Fast sampling of parameterised Gaussian random fields

Latz, Jonas; Eisenberger, Marvin; Ullmann, Elisabeth

doi:10.1016/j.cma.2019.02.003

Cited by 21 publications

(24 citation statements)

References 65 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By virtue of their practicality owing to the full characterization by their mean and covariance structure, Gaussian random fields (GRFs for short) are popular models for many applications in spatial statistics and uncertainty quantification, e.g., [4,7,19,32,39,41]. As a result, several methodologies in these disciplines require the efficient simulation of GRFs at unstructured locations in various possibly non-convex Euclidean domains, and this topic has been intensively discussed in both areas, spatial statistics and computational mathematics, see, e.g., [3,8,14,18,21,28,31,36]. In particular, sampling from non-stationary GRFs, for which methods based on circulant embedding are inapplicable, has become a central topic of current research, see, e.g., [3,9,18].…”

Section: Motivation and Backgroundmentioning

confidence: 99%

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

Cox

Kirchner

2020

Numer. Math.

View full text Add to dashboard Cite

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.

show abstract

Section: Motivation and Backgroundmentioning

confidence: 99%

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

Cox

Kirchner

2020

Numer. Math.

View full text Add to dashboard Cite

show abstract

“…A practically more relevant problem is the Bayesian elliptic inverse problem. It is the prototype example in the context of partial differential equations and has been investigated by various authors, e.g., [8,9,25,30,34].…”

Section: Relaxing the Lipschitz Conditionmentioning

confidence: 99%

“…Hierachical prior measures are used to construct more complex and flexible prior models. In Bayesian inverse problems, such are discussed in [11,12,25]. The basic idea is to employ a prior measure depending on a so-called hyperparameter.…”

Section: Hierarchical Priormentioning

confidence: 99%

“…Several authors have discussed, what we choose to call, (Lipschitz, Hellinger) well-posedness for a variety of Bayesian inverse problems. For example, elliptic partial differential equations [8,22], level-set inversion [23], Helmholtz source identification with Dirac sources [13], a Cahn-Hilliard model for tumour growth [24], hierarchical prior measures [25], stable priors in quasi-Banach spaces [36,37], convex and heavy-tailed priors [20,21]. Moreover, to show well-posedness, Stuart [34] has proposed a set of sufficient but not necessary assumptions.…”

mentioning

confidence: 99%

See 1 more Smart Citation

On the Well-posedness of Bayesian Inverse Problems

Latz¹

2020

SIAM/ASA J. Uncertainty Quantification

Self Cite

View full text Add to dashboard Cite

The subject of this article is the introduction of a new concept of well-posedness of Bayesian inverse problems. The conventional concept of (Lipschitz, Hellinger) well-posedness in [Stuart 2010, Acta Numerica 19, pp. 451-559] is difficult to verify in practice and may be inappropriate in some contexts. Our concept simply replaces the Lipschitz continuity of the posterior measure in the Hellinger distance by continuity in an appropriate distance between probability measures. Aside from the Hellinger distance, we investigate well-posedness with respect to weak convergence, the total variation distance, the Wasserstein distance, and also the Kullback-Leibler divergence. We demonstrate that the weakening to continuity is tolerable and that the generalisation to other distances is important. The main results of this article are proofs of wellposedness with respect to some of the aforementioned distances for large classes of Bayesian inverse problems. Here, little or no information about the underlying model is necessary; making these results particularly interesting for practitioners using black-box models. We illustrate our findings with numerical examples motivated from machine learning and image processing.

show abstract

“…Sampling-free approaches often involve surrogates for the forward response operator to decrease the computational cost. Typical surrogates are based on generalized polynomial chaos [10,26,30,47], sparse grids [3,5,28,36,37,38], Gaussian process regression [22,42], model reduction [13,24,27], and combinations, e.g., sparse grids and reduced bases [3,4]. To obtain an accurate (and convergent) approximation, these surrogates require certain types of smoothness of the response surface or likelihood function w.r.t.…”

mentioning

confidence: 99%

Multilevel Adaptive Sparse Leja Approximations for Bayesian Inverse Problems

Farcaș¹,

Latz²,

Ullmann³

et al. 2020

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

Deterministic interpolation and quadrature methods are often unsuitable to address Bayesian inverse problems depending on computationally expensive forward mathematical models. While interpolation may give precise posterior approximations, deterministic quadrature is usually unable to efficiently investigate an informative and thus concentrated likelihood. This leads to a large number of required expensive evaluations of the mathematical model. To overcome these challenges, we formulate and test a multilevel adaptive sparse Leja algorithm. At each level, adaptive sparse grid interpolation and quadrature are used to approximate the posterior and perform all quadrature operations, respectively. Specifically, our algorithm uses coarse discretizations of the underlying mathematical model to investigate the parameter space and to identify areas of high posterior probability. Adaptive sparse grid algorithms are then used to place points in these areas, and ignore other areas of small posterior probability. The points are weighted Leja points. As the model discretization is coarse, the construction of the sparse grid is computationally efficient. On this sparse grid, the posterior measure can be approximated accurately with few expensive, fine model discretizations. The efficiency of the algorithm can be enhanced further by exploiting more than two discretization levels. We apply the proposed multilevel adaptive sparse Leja algorithm in numerical experiments involving elliptic inverse problems in 2D and 3D space, in which we compare it with Markov chain Monte Carlo sampling and a standard multilevel approximation.

show abstract

Fast sampling of parameterised Gaussian random fields

Cited by 21 publications

References 65 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

On the Well-posedness of Bayesian Inverse Problems

Multilevel Adaptive Sparse Leja Approximations for Bayesian Inverse Problems

Contact Info

Product

Resources

About