R-3 Repository :: Browsing by Author "Riviere, Beatrice M."

Browsing by Author "Riviere, Beatrice M."

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A coupled finite volume and discontinuous Galerkin method for convection-diffusion problems
(2012) Yang, Xin; Riviere, Beatrice M.
This work formulates and analyzes a new coupled finite volume (FV) and discontinuous Galerkin (DG) method for convection-diffusion problems. DG methods, though costly, have proved to be accurate for solving convection-diffusion problems and capable of handling discontinuous and tensor coefficients. FV methods have proved to be very efficient but they are only of first order accurate and they become ineffective for tensor coefficient problems. The coupled method takes advantage of both the accuracy of DG methods in the regions containing heterogeneous coefficients and the efficiency of FV methods in other regions. Numerical results demonstrate that this coupled method is able to resolve complicated coefficient problems with a decreased computational cost compared to DG methods. This work can be applied to problems such as the transport of contaminant underground, the CO 2 sequestration and the transport of cells in the body.
A new filtration of the Magnus kernel
(2013-09-16) McNeill, Reagin; Harvey, Shelly; Cochran, Tim D.; Riviere, Beatrice M.
For a oriented genus g surface with one boundary component, S_g, the Torelli group is the group of orientation preserving homeomorphisms of S_g that induce the identity on homology. The Magnus representation of the Torelli group represents the action on F/F'' where F=π_1(S_g) and F'' is the second term of the derived series. I show that the kernel of the Magnus representation, Mag(S_g), is highly non-trivial and has a rich structure as a group. Specifically, I define an infinite filtration of Mag(S_g) by subgroups, called the higher order Magnus subgroups, M_k(S_g). I develop methods for generating nontrivial mapping classes in M_k(S_g) for all k and g≥2. I show that for each k the quotient M_k(S_g)/M_{k+1}(S_g) contains a subgroup isomorphic to a lower central series quotient of free groups E(g-1)_k/E(g-1)_{k+1}. Finally I show that for g≥3 the quotient M_k(S_g)/M_{k+1}(S_g) surjects onto an infinite rank torsion free abelian group. To do this, I define a Johnson-type homomorphism on each higher order Magnus subgroup quotient and show it has a highly non-trivial image.
Accelerated Discontinuous Galerkin Solvers with the Chebyshev Iterative Method on the Graphics Processing Unit
(2011) Tullius, Toni Kathleen; Riviere, Beatrice M.; Warburton, Tim
This work demonstrates implementations of the discontinuous Galerkin (DG) method on graphics processing units (GPU), which deliver improved computational time compared to the conventional central processing unit (CPU). The linear system developed when applying the DG method to an elliptic problem is solved using the GPU. The conjugate gradient (CG) method and the Chebyshev iterative method are the linear system solvers that are compared, to see which is more efficient when computing with the CPU's parallel architecture. When applying both methods, computational times decreased for large problems executed on the GPU compared to CPU; however, CG is the more efficient method compared to the Chebyshev iterative method. In addition, a constant-free upper bound for the DC spectrum applied to the elliptic problem is developed. Few previous works combine the DG method and the GPU. This thesis will provide useful guidelines for the numerical solution of elliptic problems using DG on the GPU.
An Approach for the Adaptive Solution of Optimization Problems Governed by Partial Diﬀerential Equations with Uncertain Coeﬃcients
(2012-09-05) Kouri, Drew; Heinkenschloss, Matthias; Sorensen, Danny C.; Riviere, Beatrice M.; Cox, Dennis D.
Using derivative based numerical optimization routines to solve optimization problems governed by partial differential equations (PDEs) with uncertain coefficients is computationally expensive due to the large number of PDE solves required at each iteration. In this thesis, I present an adaptive stochastic collocation framework for the discretization and numerical solution of these PDE constrained optimization problems. This adaptive approach is based on dimension adaptive sparse grid interpolation and employs trust regions to manage the adapted stochastic collocation models. Furthermore, I prove the convergence of sparse grid collocation methods applied to these optimization problems as well as the global convergence of the retrospective trust region algorithm under weakened assumptions on gradient inexactness. In fact, if one can bound the error between actual and modeled gradients using reliable and efficient a posteriori error estimators, then the global convergence of the proposed algorithm follows. Moreover, I describe a high performance implementation of my adaptive collocation and trust region framework using the C++ programming language with the Message Passing interface (MPI). Many PDE solves are required to accurately quantify the uncertainty in such optimization problems, therefore it is essential to appropriately choose inexpensive approximate models and large-scale nonlinear programming techniques throughout the optimization routine. Numerical results for the adaptive solution of these optimization problems are presented.
Complex flow and transport phenomena in porous media
(2010) Cesmelioglu, Aycil; Riviere, Beatrice M.
