Multilevel approximation of Gaussian random fields: Fast simulation

Herrmann, Lukas; Kirchner, Kristin; Schwab, Christoph

doi:10.1142/s0218202520500050

Cited by 29 publications

(30 citation statements)

References 52 publications

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“…Proposition 2 Let Z , Z be GRFs colored by T and T , respectively, see (20), with covariance functions denoted by and , cf. (29). Then,…”

Section: Corollary 1 Let Assumptionmentioning

confidence: 98%

“…We recall that the covariance function (29) In the next lemma, this relation and (19) are exploited to characterize continuity of the covariance function in terms of the color T of the GRF Z . Proposition 2 Let Z , Z be GRFs colored by T and T , respectively, see (20), with covariance functions denoted by and , cf.…”

Section: Corollary 1 Let Assumptionmentioning

confidence: 99%

“…After having characterized (a) the Hölder regularity (in L q (Ω)-sense) of a GRF Z , and (b) continuity of the covariance function in (29), in terms of the color of Z , we now proceed with this discussion for Sobolev spaces. Specifically, we investigate the regularity of Z in L q (Ω; H σ (D)) and of the covariance function with respect to the norm on the mixed Sobolev space…”

Section: Sobolev Regularity Of Grfs and Their Covariancesmentioning

confidence: 99%

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Regularity and convergence analysis in Sobolev and Hölder spaces for generalized Whittle–Matérn fields

Cox

Kirchner

2020

Numer. Math.

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We analyze several types of Galerkin approximations of a Gaussian random field $$\mathscr {Z}:\mathscr {D}\times \varOmega \rightarrow \mathbb {R}$$ Z : D × Ω → R indexed by a Euclidean domain $$\mathscr {D}\subset \mathbb {R}^d$$ D ⊂ R d whose covariance structure is determined by a negative fractional power $$L^{-2\beta }$$ L - 2 β of a second-order elliptic differential operator $$L:= -\nabla \cdot (A\nabla ) + \kappa ^2$$ L : = - ∇ · ( A ∇ ) + κ 2 . Under minimal assumptions on the domain $$\mathscr {D}$$ D , the coefficients $$A:\mathscr {D}\rightarrow \mathbb {R}^{d\times d}$$ A : D → R d × d , $$\kappa :\mathscr {D}\rightarrow \mathbb {R}$$ κ : D → R , and the fractional exponent $$\beta >0$$ β > 0 , we prove convergence in $$L_q(\varOmega ; H^\sigma (\mathscr {D}))$$ L q ( Ω ; H σ ( D ) ) and in $$L_q(\varOmega ; C^\delta (\overline{\mathscr {D}}))$$ L q ( Ω ; C δ ( D ¯ ) ) at (essentially) optimal rates for (1) spectral Galerkin methods and (2) finite element approximations. Specifically, our analysis is solely based on $$H^{1+\alpha }(\mathscr {D})$$ H 1 + α ( D ) -regularity of the differential operator L, where $$0<\alpha \le 1$$ 0 < α ≤ 1 . For this setting, we furthermore provide rigorous estimates for the error in the covariance function of these approximations in $$L_{\infty }(\mathscr {D}\times \mathscr {D})$$ L ∞ ( D × D ) and in the mixed Sobolev space $$H^{\sigma ,\sigma }(\mathscr {D}\times \mathscr {D})$$ H σ , σ ( D × D ) , showing convergence which is more than twice as fast compared to the corresponding $$L_q(\varOmega ; H^\sigma (\mathscr {D}))$$ L q ( Ω ; H σ ( D ) ) -rate. We perform several numerical experiments which validate our theoretical results for (a) the original Whittle–Matérn class, where $$A\equiv \mathrm {Id}_{\mathbb {R}^d}$$ A ≡ Id R d and $$\kappa \equiv {\text {const.}}$$ κ ≡ const. , and (b) an example of anisotropic, non-stationary Gaussian random fields in $$d=2$$ d = 2 dimensions, where $$A:\mathscr {D}\rightarrow \mathbb {R}^{2\times 2}$$ A : D → R 2 × 2 and $$\kappa :\mathscr {D}\rightarrow \mathbb {R}$$ κ : D → R are spatially varying.

