Sparse Grids and Applications (Lecture Notes in Computational Science and Engineering)
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In the recent decade, there has been a growing interest in the numerical treatment of high-dimensional problems. It is well known that classical numerical discretization schemes fail in more than three or four dimensions due to the curse of dimensionality. The technique of sparse grids helps overcome this problem to some extent under suitable regularity assumptions. This discretization approach is obtained from a multi-scale basis by a tensor product construction and subsequent truncation of the resulting multiresolution series expansion. This volume of LNCSE is a collection of the papers from the proceedings of the workshop on sparse grids and its applications held in Bonn in May 2011. The selected articles present recent advances in the mathematical understanding and analysis of sparse grid discretization. Aspects arising from applications are given particular attention.
Introduce and discuss space- and 4 B. Bohn and M. Griebel dimension-adaptive sparse grid algorithms. In Sect. 4, we give the results of numerical experiments which illustrate the favorable properties of our new methods. 2 Takens’ Theorem and the Delay Embedding Scheme We now provide the essential theory concerning Takens’ theorem  and give a hint to some modifications from . For an arbitrary d 2 N we can create vectors tj WD sj d C1 ; sj d C2 ; : : : ; sj 1 ; sj T 2 Rd ; j d.
Centers, then the European market with London and Frankfurt dominates, while the main trading activity takes place during the overlap of the European business hours and the later starting American market with New York as the hub [18, 22]. For the following experiments with the sparse grid regression approach the associated input data set S is obtained from the given tick data. Note that the embedding operator T at a time tj depends on a certain number of delayed data positions between tj .K 1/.
Lower bounds in (11) are chosen by p T var.Xi .T //; p T var.Xi .T //; ˇi D xi C ˛i D xi (12) (13) where is a proportionality constant relating the domain size and the standard deviation of the stochastic process in that direction. Furthermore, x D .x1 : : : : ; xd / denotes a point of interest, which might be, like e.g. in (10), a transformed initial state vector s for an option we want to price by the evaluation of e rT u.T; x/ D V .0; s/. We then can write the PIDE (9) with the definition.
C C C C; C A which can solely be expressed using the prolongations Il;l 0 ; l > l 0 ; and the isotropic operator matrices Al;l . We can precalculate vectors wl for l D 2; : : : ; k using the recursive relationship wl WD l X Il;l 0 ul 0 l 0 D1 D l 1 X Il;l 1 Il 1;l 0 ul 0 C ul l 0 D1 D Il;l 1 wl 1 C ul starting with w1 WD u1 . Now (30) can be expressed by vl D l X Al;l Il;l 0 ul 0 D Al;l l 0 D1 l X Il;l 0 ul 0 D Al;l wl : (31) l 0 D1 As long as the prolongations Il;l 1 and.
Compact (cf. Theorems 1 and 2). By rescaling the resulting domain of the reconstructed process we always can obtain the domain Œ0; 1d . 4.1 Multilevel Hierarchical Bases and Regular Sparse Grids First, we recall the construction of a full grid space using a piecewise linear hierarchical basis and discuss its relation to a sparse grid space. Let the onedimensional hat function W R ! Œ0; 1 be defined by 1 .x/ WD jxj; if x 2 Œ 1; 1 0 else and let l;i .x/ WD .2l x i /jŒ0;1 for any l; i 2 N.