IGA-based Multi-Index Stochastic Collocation for random PDEs on arbitrary domains

This paper proposes an extension of the Multi-Index Stochastic Collocation (MISC) method for forward uncertainty quantification (UQ) problems in computational domains of shape other than a square or cube, by exploiting isogeometric analysis (IGA) techniques. Introducing IGA solvers to the MISC algorithm is very natural since they are tensor-based PDE solvers, which are precisely what is required by the MISC machinery. Moreover, the combination-technique formulation of MISC allows the straight-forward reuse of existing implementations of IGA solvers. We present numerical results to showcase the effectiveness of the proposed approach.Comment: version 3, version after revisio

WENO schemes on arbitrary mixed-element unstructured meshes in three space dimensions

The paper extends weighted essentially non-oscillatory (WENO) methods to three dimensional mixed-element unstructured meshes, comprising tetrahedral, hexahedral, prismatic and pyramidal elements. Numerical results illustrate the convergence rates and non-oscillatory properties of the schemes for various smooth and discontinuous solutions test cases and the compressible Euler equations on various types of grids. Schemes of up to fifth order of spatial accuracy are considered

Interactive ray tracing for volume visualization

Journal ArticleWe present a brute-force ray tracing system for interactive volume visualization, The system runs on a conventional (distributed) shared-memory multiprocessor machine. For each pixel we trace a ray through a volume to compute the color for that pixel. Although this method has high intrinsic computational cost, its simplicity and scalability make it ideal for large datasets on current high-end parallel systems

Numerical solution of 3-D electromagnetic problems in exploration geophysics and its implementation on massively parallel computers

The growing significance, technical development and employment of electromagnetic (EM) methods in exploration geophysics have led to the increasing need for reliable and fast techniques of interpretation of 3-D EM data sets acquired in complex geological environments. The first and most important step to creating an inversion method is the development of a solver for the forward problem. In order to create an efficient, reliable and practical 3-D EM inversion, it is necessary to have a 3-D EM modelling code that is highly accurate, robust and very fast. This thesis focuses precisely on this crucial and very demanding step to building a 3-D EM interpretation method. The thesis presents as its main contribution a highly accurate, robust, very fast and extremely scalable numerical method for 3-D EM modelling in geophysics that is based on finite elements (FE) and designed to run on massively parallel computing platforms. Thanks to the fact that the FE approach supports completely unstructured tetrahedral meshes as well as local mesh refinements, the presented solver is able to represent complex geometries of subsurface structures very precisely and thus improve the solution accuracy and avoid misleading artefacts in images. Consequently, it can be successfully used in geological environments of arbitrary geometrical complexities. The parallel implementation of the method, which is based on the domain decomposition and a hybrid MPI-OpenMP scheme, has proved to be highly scalable - the achieved speed-up is close to the linear for more than a thousand processors. Thanks to this, the code is able to deal with extremely large problems, which may have hundreds of millions of degrees of freedom, in a very efficient way. The importance of having this forward-problem solver lies in the fact that it is now possible to create a 3-D EM inversion that can deal with data obtained in extremely complex geological environments in a way that is realistic for practical use in industry. So far, such imaging tool has not been proposed due to a lack of efficient, parallel FE solutions as well as the limitations of efficient solvers based on finite differences. In addition, the thesis discusses physical, mathematical and numerical aspects and challenges of 3-D EM modelling, which have been studied during my research in order to properly design the presented software for EM field simulations on 3-D areas of the Earth. Through this work, a physical problem formulation based on the secondary Coulomb-gauged EM potentials has been validated, proving that it can be successfully used with the standard nodal FE method to give highly accurate numerical solutions. Also, this work has shown that Krylov subspace iterative methods are the best solution for solving linear systems that arise after FE discretisation of the problem under consideration. More precisely, it has been discovered empirically that the best iterative method for this kind of problems is biconjugate gradient stabilised with an elaborate preconditioner. Since most commonly used preconditioners proved to be either unable to improve the convergence of the implemented solvers to the desired extent, or impractical in the parallel context, I have proposed a preconditioning technique for Krylov methods that is based on algebraic multigrid. Tests for various problems with different conductivity structures and characteristics have shown that the new preconditioner greatly improves the convergence of different Krylov subspace methods, which significantly reduces the total execution time of the program and improves the solution quality. Furthermore, the preconditioner is very practical for parallel implementation. Finally, it has been concluded that there are not any restrictions in employing classical parallel programming models, MPI and OpenMP, for parallelisation of the presented FE solver. Moreover, they have proved to be enough to provide an excellent scalability for it

