The Wiener-Hermite Expansion with Time-Dependent Ideal Random Function

Doi, Masaaki; Imamura, Tsutomu

doi:10.1143/ptp.41.358

Cited by 24 publications

(10 citation statements)

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“…In [26][27][28] it was proposed to make the random functions time-dependent and to set up a separate differential equation for the determination of the optimal time-dependent ideal random functions. In [27] it is stated that: ''The principle idea of the method is to choose different ideal random functions at different times in such a way that the unknown random function is expressed with good approximation by the first few terms of the Wiener-Hermite expansion for long-time duration.…”

Section: Time-dependent Wiener-hermite Expansionmentioning

confidence: 99%

Time-dependent generalized polynomial chaos

Gerritsma

Steen

Vos

et al. 2010

Journal of Computational Physics

160

148

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Section: Time-dependent Wiener-hermite Expansionmentioning

confidence: 99%

Time-dependent generalized polynomial chaos

Gerritsma

Steen

Vos

et al. 2010

Journal of Computational Physics

160

148

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“…(2) The assertion similar to Lemma 2.4 in classical probability is known (see [5]), where the non-crossing condition must be dropped.…”

Section: This Says That For Eachmentioning

confidence: 96%

Noncommutative Sobolev Spaces, $C^∞$ Algebras and Schwartz Distributions Associated with Semicircular Systems

Mizuo

2003

Publ. Res. Inst. Math. Sci.

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When the C * -algebra and the W * -algebra generated by a semicircular system are viewed from the viewpoints of noncommutative topology and noncommutative probability theory, we may consider the C * -algebra as a certain kind of a "noncommutative cubic space" and the W * -algebra as a "noncommutative cubic measure space." In this paper we introduce the Sobolev spaces W p n associated with the W * -algebra generated by a semicircular system, and the C ∞ algebra S is defined as the projective limit of W p n . The Schwartz distribution space is then defined as the dual space of S and the Fourier representation theorem is obtained for Schwartz distributions. We furthermore discuss vector fields on the C ∞ algebra S. Appendix treats the K-theory of the noncommutative cubic space.

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“…where the subscripts ''u'' in both (14) and (15) emphasize the fact that such forms depend also and possibly in a nonlinear way on the random field u. We will also assume that (14) and (15) are nondegenerate and satisfy the requirements of symmetry and positive definiteness, i.e.…”

Section: Local Inner Productsmentioning

confidence: 99%

“…We will also assume that (14) and (15) are nondegenerate and satisfy the requirements of symmetry and positive definiteness, i.e. they define two local inner products in H 1 and H 2 , respectively.…”

Section: Local Inner Productsmentioning

confidence: 99%

“…time-dependent bases [12][13][14][15], in order to refer the statistical description of a time-evolving stochastic system to an epoch that is not continuously escaping in the past (see [16] for a more recent account and [17] for an interesting alternative representation of time-evolving random spaces). All these pioneering contributions revealed a close connection between random input processes and orthogonal polynomial functionals [18].…”

Section: Introductionmentioning

confidence: 99%

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