Analysis of Boundary Effects on PDE-Based Sampling of Whittle--Matérn Random Fields

Khristenko, Ustim; Scarabosio, Laura; Swierczynski, Piotr; Ullmann, Elisabeth; Wohlmuth, Barbara

doi:10.1137/18m1215700

Cited by 52 publications

(42 citation statements)

References 57 publications

(101 reference statements)

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“…The variance with the SPDE method ( Figure 3B) showed instead a strong boundary effect, being significantly larger close to the boundary. This is a known effect of Neumann boundary conditions [12]. Finally, we computed the correlation with respect to the geodesic distance from point (0.25, 0.25) towards point (0.75, 0.25): see Figure 3C, reporting excellent agreement.…”

Section: Comparison Of the Two Methodsmentioning

confidence: 78%

On Sampling Spatially-Correlated Random Fields for Complex Geometries

Pezzuto

Quaglino

Potse

2019

Lecture Notes in Computer Science

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Extracting spatial heterogeneities from patient-specific data is challenging. In most cases, it is unfeasible to achieve an arbitrary level of detail and accuracy. This lack of perfect knowledge can be treated as an uncertainty associated with the estimated parameters and thus be modeled as a spatiallycorrelated random field superimposed to them. In order to quantify the effect of this uncertainty on the simulation outputs, it is necessary to generate several realizations of these random fields. This task is far from trivial, particularly in the case of complex geometries. Here, we present two different approaches to achieve this. In the first method, we use a stochastic partial differential equation, yielding a method which is general and fast, but whose underlying correlation function is not readily available. In the second method, we propose a geodesic-based modification of correlation kernels used in the truncated Karhunen-Loève expansion with pivoted Cholesky factorization, which renders the method efficient even for complex geometries, provided that the correlation length is not too small. Both methods are tested on a few examples and cardiac applications.

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Section: Comparison Of the Two Methodsmentioning

confidence: 78%

On Sampling Spatially-Correlated Random Fields for Complex Geometries

Pezzuto

Quaglino

Potse

2019

Lecture Notes in Computer Science

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“…such that , ( b ) is small) will be largely unaffected by the artificial boundaries. This point has been analyzed in mathematical detail in a recent article by Khristenko et al (2018). From Figure 7, it is interesting to notice that, apart from a few isolated points in the interior of the domain, the amplitude errors are negative; that is, the amplitude is mostly underestimated.…”

Section: Accuracy Of the Diagonal Elements Ofmentioning

confidence: 88%

“…such that c m , ℓ ( r b ) is small) will be largely unaffected by the artificial boundaries. This point has been analyzed in mathematical detail in a recent article by Khristenko et al ().…”

Section: Application To Unstructured Satellite Observationsmentioning

confidence: 99%

Modelling spatially correlated observation errors in variational data assimilation using a diffusion operator on an unstructured mesh

Guillet

Weaver

Vasseur

et al. 2019

Quart J Royal Meteoro Soc

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We propose a method for representing spatially correlated observation errors in variational data assimilation. The method is based on the numerical solution of a diffusion equation, a technique commonly used for representing spatially correlated background errors. The discretization of the pseudo‐time derivative of the diffusion equation is done implicitly using a backward Euler scheme. The solution of the resulting elliptic equation can be interpreted as a correlation operator whose kernel is a correlation function from the Matérn family. In order to account for the possibly heterogeneous distribution of observations, a spatial discretization technique based on the finite element method (FEM) is chosen where the observation locations are used to define the nodes of an unstructured mesh on which the diffusion equation is solved. By construction, the method leads to a convenient operator for the inverse of the observation‐error correlation matrix, which is an important requirement when applying it with standard minimization algorithms in variational data assimilation. Previous studies have shown that spatially correlated observation errors can also be accounted for by assimilating the observations together with their directional derivatives up to arbitrary order. In the continuous framework, we show that the two approaches are formally equivalent for certain parameter specifications. The FEM provides an appropriate framework for evaluating the derivatives numerically, especially when the observations are heterogeneously distributed. Numerical experiments are performed using a realistic data distribution from the Spinning Enhanced Visible and InfraRed Imager (SEVIRI). Correlations obtained with the FEM‐discretized diffusion operator are compared with those obtained using the analytical Matérn correlation model. The method is shown to produce an accurate representation of the target Matérn function in regions where the data are densely distributed. The presence of large gaps in the data distribution degrades the quality of the mesh and leads to numerical errors in the representation of the Matérn function. Strategies to improve the accuracy of the method in the presence of such gaps are discussed.

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“…In practice, equation (2.8) is solved on a bounded domain with arbitrary boundary conditions [25,31,41].…”

Section: Matérn Covariancementioning

confidence: 99%

A Statistical Framework for Generating Microstructures of Two-Phase Random Materials: Application to Fatigue Analysis

Khristenko¹,

Constantinescu²,

Tallec³

et al. 2020

Multiscale Model. Simul.

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Random microstructures of heterogeneous materials play a crucial role in the material macroscopic behavior and in predictions of its effective properties. A common approach to modeling random multiphase materials is to develop so-called surrogate models approximating statistical features of the material. However, the surrogate models used in fatigue analysis usually employ simple microstructure, consisting of ideal geometries such as ellipsoidal inclusions, which generally does not capture complex geometries. In this paper, we introduce a simple but flexible surrogate microstructure model for two-phase materials through a level-cut of a Gaussian random field with covariance of Matérn class. Such parametrization of the covariance function allows for the representation of a few key design parameters while representing the geometry of inclusions in a more general setting for a large class of random heterogeneous two-phase media. In addition to the traditional morphology descriptors such as porosity, size and aspect ratio, it provides control of the regularity of the inclusions interface and sphericity. These parameters are estimated from a small number of real material images using Bayesian inversion. An efficient process of evaluating the samples, based on the Fast Fourier Transform, makes possible the use of Monte-Carlo methods to estimate statistical properties for the quantities of interest in a given material class. We demonstrate the overall framework of the use of the surrogate material model in application to the uncertainty quantification in fatigue analysis, its feasibility and efficiency, and its role in the microstructure design.

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Analysis of Boundary Effects on PDE-Based Sampling of Whittle--Matérn Random Fields

Cited by 52 publications

References 57 publications

On Sampling Spatially-Correlated Random Fields for Complex Geometries

On Sampling Spatially-Correlated Random Fields for Complex Geometries

Modelling spatially correlated observation errors in variational data assimilation using a diffusion operator on an unstructured mesh

A Statistical Framework for Generating Microstructures of Two-Phase Random Materials: Application to Fatigue Analysis

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