Difference Schemes on Uniform Grids for an Initial-Boundary Value Problem for a Singularly Perturbed Parabolic Convection-Diffusion Equation

The convergence of difference schemes on uniform grids for an initial-boundary value problem for a singularly perturbed parabolic convection-diffusion equation is studied; the highest x-derivative in the equation is multiplied by a perturbation parameter ε taking arbitrary values in the interval {(0,1]}. For small ε, the problem involves a boundary layer of width {\mathcal{O}(\varepsilon)}, where the solution changes by a finite value, while its derivative grows unboundedly as ε tends to zero. We construct a standard difference scheme on uniform meshes based on the classical monotone grid approximation (upwind approximation of the first-order derivatives). Using a priori estimates, we show that such a scheme converges as {\{\varepsilon N\},N_{0}\to\infty} in the maximum norm with first-order accuracy in {\{\varepsilon N\}} and {N_{0}}; as {N,N_{0}\to\infty}, the convergence is conditional with respect toN, where {N+1} and {N_{0}+1} are the numbers of mesh points in x and t, respectively. We develop an improved difference scheme on uniform meshes using the grid approximation of the first x-derivative in the convective term by the central difference operator under the condition{h\leq m\varepsilon}, which ensures the monotonicity of the scheme; here m is some rather small positive constant. It is proved that this scheme converges in the maximum norm at a rate of {\mathcal{O}(\varepsilon^{-2}N^{-2}+N^{-1}_{0})}. We compare the convergence rate of the developed scheme with the known Samarskii scheme for a regular problem. It is found that the improved scheme (for {\varepsilon=1}), as well as the Samarskii scheme, converges in the maximum norm with second-order accuracy inx and first-order accuracy int.

A sparse grid approach to balance sheet risk measurement

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.

Singular value decomposition versus sparse grids: refined complexity estimates

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.