Towards Data-Driven Large Scale Scientific Visualization and Exploration

Technological advances have enabled us to acquire extremely large datasets but it remains a challenge to store, process, and extract information from them. This dissertation builds upon recent advances in machine learning, visualization, and user interactions to facilitate exploration of large-scale scientific datasets. First, we use data-driven approaches to computationally identify regions of interest in the datasets. Second, we use visual presentation for effective user comprehension. Third, we provide interactions for human users to integrate domain knowledge and semantic information into this exploration process. Our research shows how to extract, visualize, and explore informative regions on very large 2D landscape images, 3D volumetric datasets, high-dimensional volumetric mouse brain datasets with thousands of spatially-mapped gene expression profiles, and geospatial trajectories that evolve over time. The contribution of this dissertation include: (1) We introduce a sliding-window saliency model that discovers regions of user interest in very large images; (2) We develop visual segmentation of intensity-gradient histograms to identify meaningful components from volumetric datasets; (3) We extract boundary surfaces from a wealth of volumetric gene expression mouse brain profiles to personalize the reference brain atlas; (4) We show how to efficiently cluster geospatial trajectories by mapping each sequence of locations to a high-dimensional point with the kernel distance framework. We aim to discover patterns, relationships, and anomalies that would lead to new scientific, engineering, and medical advances. This work represents one of the first steps toward better visual understanding of large-scale scientific data by combining machine learning and human intelligence

On the predictivity of pore-scale simulations : estimating uncertainties with multilevel Monte Carlo

A fast method with tunable accuracy is proposed to estimate errors and uncertainties in pore-scale and Digital Rock Physics (DRP) problems. The overall predictivity of these studies can be, in fact, hindered by many factors including sample heterogeneity, computational and imaging limitations, model inadequacy and not perfectly known physical parameters. The typical objective of pore-scale studies is the estimation of macroscopic effective parameters such as permeability, effective diffusivity and hydrodynamic dispersion. However, these are often non-deterministic quantities (i.e., results obtained for specific pore-scale sample and setup are not totally reproducible by another “equivalent” sample and setup). The stochastic nature can arise due to the multi-scale heterogeneity, the computational and experimental limitations in considering large samples, and the complexity of the physical models. These approximations, in fact, introduce an error that, being dependent on a large number of complex factors, can be modeled as random. We propose a general simulation tool, based on multilevel Monte Carlo, that can reduce drastically the computational cost needed for computing accurate statistics of effective parameters and other quantities of interest, under any of these random errors. This is, to our knowledge, the first attempt to include Uncertainty Quantification (UQ) in pore-scale physics and simulation. The method can also provide estimates of the discretization error and it is tested on three-dimensional transport problems in heterogeneous materials, where the sampling procedure is done by generation algorithms able to reproduce realistic consolidated and unconsolidated random sphere and ellipsoid packings and arrangements. A totally automatic workflow is developed in an open-source code [1], that include rigid body physics and random packing algorithms, unstructured mesh discretization, finite volume solvers, extrapolation and post-processing techniques. The proposed method can be efficiently used in many porous media applications for problems such as stochastic homogenization/upscaling, propagation of uncertainty from microscopic fluid and rock properties to macro-scale parameters, robust estimation of Representative Elementary Volume size for arbitrary physics

Efficient and robust monolithic finite element multilevel Krylov subspace solvers for the solution of stationary incompressible Navier-Stokes equations

Multigrid methods belong to the best-known methods for solving linear systems arising from the discretization of elliptic partial differential equations. The main attraction of multigrid methods is that they have an asymptotically meshindependent convergence behavior. Multigrid with Vanka (or local multilevel pressure Schur complement method) as smoother have been frequently used for the construction of very effcient coupled monolithic solvers for the solution of the stationary incompressible Navier-Stokes equations in 2D and 3D. However, due to its innate Gauß-Seidel/Jacobi character, Vanka has a strong influence of the underlying mesh, and therefore, coupled multigrid solvers with Vanka smoothing very frequently face convergence issues on meshes with high aspect ratios. Moreover, even on very nice regular grids, these solvers may fail when the anisotropies are introduced from the differential operator. In this thesis, we develop a new class of robust and efficient monolithic finite element multilevel Krylov subspace methods (MLKM) for the solution of the stationary incompressible Navier-Stokes equations as an alternative to the coupled multigrid-based solvers. Different from multigrid, the MLKM utilizes a Krylov method as the basis in the error reduction process. The solver is based on the multilevel projection-based method of Erlangga and Nabben, which accelerates the convergence of the Krylov subspace methods by shifting the small eigenvalues of the system matrix, responsible for the slow convergence of the Krylov iteration, to the largest eigenvalue. Before embarking on the Navier-Stokes equations, we first test our implementation of the MLKM solver by solving scalar model problems, namely the convection-diffusion problem and the anisotropic diffusion problem. We validate the method by solving several standard benchmark problems. Next, we present the numerical results for the solution of the incompressible Navier-Stokes equations in two dimensions. The results show that the MLKM solvers produce asymptotically mesh-size independent, as well as Reynolds number independent convergence rates, for a moderate range of Reynolds numbers. Moreover, numerical simulations also show that the coupled MLKM solvers can handle (both mesh and operator based) anisotropies better than the coupled multigrid solvers