Multilevel Methods for Uncertainty Quantification of Elliptic PDEs with Random Anisotropic Diffusion

We consider elliptic diffusion problems with a random anisotropic diffusion coefficient, where, in a notable direction given by a random vector field, the diffusion strength differs from the diffusion strength perpendicular to this notable direction. The Karhunen-Lo\`eve expansion then yields a parametrisation of the random vector field and, therefore, also of the solution of the elliptic diffusion problem. We show that, given regularity of the elliptic diffusion problem, the decay of the Karhunen-Lo\`eve expansion entirely determines the regularity of the solution's dependence on the random parameter, also when considering this higher spatial regularity. This result then implies that multilevel collocation and multilevel quadrature methods may be used to lessen the computation complexity when approximating quantities of interest, like the solution's mean or its second moment, while still yielding the expected rates of convergence. Numerical examples in three spatial dimensions are provided to validate the presented theory

Multilevel quadrature for elliptic problems on random domains by the coupling of FEM and BEM

Elliptic boundary value problems which are posed on a random domain can be mapped to a fixed, nominal domain. The randomness is thus transferred to the diffusion matrix and the loading. While this domain mapping method is quite efficient for theory and practice, since only a single domain discretisation is needed, it also requires the knowledge of the domain mapping. However, in certain applications, the random domain is only described by its random boundary, while the quantity of interest is defined on a fixed, deterministic subdomain. In this setting, it thus becomes necessary to compute a random domain mapping on the whole domain, such that the domain mapping is the identity on the fixed subdomain and maps the boundary of the chosen fixed, nominal domain on to the random boundary. To overcome the necessity of computing such a mapping, we therefore couple the finite element method on the fixed subdomain with the boundary element method on the random boundary. We verify the required regularity of the solution with respect to the random domain mapping for the use of multilevel quadrature, derive the coupling formulation, and show by numerical results that the approach is feasible

Multilevel methods for uncertainty quantification of elliptic PDEs with random anisotropic diffusion

We consider elliptic diffusion problems with a random anisotropic diffusion coefficient, where, in a notable direction given by a random vector field, the diffusion strength differs from the diffusion strength perpendicular to this notable direction. The Karhunen-Loève expansion then yields a parametrisation of the random vector field and, therefore, also of the solution of the elliptic diffusion problem. We show that, given regularity of the elliptic diffusion problem, the decay of the Karhunen-Loève expansion entirely determines the regularity of the solution's dependence on the random parameter, also when considering this higher spatial regularity. This result then implies that multilevel quadrature methods may be used to lessen the computation complexity when approximating quantities of interest, like the solution's mean or its second moment, while still yielding the expected rates of convergence. Numerical examples in three spatial dimensions are provided to validate the presented theory

Quantifying domain uncertainty in linear elasticity

The present article considers the quantification of uncertainty for the equations of linear elasticity on random domains. To this end, we model the random domains as the images of some given fixed, nominal domain under random domain mappings, which are defined by a Karhunen-Loève expansion. We then prove the analytic regularity of the random solution with respect to the countable random input parameters which enter the problem through the Karhunen-Loève expansion of the random domain mappings. In particular, we provide appropriate bounds on arbitrary derivatives of the random solution with respect to those input parameters, which enable the use of state-of-the-art quadrature methods to compute deterministic statistics such as the mean and variance of quantities of interest such as the random solution itself or the random von Mises stress as integrals over the countable random input parameters in a dimensionally robust way. Numerical examples qualify and quantify the theoretical findings

Identification of sparsely representable diffusion parameters in elliptic problems

We consider the task of estimating the unknown diffusion parameter in an elliptic PDE as a model problem to develop and test the effectiveness and robustness to noise of reconstruction schemes with sparsity regularisation. To this end, the model problem is recasted as a nonlinear optimal control problem, where the unknown diffusion parameter is modelled using a linear combination of the elements of a known bounded sequence of functions with unknown coefficients. We show that the regularisation of this nonlinear optimal control problem using a weighted $\ell^1$-norm has minimisers that are finitely supported. We then propose modifications of well-known algorithms (ISTA and FISTA) to find a minimiser of this weighted $\ell^1$-norm regularised nonlinear optimal control problem that account for the fact that in general the coefficients need to be $\ell^1$ and not only $\ell^2$ summable. We also introduce semismooth methods (ASISTA and FASISTA) for finding a minimiser, which locally use Gauss-Newton type surrogate models that additionally are stabilised by means of a Levenberg-Marquardt type approach. Our numerical examples show that the regularisation with the weighted $\ell^1$-norm indeed does make the estimation more robust with respect to noise. Moreover, the numerical examples also demonstrate that the ASISTA and FASISTA methods are quite efficient, outperforming both ISTA and FISTA

