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Sparse grid
About: Sparse grid is a research topic. Over the lifetime, 1013 publications have been published within this topic receiving 20664 citations.
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TL;DR: This work presents reduced models for pricing basket options with the Black-Scholes and the Heston model, and results are presented for a single-asset option that leads to a two-dimensional pricing problem, where the model reduction approach is based on high-fidelity sparse grid models.
Abstract: This work presents reduced models for pricing basket options with the Black-Scholes and the Heston model. Basket options lead to multi-dimensional partial differential equations (PDEs) that quickly become computationally infeasible to discretize on full tensor grids. We therefore rely on sparse grid discretizations of the PDEs, which allow us to cope with the curse of dimensionality to some extent. We then derive reduced models with proper orthogonal decomposition. Our numerical results with the Black-Scholes model show that sufficiently accurate results are achieved while gaining speedups between 80 and 160 compared to the high-fidelity sparse grid model for 2-, 3-, and 4-asset options. For the Heston model, results are presented for a single-asset option that leads to a two-dimensional pricing problem, where we achieve significant speedups with our model reduction approach based on high-fidelity sparse grid models.
10 citations
14 Sep 2009
TL;DR: The main idea proposed in this work is to avoid the application of a two-dimensional wavelet transform in order to construct an adaptive scheme for solving systems of type given by (1), keeping however the SPR approach.
Abstract: for which the grid adaptivity is controlled by the application of a wavelet transform to the numerical solution at each time step combined with a splitting method. Wavelet based adaptive schemes for uni-dimensional problems go back to Harten [4], who was one of the precursors in applying interpolating wavelet transforms to obtain a multiscale decomposition of the numerical solution. This multiresolution decomposition was used to decide whether the numerical flux function should be evaluated by an expensive high resolution scheme or it could be computed from a polynomial interpolation process in a multilevel fashion. For two-dimensional problems, the ideas proposed in [4] were extended in [6] where a two-dimensional wavelet transform is applied to the numerical divergence of (1). Again the multiresolution transform is used to analyze the smoothness of the numerical solution to decide where the numerical quantities can be obtained by interpolation or if they must be evaluated exactly by the numerical scheme. In reality, Marten's approach provides a sensor to manage the flux (or divergence) computation but is far from being an adaptive mesh refinement technique since the numerical values on the highest resolution level (the finest dyadic grid of all multiresolution set) must be always available. An interesting alternative has been proposed by Holmstrom [2] in order to indeed produce an adaptive mesh refinement procedure based on a wavelet transform. Holmstrom introduced the concept of sparse point representation (SPR), which is a set of points obtained from the thresholded wavelet transform. The differential operator is then solved by a finite difference scheme over the sparse grid associated to the sparse point representation of the solution. For the SPR of uni-dimensional data the interpolating wavelet transform is considered and for the two-dimensional problems the basis for the bi-dimensional transform is obtained by the tensor product of one-dimensional wavelet and scaling function spaces [2]. The main idea proposed in this work is to avoid the application of a two-dimensional wavelet transform in order to construct an adaptive scheme for solving systems of type given by (1), keeping however the SPR approach. One possibihty for reducing the multidimensional problem to a sequence of one-dimensional problems is to consider a sphtting method (for example, here we have considered a time split MacCormack scheme [8]). Once the multidimensional problem is sphtted in many uni-dimensional sub-problems, each one, associated to a spatial direction, can be solved with an adaptive scheme in each slice of the spatial domain. The following section is reserved for presenting the wavelet time sphtting scheme, whose formulation is inspired by the standard MacCormack method [8]. The discretization is given with respect to a sparse grid per direction. In this sense, the adaptivity is performed for each spatial direction separately, meaning that no tensor product will be necessary to construct the wavelet transform. Each component of the sphtting method is solved by a Lax-Friedrichs scheme, in which numerical flux function is computed by an essentially non-oscillatory (ENO) reconstruction, as done for one-dimensional problems in [1]. The time evolution is done according to the time splitting formulation. Finally, in the last section some numerical simulations are presented.
9 citations
TL;DR: A sparse grid collocation method which adopts nested quadrature rules with delay and transformation to quantify the uncertainty of model solutions and is found to be much more efficient than the Monte-Carlo simulations.
9 citations
01 Jan 1998
TL;DR: It is suggested to use sparse girds as a data compression method in order to visualize huge data sets even on small workstations because the size of data sets used in numerical simulations is still growing, this feature makes it possible that workst stations can continue to handle these data sets.
Abstract: These days sparse grids are of increasing interest in numerical simulations. Based upon hierarchical tensor product bases, the sparse grid approach is a very efficient one improving the ratio of invested storage and computing time to the achieved accuracy for many problems in the area of numerical solution of differential equations, for instance in numerical fluid mechanics. The particle tracing algorithms that are available so far cannot cope with sparse grids. Now we present an approach that directly works on sparse grids. As a second aspect in this paper, we suggest to use sparse girds as a data compression method in order to visualize huge data sets even on small workstations. Because the size of data sets used in numerical simulations is still growing, this feature makes it possible that workstations can continue to handle these data sets.
9 citations
TL;DR: It is found that this proposed global topological approach is capable of handling noise levels exceeding the range of the signal with minimal loss of accuracy and works successfully with data sampled on sparse grids with spacing comparable to the mean separation between the phase singularities for complex patterns featuring multiple interacting spiral waves.
Abstract: The motion of and interaction between phase singularities that lie at the centers of spiral waves capture many qualitative and, in some cases, quantitative features of complex dynamics in excitable systems. Being able to accurately reconstruct their position is thus quite important, even if the data are noisy and sparse, as in electrophysiology studies of cardiac arrhythmias, for instance. A recently proposed global topological approach [Marcotte and Grigoriev, Chaos 27, 093936 (2017)] promises to meaningfully improve the quality of the reconstruction compared with traditional, local approaches. Indeed, we found that this approach is capable of handling noise levels exceeding the range of the signal with minimal loss of accuracy. Moreover, it also works successfully with data sampled on sparse grids with spacing comparable to the mean separation between the phase singularities for complex patterns featuring multiple interacting spiral waves.
9 citations