Adaptive stochastic Galerkin FEM for lognormal coefficients in hierarchical tensor representations

Stochastic Galerkin methods for non-affine coefficient representations are known to cause major difficulties from theoretical and numerical points of view. In this work, an adaptive Galerkin FE method for linear parametric PDEs with lognormal coefficients discretized in Hermite chaos polynomials is derived. It employs problem-adapted function spaces to ensure solvability of the variational formulation. The inherently high computational complexity of the parametric operator is made tractable by using hierarchical tensor representations. For this, a new tensor train format of the lognormal coefficient is derived and verified numerically. The central novelty is the derivation of a reliable residual-based a posteriori error estimator. This can be regarded as a unique feature of stochastic Galerkin methods. It allows for an adaptive algorithm to steer the refinements of the physical mesh and the anisotropic Wiener chaos polynomial degrees. For the evaluation of the error estimator to become feasible, a numerically efficient tensor format discretization is developed. Benchmark examples with unbounded lognormal coefficient fields illustrate the performance of the proposed Galerkin discretization and the fully adaptive algorithm

A partitioned model order reduction approach to rationalise computational expenses in multiscale fracture mechanics

We propose in this paper an adaptive reduced order modelling technique based on domain partitioning for parametric problems of fracture. We show that coupling domain decomposition and projection-based model order reduction permits to focus the numerical effort where it is most needed: around the zones where damage propagates. No \textit{a priori} knowledge of the damage pattern is required, the extraction of the corresponding spatial regions being based solely on algebra. The efficiency of the proposed approach is demonstrated numerically with an example relevant to engineering fracture.Comment: Submitted for publication in CMAM

A streamline derivative POD-ROM for advection-diffusion-reaction equations

We introduce a new streamline derivative projection-based closure modeling strategy for the numerical stabilization of Proper Orthogonal Decomposition-Reduced Order Models (PODROM). As a first preliminary step, the proposed model is analyzed and tested for advection-dominated advection-diffusion-reaction equations. In this framework, the numerical analysis for the Finite Element (FE) discretization of the proposed new POD-ROM is presented, by mainly deriving the corresponding error estimates. Numerical tests for advection-dominated regime show the efficiency of the proposed method, as well the increased accuracy over the standard POD-ROM that discovers its well-known limitations very soon in the numerical settings considered, i.e. for low diffusion coefficients.Nous introduisons une nouvelle stratégie de modélisation de type streamline derivative basée sur projection pour la stabilisation numérique de modèles d’ordre réduit de type POD (PODROM). Comme première étape préliminaire, le modèle proposé est analysé et testé pour les équations d’advection-diffusion-réaction dominées par l’advection. Dans ce cadre, l’analyse numérique de la discrétisation par éléments finis (FE) du nouveau POD-ROM proposé est présentée, en dérivant principalement les estimations d’erreur correspondantes. Des tests numériques pour le régime dominé par l’advection montrent l’efficacité de la méthode proposée, ainsi que la précision accrue par rapport à la méthode POD-ROM standard qui d´ecouvre très rapidement ses limites bien connues dans le cas des paramètres numériques considérés, c’est-à-dire pour de faibles coefficients de diffusion

Stochastic kinetic models: Dynamic independence, modularity and graphs

The dynamic properties and independence structure of stochastic kinetic models (SKMs) are analyzed. An SKM is a highly multivariate jump process used to model chemical reaction networks, particularly those in biochemical and cellular systems. We identify SKM subprocesses with the corresponding counting processes and propose a directed, cyclic graph (the kinetic independence graph or KIG) that encodes the local independence structure of their conditional intensities. Given a partition $[A,D,B]$ of the vertices, the graphical separation $A\perp B|D$ in the undirected KIG has an intuitive chemical interpretation and implies that $A$ is locally independent of $B$ given $A\cup D$. It is proved that this separation also results in global independence of the internal histories of $A$ and $B$ conditional on a history of the jumps in $D$ which, under conditions we derive, corresponds to the internal history of $D$. The results enable mathematical definition of a modularization of an SKM using its implied dynamics. Graphical decomposition methods are developed for the identification and efficient computation of nested modularizations. Application to an SKM of the red blood cell advances understanding of this biochemical system.Comment: Published in at http://dx.doi.org/10.1214/09-AOS779 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Spectral-Element and Adjoint Methods in Seismology

We provide an introduction to the use of the spectral-element method (SEM) in seismology. Following a brief review of the basic equations that govern seismic wave propagation, we discuss in some detail how these equations may be solved numerically based upon the SEM to address the forward problem in seismology. Examples of synthetic seismograms calculated based upon the SEM are compared to data recorded by the Global Seismographic Network. Finally, we discuss the challenge of using the remaining differences between the data and the synthetic seismograms to constrain better Earth models and source descriptions. This leads naturally to adjoint methods, which provide a practical approach to this formidable computational challenge and enables seismologists to tackle the inverse problem