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84
DimensionAdaptive TensorProduct Quadrature
 Computing
, 2003
"... We consider the numerical integration of multivariate functions defined over the unit hypercube. Here, we especially address the highdimensional case, where in general the curse of dimension is encountered. Due to the concentration of measure phenomenon, such functions can often be well approxi ..."
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Cited by 74 (12 self)
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We consider the numerical integration of multivariate functions defined over the unit hypercube. Here, we especially address the highdimensional case, where in general the curse of dimension is encountered. Due to the concentration of measure phenomenon, such functions can often be well approximated by sums of lowerdimensional terms. The problem, however, is to find a good expansion given little knowledge of the integrand itself.
Recent developments in spectral stochastic methods for the numerical solution of stochastic partial differential equations
, 2009
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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 ..."
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Cited by 22 (1 self)
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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 ..."
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Cited by 18 (0 self)
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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.
Numerical methods and Smolyak quadrature for nonlinear stochastic partial differential equations
, 2003
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A general assetliability management model for the efficient simulation of portfolios of life insurance policies
 Insurance: Math. Ecomonics
"... New regulations and a stronger competition have increased the importance of stochastic assetliability management models for insurance companies in recent years. In this paper, we propose a discrete time assetliability management model for the simulation of simplified balance sheets of life insuran ..."
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Cited by 17 (4 self)
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New regulations and a stronger competition have increased the importance of stochastic assetliability management models for insurance companies in recent years. In this paper, we propose a discrete time assetliability management model for the simulation of simplified balance sheets of life insurance products. The model incorporates the most important life insurance product characteristics, the surrender of contracts, a reservedependent surplus declaration, a dynamic asset allocation and a twofactor stochastic capital market. All arising terms in the model can be calculated recursively which allows an easy implementation and efficient simulation. Furthermore, the model is designed to have a modular organisation which permits straightforward modifications and extensions to handle specific requirements. In a sensitivity analysis for example portfolios and parameters we investigate the impact of the most important product and management parameters on the risk exposure of the insurance company and show that the model captures the main behaviour patterns of the balance sheet development of life insurance products. Keywords: assetliability management, participating policies, numerical simulation
Adaptive wavelet algorithms for elliptic PDEs on product domains
 Math. Comp
"... Abstract. With standard isotropic approximation by (piecewise) polynomials of fixed order in a domain D ⊂ Rd, the convergence rate in terms of the number N of degrees of freedom is inversely proportional to the space dimension d. This socalled curse of dimensionality can be circumvented by applying ..."
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Cited by 14 (4 self)
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Abstract. With standard isotropic approximation by (piecewise) polynomials of fixed order in a domain D ⊂ Rd, the convergence rate in terms of the number N of degrees of freedom is inversely proportional to the space dimension d. This socalled curse of dimensionality can be circumvented by applying sparse tensor product approximation, when certain high order mixed derivatives of the approximated function happen to be bounded in L2. Itwasshown by Nitsche (2006) that this regularity constraint can be dramatically reduced by considering best Nterm approximation from tensor product wavelet bases. When the function is the solution of some wellposed operator equation, dimension independent approximation rates can be practically realized in linear complexity by adaptive wavelet algorithms, assuming that the infinite stiffness matrix of the operator with respect to such a basis is highly compressible. Applying piecewise smooth wavelets, we verify this compressibility for general, nonseparable elliptic PDEs in tensor domains. Applications of the general theory developed include adaptive Galerkin discretizations of multiple scale homogenization problems and of anisotropic equations which are robust, i.e., independent of the scale parameters, resp. of the size of the anisotropy. 1. Motivation and
Spacetime approximation with sparse grids
"... In this article we introduce approximation spaces for 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 socalled spacetime sparse grid spaces. For a uniform discretization of the spa ..."
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Cited by 13 (1 self)
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In this article we introduce approximation spaces for 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 socalled spacetime sparse grid spaces. For a uniform discretization of the spatial space of dimension d with O(N d) degrees of freedom, these spaces involve for d> 1 also only O(N d) degrees of freedom for the discretization of the whole spacetime problem. But they provide the same approximation rate as classical spacetime Finite Element spaces which need O(N d+1) degrees of freedoms. This makes these approximation spaces well suited for conventional parabolic and for timedependent optimization problems. We analyze the approximation properties and the dimension of these sparse grid spacetime 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 Ω, 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 spacetime sparse grid approach. We give numerical results on the convergence rate of the interpolation error of these spaces for various spacetime problems with two spatial dimensions. Also implementational issues, data structures and questions of adaptivity are addressed to some extent.
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 ..."
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Cited by 13 (3 self)
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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