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“…Proposition 2 Let Z , Z be GRFs colored by T and T , respectively, see (20), with covariance functions denoted by and , cf. (29). Then,…”

Section: Corollary 1 Let Assumptionmentioning

confidence: 98%

Section: Corollary 1 Let Assumptionmentioning

confidence: 99%

Section: Sobolev Regularity Of Grfs and Their Covariancesmentioning

confidence: 99%

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Regularity and convergence analysis in Sobolev and Hölder spaces for generalized Whittle–Matérn fields

Cox

Kirchner

2020

Numer. Math.

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“…Indeed, the computational benefits of the stochastic partial differential equation approach in the Gaussian case with differential operators as in equation 5 k ∈ N have already been emphasized by . Recent developments in numerical methods for fractional operators furthermore facilitate treating the whole admissible parameter range φ > 0 while maintaining low computational complexity Herrmann et al, 2020). Specifically, it is a well-known result that, in the non-fractional case D 1 = κ 2 − @ 2 =@t 2 , inverses of the symmetric, elliptic differential operator D 1 , when discretized by means of a finite element method with continuous piecewise linear basis functions on the mesh 0 = s 1 < : : : < s K = t max as used in equation 14, can be realized numerically with multilevel preconditioned iterative solvers in a complexity which is (essentially) linear in the number K of mesh nodes; see, for example Xu (1992).…”

Section: Kristin Kirchner (Delft University Of Technology)mentioning

confidence: 99%

“…Specifically, it is a well-known result that, in the non-fractional case D 1 = κ 2 − @ 2 =@t 2 , inverses of the symmetric, elliptic differential operator D 1 , when discretized by means of a finite element method with continuous piecewise linear basis functions on the mesh 0 = s 1 < : : : < s K = t max as used in equation 14, can be realized numerically with multilevel preconditioned iterative solvers in a complexity which is (essentially) linear in the number K of mesh nodes; see, for example Xu (1992). Recently, it has been shown that, given a prescribed accuracy, this complexity can also be achieved in the fractional case (Herrmann et al, 2020). The numerical inversion of the operator D 2 = κ + @=@t introduced in Section 3.2.1 and treated in a variational framework with a Petrov-Galerkin discretization in Appendix B, however, requires an additional discussion due to its asymmetry.…”

Section: Kristin Kirchner (Delft University Of Technology)mentioning

confidence: 99%

Linear Mixed Effects Models for Non-Gaussian Continuous Repeated Measurement Data

Asar

Bolin

Diggle

et al. 2020

Journal of the Royal Statistical Society Series C: Applied Statistics

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We consider the analysis of continuous repeated measurement outcomes that are collected longitudinally. A standard framework for analysing data of this kind is a linear Gaussian mixed effects model within which the outcome variable can be decomposed into fixed effects, time invariant and time-varying random effects, and measurement noise. We develop methodology that, for the first time, allows any combination of these stochastic components to be non-Gaussian, using multivariate normal variance-mean mixtures. To meet the computational challenges that are presented by large data sets, i.e. in the current context, data sets with many subjects and/or many repeated measurements per subject, we propose a novel implementation of maximum likelihood estimation using a computationally efficient subsampling-based stochastic gradient algorithm. We obtain standard error estimates by inverting the observed Fisher information matrix and obtain the predictive distributions for the random effects in both filtering (conditioning on past and current data) and smoothing (conditioning on all data) contexts. To implement these procedures, we introduce an R package: ngme. We reanalyse two data sets, from cystic fibrosis and nephrology research, that were previously analysed by using Gaussian linear mixed effects models.

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Spatial Field

Bolin¹

2022

Wiley StatsRef: Statistics Reference Online

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Spatial fields, aka random fields, are important models for analyzing data collected at different spatial locations. This article provides a short introduction to spatial fields for continuously indexed data, focusing on how to construct flexible Gaussian and non‐Gaussian fields. A brief introduction to spatial prediction of random fields is given, and approaches for dealing with the computational problems that arise when using spatial fields to analyze large data sets are discussed.

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Multilevel approximation of Gaussian random fields: Fast simulation

Cited by 29 publications

References 52 publications

Regularity and convergence analysis in Sobolev and Hölder spaces for generalized Whittle–Matérn fields

Regularity and convergence analysis in Sobolev and Hölder spaces for generalized Whittle–Matérn fields

Linear Mixed Effects Models for Non-Gaussian Continuous Repeated Measurement Data

Spatial Field

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