MULTIDIMENSIONAL PARAMETER WHITE NOISE AND GAUSSIAN RANDOM FIELDS

2001 ◽  
pp. 375-381
Author(s):  
Takeyuki Hida ◽  
Ke-Seung Lee ◽  
SI SI
Keyword(s):  
White Noise ◽  
Random Fields ◽  
Gaussian Random Fields ◽  
Multidimensional Parameter

Multilevel approximation of Gaussian random fields: Fast simulation

2019 ◽  
Vol 30 (01) ◽  
pp. 181-223 ◽  
Author(s):  
Lukas Herrmann ◽  
Kristin Kirchner ◽  
Christoph Schwab
Keyword(s):  
White Noise ◽  
Fractional Order ◽  
Random Fields ◽  
Computational Cost ◽  
Gaussian Random Fields ◽  
Fast Simulation ◽  
Covariance Operators ◽  
Multilevel Approximation ◽  
Sinc Quadrature ◽  
Consistency Error

We propose and analyze several multilevel algorithms for the fast simulation of possibly nonstationary Gaussian random fields (GRFs) indexed, for example, by the closure of a bounded domain [Formula: see text] or, more generally, by a compact metric space [Formula: see text] such as a compact [Formula: see text]-manifold [Formula: see text]. A colored GRF [Formula: see text], admissible for our algorithms, solves the stochastic fractional-order equation [Formula: see text] for some [Formula: see text], where [Formula: see text] is a linear, local, second-order elliptic self-adjoint differential operator in divergence form and [Formula: see text] is white noise on [Formula: see text]. We thus consider GRFs on [Formula: see text] with covariance operators of the form [Formula: see text]. The proposed algorithms numerically approximate samples of [Formula: see text] on nested sequences [Formula: see text] of regular, simplicial partitions [Formula: see text] of [Formula: see text] and [Formula: see text], respectively. Work and memory to compute one approximate realization of the GRF [Formula: see text] on the triangulation [Formula: see text] of [Formula: see text] with consistency [Formula: see text], for some consistency order [Formula: see text], scale essentially linearly in [Formula: see text], independent of the possibly low regularity of the GRF. The algorithms are based on a sinc quadrature for an integral representation of (the application of) the negative fractional-order elliptic “coloring” operator [Formula: see text] to white noise [Formula: see text]. For the proposed numerical approximation, we prove bounds of the computational cost and the consistency error in various norms.

On the comparison of the distribution of the supremum of random fields represented by stochastic integrals

1989 ◽  
Vol 21 (4) ◽  
pp. 770-780 ◽  
Author(s):  
Enzo Orsingher ◽  
Bruno Bassan
Keyword(s):  
White Noise ◽  
Random Fields ◽  
Upper Bounds ◽  
Gaussian Random Fields ◽  
Stochastic Integrals ◽  
Response Functions ◽  
Noise Field ◽  
Fixed Radius ◽  
Different Response

In this paper we compare the distribution of the supremum of the Gaussian random fields Z(P) = ∫CpG(P, P′) dW(P′) and U(P) = ∫CpdW(P'), where CP are circles of fixed radius, dW is a white noise field and G are special deterministic response functions.The results obtained permit us to establish upper bounds for the distribution of the supremum of Z(P) by applying some well-known inequalities on U(P).The comparison of the suprema is carried out also, when CP = ℝ2, between fields with different response functions.

scholarly journals White noise approach to Gaussian random fields

1990 ◽  
Vol 119 ◽  
pp. 93-106 ◽  
Author(s):  
Ke-Seung Lee
Keyword(s):  
White Noise ◽  
Random Fields ◽  
Integrable Function ◽  
Smooth Boundary ◽  
Noise Analysis ◽  
Variational Calculus ◽  
Main Tool ◽  
Gaussian Random Fields ◽  
Dimensional Euclidean Space ◽  
Square Integrable Function

The purpose of this paper is to investigate way of dependency of Gaussian random fields X(D) indexed by a domain D in d-dimensional Euclidean space Rd. Our main tool is variational calculus, where the boundary of a domain varies and deforms and we appeal to the white noise analysis. We therefore assume that X(D) is expressed white noise integral of the form(0.1) X(D) = X(D, W)=∫D F(D, u)W(u)du,where W is the Rd-parameter white noise and the kernel F(D, u) is a square integrable function over Rd, and where D is a bounded domain with smooth boundary.

On the comparison of the distribution of the supremum of random fields represented by stochastic integrals

1989 ◽  
Vol 21 (04) ◽  
pp. 770-780
Author(s):  
Enzo Orsingher ◽  
Bruno Bassan
Keyword(s):  
White Noise ◽  
Random Fields ◽  
Upper Bounds ◽  
Gaussian Random Fields ◽  
Stochastic Integrals ◽  
Response Functions ◽  
Noise Field ◽  
Fixed Radius ◽  
Different Response

In this paper we compare the distribution of the supremum of the Gaussian random fields Z(P) = ∫ Cp G(P, P′) dW(P′) and U(P) = ∫ Cp dW(P'), where CP are circles of fixed radius, dW is a white noise field and G are special deterministic response functions. The results obtained permit us to establish upper bounds for the distribution of the supremum of Z(P) by applying some well-known inequalities on U(P). The comparison of the suprema is carried out also, when C P = ℝ2, between fields with different response functions.

Asymptotic Properties for Increments of l∞-Valued Gaussian Random Fields

2008 ◽  
Vol 60 (2) ◽  
pp. 313-333
Author(s):  
Yong-Kab Choi ◽  
Miklós Csörgő
Keyword(s):  
Random Fields ◽  
Asymptotic Properties ◽  
Gaussian Random Fields ◽  
Moduli Of Continuity ◽  
Multidimensional Parameter

AbstractThis paper establishes general theorems which contain both moduli of continuity and large incremental results for l∞-valued Gaussian random fields indexed by a multidimensional parameter under explicit conditions.

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