Dimensionality Reducing Expansion of Multivariate by Tian-Xiao He

By Tian-Xiao He

Multivariate integration has been a primary topic in arithmetic, with extensive connections to a couple of components: numerical research, approximation idea, partial differential equations, fundamental equations, harmonic research, and so on. during this paintings the exposition focuses totally on a robust software that has develop into specifically very important in our automatic age, specifically, dimensionality lowering growth (DRE). the tactic of DRE is a method for altering a better dimensional integration to a decrease dimensional one without or with the rest. up to now, there's no entire therapy of this topic in monograph or textbook shape.

Key beneficial properties of this self-contained monograph comprise:

* advantageous exposition masking the historical past of the topic

* up to date new effects, with regards to many fields of present study equivalent to boundary aspect equipment for fixing PDEs and wavelet research

* presentation of DRE innovations utilizing a large array of examples

* solid stability among idea and alertness

* insurance of such comparable issues as boundary variety quadratures and asymptotic expansions of oscillatory integrals

* very good and complete bibliography and index

This paintings will entice a wide viewers of scholars and researchers in natural and utilized arithmetic, information, and physics, and will be utilized in a graduate/advanced undergraduate path or as a customary reference text.

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Extra resources for Dimensionality Reducing Expansion of Multivariate Integration

Example text

4. If J~oo t k1/l(t) dt = °for k = 0, 1, ... ,r, ... ,m, but J~oo xm+l 1/1 (x) dx :I 0, then we can increase the number of evaluation points to m + 2 and construct the corresponding quadrature formula similarly. 5. 1 in Daubechies [11]). 4 Let 1/1 E Sr, with 1/Imn(t) = 2m/2 1/1(2mt - n) being an orthonormal system in L2(lR), and J~oo t r+11/l(t) dt = A :I 0. Also let 1/I*(t) = Lk 1/I(t - k) be the periodized version 0/1/1 with period 1. 16) where C;=A and {tj : i n r+! Jr' tj - tj = 0,1, ... , r + I} are r + 2 distinct real numbers.

2 -I ~1 dkg' dm-k-1f ]x=! 2. ) K 3 3 3m G(X )il3 F(X )dX , respectively. 6). First define the following classes of functions with respect to a given constant M > 0: E2 E3 == E2(M):= {f(X2) : il'2 f E C(Q), lil'2 fl ::: M}, == E3(M):= {F(X 3) : ilT FE C(K), lilT FI ::: M}. 8) become optimal DREs with respect to E2 and E3 respectively. Hence, we have the following result.. 1 For any given M > 0 we have . 1 M mf sup Ipm(f, g)1 = 4m-l( ,)2 m. 11) and . 6) respectively. Moreover, the infimums can be precisely attained with g(X2) = Qm(x)Qm(y) and G(X3) = Qm(x)Qm(y)Qm(Z).

First, we note that, if ti f/. [0, 1), we can always find t; E [0, 1) such that f(ti) = f(t;) because of the periodicity of f. Second, we have 10 1 f(t)1/I*(t) dt = 10 1 f(t) ~ 1/I(t = LA (HI f(t k k) dt + k)1/I(t) dt = 1 00 f(t)1/I(t) dt. 15) respectively. 11). The weight function in each integra1 can be expanded in tenns of sufficiently smooth periodic wavelet functions, such as the C r continuous periodic wavelet 1/1* = 1/I(t -k), 1/1 E Sr. 2) with the quadrature constructed through the above process.

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