Isotropic Gaussian random fields on the sphere: regularity, fast simulation, and stochastic partial differential equations
Annika Lang,Christoph Schwab +1 more
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In this article, rates of convergence of their finitely truncated Karhunen-Lo-ve expansions in terms of the covariance spectrum are established, and algorithmic aspects of fast sample generation via fast Fourier transforms on the sphere are indicated.Abstract:
Isotropic Gaussian random fields on the sphere are characterized by Karhunen-Lo\`{e}ve expansions with respect to the spherical harmonic functions and the angular power spectrum The smoothness of the covariance is connected to the decay of the angular power spectrum and the relation to sample H\"{o}lder continuity and sample differentiability of the random fields is discussed Rates of convergence of their finitely truncated Karhunen-Lo\`{e}ve expansions in terms of the covariance spectrum are established, and algorithmic aspects of fast sample generation via fast Fourier transforms on the sphere are indicated The relevance of the results on sample regularity for isotropic Gaussian random fields and the corresponding lognormal random fields on the sphere for several models from environmental sciences is indicated Finally, the stochastic heat equation on the sphere driven by additive, isotropic Wiener noise is considered, and strong convergence rates for spectral discretizations based on the spherical harmonic functions are provenread more
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Padé-parametric FEM approximation for fractional powers of elliptic operators on manifolds
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Surface Finite Element Approximation of Spherical Whittle--Matérn Gaussian Random Fields
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A Note on the Derivatives of Isotropic Positive Definite Functions on the Hilbert Sphere
TL;DR: In this article, a recursive formula for the derivatives of isotropic positive definite functions on the Hilbert sphere was given, and it was used to prove a conjecture stated by Trubner and Ziegel.
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Gaussian Process for Radiance Functions on the S2$\mathbb {S}^2$ Sphere
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Strong Local Nondeterminism and Exact Modulus of Continuity for Isotropic Gaussian Random Fields on Compact Two-Point Homogeneous Spaces
TL;DR: In this paper , sample path properties of real-valued isotropic Gaussian fields on compact two-point homogeneous spaces were studied and an exact uniform modulus of continuity for its sample paths was established.
References
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Orthogonal Polynomials
TL;DR: In this paper, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed and orthogonality on the unit circle is not discussed.
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Higher Transcendental Functions
TL;DR: Higher Transcendental Functions Based on notes left by the late Prof. Harry Bateman, and compiled by the Staff of the Bateman Project as discussed by the authors, are presented in Table 1.
Book
Theory of function spaces
TL;DR: In this article, the authors measure smoothness using Atoms and Pointwise Multipliers, Wavelets, Spaces on Lipschitz Domains, Wavelet and Sampling Numbers.
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Isotropic Gaussian random fields on the sphere: Regularity, fast simulation and stochastic partial differential equations
Annika Lang,Christoph Schwab +1 more