded mixed derivatives on the unit hypercube, the associated approximation rate in terms of the involved degrees of freedom shows no dependence on the dimension at all, neither in the approximation order nor in the order constant.
The MultiElement Probabilistic Collocation Method: Error Analysis and Applications
 J Comp Physics
"... Stochastic spectral methods are numerical techniques for approximating solutions to partial differential equations with random parameters. In this work, we present and examine the multielement probabilistic collocation method (MEPCM), which is a generalized form of the probabilistic collocation me ..."
Abstract

Cited by 36 (4 self)
 Add to MetaCart
(Show Context)
Stochastic spectral methods are numerical techniques for approximating solutions to partial differential equations with random parameters. In this work, we present and examine the multielement probabilistic collocation method (MEPCM), which is a generalized form of the probabilistic collocation method. In the MEPCM, the parametric space is discretized and a collocation/cubature grid is prescribed on each element. Both full and sparse tensor product grids based on Gauss and ClenshawCurtis quadrature rules are considered. We prove analytically and observe in numerical tests that as the parameter space mesh is refined, the convergence rate of the solution depends on the quadrature rule of each element only through its degree of exactness. In addition, the L2 error of the tensor product interpolant is examined and an adaptivity algorithm is provided. Numerical examples demonstrating adaptive MEPCM are shown, including lowregularity problems and longtime integration. We test the MEPCM on twodimensional Navier Stokes examples and a stochastic diffusion problem with various random input distributions and up to 50 dimensions. While the convergence rate of MEPCM deteriorates in 50 dimensions, the error in the mean and variance is two orders of magnitude lower than the error obtained with the Monte Carlo method using only a small number of samples (e.g., 100). The computational cost of MEPCM is found to be favorable when compared to the cost of other methods including stochastic Galerkin, Monte Carlo and quasirandom sequence methods. 1
Recent developments in spectral stochastic methods for the numerical solution of stochastic partial differential equations
, 2009
"... ..."
Why are highdimensional finance problems often of low effective dimension?
 SIAM J. Sci. Comput
, 2003
"... Many problems in mathematical finance can be formulated as highdimensional integrals, where the large number of dimensions arises from small time steps in time discretization and/or a large number of state variables. QuasiMonte Carlo (QMC) methods have been successfully used for approximating such ..."
Abstract

Cited by 27 (2 self)
 Add to MetaCart
Many problems in mathematical finance can be formulated as highdimensional integrals, where the large number of dimensions arises from small time steps in time discretization and/or a large number of state variables. QuasiMonte Carlo (QMC) methods have been successfully used for approximating such integrals. To understand this success, this paper focuses on investigating the special features of some typical highdimensional finance problems, namely option pricing and bond valuation. We provide new insight into the connection between the effective dimension and the efficiency of QMC, and present methods to analyze the dimension structure of a function. We confirm the observation of Caflisch, Morokoff and Owen that functions from finance are often of low effective dimension, in the sense that they can be well approximated by their loworder ANOVA (analysis of variance) terms, usually just the order1 and order2 terms. We explore why the effective dimension is small for many integrals from finance. By deriving explicit forms of the ANOVA terms in simple cases, we find that the importance of each dimension is naturally weighted, by certain hidden weights. These weights characterize the relative importance of different variables or groups of variables, and limit the importance of the higherorder ANOVA terms. We study the variance ratios captured by loworder ANOVA terms and their asymptotic properties as the dimension tends to infinity, and show that with the increase of dimension the lowerorder terms continue to play a significant role and the higherorder terms tend to be negligible. This provides some insight into highdimensional problems from finance and explains why QMC algorithms are efficient for problems of this kind.
Dimensionwise Integration of Highdimensional Functions with Applications to Finance
, 2009
"... We present a new general class of methods for the computation of highdimensional integrals. The quadrature schemes result by truncation and discretization of the anchoredANOVA decomposition. They are designed to exploit low effective dimensions and include sparse grid methods as special case. To d ..."
Abstract

Cited by 23 (1 self)
 Add to MetaCart
We present a new general class of methods for the computation of highdimensional integrals. The quadrature schemes result by truncation and discretization of the anchoredANOVA decomposition. They are designed to exploit low effective dimensions and include sparse grid methods as special case. To derive bounds for the resulting modelling and discretization errors, we introduce effective dimensions for the anchoredANOVA decomposition. We show that the new methods can be applied in a locallyadaptive and dimensionadaptive way and demonstrate their efficiency by numerical experiments with highdimensional integrals from finance.
Semisupervised learning with sparse grids
 Proc. of the 22nd ICML Workshop on Learning with Partially Classified Training Data
, 2005
"... Sparse grids were recently introduced for classification and regression problems. In this article we apply the sparse grid approach to semisupervised classification. We formulate the semisupervised learning problem by a regularization approach. Here, besides a regression formulation for the labele ..."
Abstract

Cited by 18 (0 self)
 Add to MetaCart
(Show Context)
Sparse grids were recently introduced for classification and regression problems. In this article we apply the sparse grid approach to semisupervised classification. We formulate the semisupervised learning problem by a regularization approach. Here, besides a regression formulation for the labeled data, an additional term is involved which is based on the graph Laplacian for an adjacency graph of all, labeled and unlabeled data points. It reflects the intrinsic geometric structure of the data distribution. We discretize the resulting problem in function space by the sparse grid method and solve the arising equations using the socalled combination technique. In contrast to recently proposed kernel based methods which currently scale cubic in regard to the number of overall data, our method scales only linear, provided that a sparse graph Laplacian is used. This allows to deal with huge data sets which involve millions of points. We show experimental results with the new approach. 1.
Simulationbased optimal Bayesian experimental design for nonlinear systems
 Journal of Computational Physics
, 2012
"... iv ..."
(Show Context)
Efficient hierarchical approximation of highdimensional option pricing problems
, 2006
"... A major challenge in computational finance is the pricing of options that depend on a large number of risk factors. Prominent examples are basket or index options where dozens or even hundreds of stocks constitute the underlying asset and determine the dimensionality of the corresponding degenerate ..."
Abstract

Cited by 13 (3 self)
 Add to MetaCart
(Show Context)
A major challenge in computational finance is the pricing of options that depend on a large number of risk factors. Prominent examples are basket or index options where dozens or even hundreds of stocks constitute the underlying asset and determine the dimensionality of the corresponding degenerate parabolic equation. The objective of this article is to show how an efficient discretisation can be achieved by hierarchical approximation as well as asymptotic expansions of the underlying continuous problem. The relation to a number of stateoftheart methods is highlighted. 1