The Use of Sparse Grid Approximation for the r-Term Tensor Representation

In this work, we present a numerical method based on a sparse grid approximation to compute the loss distribution of the balance sheet of a financial or an insurance company. We first describe, in a stylised way, the assets and liabilities dynamics that are used for the numerical estimation of the balance sheet distribution. For the pricing and hedging model, we chose a classical Black & choles model with a stochastic interest rate following a Hull & White model. The risk management model describing the evolution of the parameters of the pricing and hedging model is a Gaussian model. The new numerical method is compared with the traditional nested simulation approach. We review the convergence of both methods to estimate the risk indicators under consideration. Finally, we provide numerical results showing that the sparse grid approach is extremely competitive for models with moderate dimension.

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Sparse grid approximation spaces for space–time boundary integral formulations of the heat equation

Computers & Mathematics with Applications ◽

10.1016/j.camwa.2019.06.036 ◽

2019 ◽

Vol 78 (11) ◽

pp. 3605-3619

Author(s):

Alexey Chernov ◽

Anne Reinarz

Keyword(s):

Heat Equation ◽

Space Time ◽

Boundary Integral ◽

Sparse Grid ◽

Approximation Spaces ◽

Grid Approximation ◽

Time Boundary

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Singular value decomposition versus sparse grids: refined complexity estimates

IMA Journal of Numerical Analysis ◽

10.1093/imanum/dry039 ◽

2018 ◽

Vol 39 (4) ◽

pp. 1652-1671 ◽

Cited By ~ 2

Author(s):

Michael Griebel ◽

Helmut Harbrecht

Keyword(s):

Singular Value Decomposition ◽

Sparse Grids ◽

Singular Value ◽

Sparse Grid ◽

Approximation Schemes ◽

Approximation Of Functions ◽

Grid Approximation ◽

Numer Anal ◽

Value Decomposition ◽

The Cost

Abstract We compare the cost complexities of two approximation schemes for functions that live on the product domain $\varOmega _1\times \varOmega _2$ of sufficiently smooth domains $\varOmega _1\subset \mathbb{R}^{n_1}$ and $\varOmega _2\subset \mathbb{R}^{n_2}$, namely the singular value / Karhunen–Lòeve decomposition and the sparse grid representation. We assume that appropriate finite element methods with associated orders $r_1$ and $r_2$ of accuracy are given on the domains $\varOmega _1$ and $\varOmega _2$, respectively. This setting reflects practical needs, since often black-box solvers are used in numerical simulation, which restrict the freedom in the choice of the underlying discretization. We compare the cost complexities of the associated singular value decomposition and the associated sparse grid approximation. It turns out that, in this situation, the approximation by the sparse grid is always equal or superior to the approximation by the singular value decomposition. The results in this article improve and generalize those from the study by Griebel & Harbrecht (2014, Approximation of bi-variate functions. Singular value decomposition versus sparse grids. IMA J. Numer. Anal., 34, 28–54). Especially, we consider the approximation of functions from generalized isotropic and anisotropic Sobolev spaces.

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