In vitro tooth cleaning efficacy of manual toothbrushes around brackets

The purpose of this laboratory study was to assess the potential cleaning efficacy of nine different toothbrushes around brackets in vitro. Standard and Mini Diamond™ brackets were fixed on coloured teeth in a special model, coated with white titanium oxide, brushed in a machine with different manual toothbrushes (three different types: planar, staged, and v-shaped bristle field), and tested with a horizontal motion for 1 minute. After brushing, the teeth were scanned and the black surfaces were planimetrically assessed using a grey scale. Tooth areas which were black again after brushing indicated tooth surface contact of the filaments. The remaining white tooth areas around the brackets indicated ‘plaque-retentive' niches. Statistical analysis was carried out using the Kruskal-Wallis one-way test of variance for individual comparison. Bonferroni adjustment was used for multiple testing, and comparison of bracket size with Wilcoxon signed rank test. In the most critical area of 2 mm around the brackets, there was no statistically significant difference between the different toothbrushes evaluated. The untouched area ranged from 11 to 26 per cent of the initially whitened tooth surface. By pooling the toothbrushes according to their design, the median cleaning efficacy of the v-shaped (73.1 per cent) and staged (75.6 per cent) toothbrushes resulted in significantly superior cleaning efficacy than planar toothbrushes (60.7 per cent) for standard brackets. For mini bracket type, staged toothbrushes showed a significantly better mean cleaning efficacy (77.8 per cent) than planar (65 per cent) and v-shaped (72.4 per cent) toothbrushes. Staged and v-shaped brush designs resulted in superior cleaning efficacy of teeth with fixed orthodontic attachments than toothbrushes with a planar bristle field. None of the tested toothbrushes showed a consistent, significantly higher cleaning efficacy than the others in this in vitro experimen

Space-time multilevel quadrature methods and their application for cardiac electrophysiology

We present a novel approach which aims at high-performance uncertainty quantification for cardiac electrophysiology simulations. Employing the monodomain equation to model the transmembrane potential inside the cardiac cells, we evaluate the effect of spatially correlated perturbations of the heart fibers on the statistics of the resulting quantities of interest. Our methodology relies on a close integration of multilevel quadrature methods, parallel iterative solvers and space-time finite element discretizations, allowing for a fully parallelized framework in space, time and stochastics. Extensive numerical studies are presented to evaluate convergence rates and to compare the performance of classical Monte Carlo methods such as standard Monte Carlo (MC) and quasi-Monte Carlo (QMC), as well as multilevel strategies, i.e. multilevel Monte Carlo (MLMC) and multilevel quasi-Monte Carlo (MLQMC) on hierarchies of nested meshes. Finally, we employ a recently suggested variant of the multilevel approach for non-nested meshes to deal with a realistic heart geometry

Islet Endothelial Activation and Oxidative Stress Gene Expression Is Reduced by IL-1Ra Treatment in the Type 2 Diabetic GK Rat

Inflammation followed by fibrosis is a component of islet dysfunction in both rodent and human type 2 diabetes. Because islet inflammation may originate from endothelial cells, we assessed the expression of selected genes involved in endothelial cell activation in islets from a spontaneous model of type 2 diabetes, the Goto-Kakizaki (GK) rat. We also examined islet endotheliuml/oxidative stress (OS)/inflammation-related gene expression, islet vascularization and fibrosis after treatment with the interleukin-1 (IL-1) receptor antagonist (IL-1Ra)

Multilevel quadrature for elliptic problems on random domains by the coupling of FEM and BEM

Elliptic boundary value problems which are posed on a random domain can be mapped to a fixed, nominal domain. The randomness is thus transferred to the diffusion matrix and the loading. While this domain mapping method is quite efficient for theory and practice, since only a single domain discretisation is needed, it also requires the knowledge of the domain mapping. However, in certain applications, the random domain is only described by its random boundary, while the quantity of interest is defined on a fixed, deterministic subdomain. In this setting, it thus becomes necessary to compute a random domain mapping on the whole domain, such that the domain mapping is the identity on the fixed subdomain and maps the boundary of the chosen fixed, nominal domain on to the random boundary. To overcome the necessity of computing such a mapping, we therefore couple the finite element method on the fixed subdomain with the boundary element method on the random boundary. We verify on one hand the regularity of the solution with respect to the random domain mapping required for many multilevel quadrature methods, such as the multilevel quasi-Monte Carlo quadrature using Halton points, the multilevel sparse anisotropic Gauss-Legendre and Clenshaw-Curtis quadratures and multilevel interlaced polynomial lattice rules. On the other hand, we derive the coupling formulation and show by numerical results that the approach is feasible

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