This thesis analyzes partial differential equations related to the coupled surface and subsurface flows and develops efficient high order discontinuous Galerkin (DG) methods to solve them numerically. Specifically, the coupling of the Navier-Stokes and the Darcy's equations, which is encountered in the environmental problem of groundwater contamination through lakes and rivers, is considered. Predicting accurately the transport of contaminants by this coupled flow is of great importance for the remediation strategies. The first part of this thesis analyzes a weak formulation of the time-dependent Navier-Stokes equation coupled with the Darcy's equation through the Beavers-Joseph-Saffman condition. The analysis changes depending on whether the inertial forces are included in the interface conditions or not. The inclusion of the inertial forces (Model I) remedies the difficulty in the analysis caused by the nonlinear convection term; however, it does not reflect the physical interactions on the interface correctly. Hence, I also analyze the weak problem by omitting these forces (Model II) which complicates the analysis and necessitates an extra small data condition. For Model I, a fully discrete scheme based on the DG method and the Crank-Nicolson method is introduced. The convergence of the scheme is proven with optimal error estimates. The second part couples the surface flow and a convection-diffusion type transport with miscible displacement in the subsurface. Initially, I consider the coupled stationary Stokes and Darcy's equations for the flow and establish the existence of a weak solution. Next, imposing additional assumptions on the data, I extend the result to the nonlinear case where the surface flow is given by the Navier-Stokes equation. The analysis also applies to the particular case where the flow is loosely coupled to the transport, that is, the velocity field obtained from the flow is an input for the transport equation. The flow is discretized by combinations of the continuous finite element method and the DG method whereas the discretization of the transport is done by a combined DG and backward Euler methods. The scheme yields optimal error estimates and its robustness for fractured porous media is shown by a numerical example.
Convergence Analysis of Discontinuous Galerkin Methods for Poroelasticity Equations
(2013-09-23) Tan, Jun; Riviere, Beatrice M.; Heinkenschloss, Matthias; Symes, William W.
This thesis analyzes a numerical method for solving the poroelasticity equations. The model incorporating the poroelasticity equations in this thesis can be applied in intestinal edema, which is a medical condition referring to the accumulation of excess fluid in the spaces between cells of intestinal wall tissue. The model has a dilatation term and can give a comprehensive prediction of pressure and displacement for intestinal edema. I approximate the pressure, displacement and dilatation by the discontinuous Galerkin method, which includes symmetric, nonsymmetric and incomplete interior penalty Galerkin cases. Moreover, in order to solve for the nonsymmetric case, I introduce an additional penalty term in the scheme. Theoretical convergence error estimates derived in a discrete-in-time setting show the a priori error can be bounded by some constant, which is related to the pressure, displacement, dilatation and the mesh size.
Coupling surface flow with porous media flow
(2010) Chidyagwai, Prince; Riviere, Beatrice M.
This thesis proposes a model for the interaction between ground flow and surface flow using a coupled system of the Navier-Stokes and Darcy equations. The coupling of surface flow with porous media flow has important applications in science and engineering. This work is motivated by applications to geo-sciences. This work couples the two flows using interface conditions that incorporate the continuity of the normal component, the balance of forces and the Beaver-Joseph-Saffman Law. The balance of forces condition can be written with or without inertial forces from the free fluid region. This thesis provides both theoretical and numerical analysis of the effect of the inertial forces on the model. Flow in porous media is often simulated over large domains in which the actual permeability is heterogeneous with discontinuities across the domain. The discontinuous Galerkin method is well suited to handle this problem. On the other hand, the continuous finite element is adequate for the free flow problems considered in this work. As a result this thesis proposes coupling the continuous finite element method in the free flow region with the discontinuous Galerkin method in the porous medium. Existence and uniqueness results of a weak solution and numerical scheme are proved. This work also provides derivations of optimal a priori error estimates for the numerical scheme. A two-grid approach to solving the coupled problem is analyzed. This method will decouple the problem naturally into two problems, one in the free flow domain and other in the porous medium. In applications for this model, it is often the case that the areas of interest (faults, kinks) in the porous medium are small compared to the rest of the domain. In view of this fact, the rest of the thesis is dedicated to a coupling of the Discontinuous Galerkin method in the problem areas with a cheaper method on the rest of the domain. The finite volume method will be coupled with the Discontinuous Galerkin method on parts of the domain on which the permeability field varies gradually to decrease the problem sizes and thus make the scheme more efficient.
Deformations of Hilbert Schemes of Points on K3 Surfaces and Representation Theory
(2014-04-22) Zhang, Letao; Hassett, Brendan E.; Hardt, Robert M.; Riviere, Beatrice M.
We study the cohomology rings of Kaehler deformations X of Hilbert schemes of points on K3 surfaces by representation theory. We compute the graded character formula of the Mumford - Tate group representation on the cohomology ring of X. Furthermore, we also study the Hodge structure of X, and find the generating series for deducting the number of canaonical Hodge classes in the middle cohomology.
Density Functional Theory Study of Microstructure and Phase Behavior of Stimuli-Responsive Polymer Brushes
(2013-12-04) Gong, Kai; Chapman, Walter G.; Verduzco, Rafael; Riviere, Beatrice M.
Stimuli-responsive polymer materials can change their structure and physical properties drastically on external signals like a change in temperature, solvent properties (pH, ionic strength), the magnetic or electrical field etc. Such "smart" polymer materials play an important role in various fields such as biology, medicine, and soft materials. However, it is a great challenge to investigate such "smart" polymer materials due to highly inhomogeneous structure at the molecular scale and the complex interactions. In this thesis, we have systematically studied three common types of stimuli-responsive polymer brushes such as temperature responsive polymer brushes, copolymer brushes, and mixed polymer brushes by using classical density functional theory. We find a surface outer layer switch for both copolymer brushes and mixed polymer brushes with a selective solvent. Without using any temperature-dependent parameter, our theory successfully captures the lower critical solution temperature behavior of the associating polymer brushes. Related parameters such as molecular weight, grafting density, and solvent properties that affect the phase behavior of these stimuli-responsive polymer brushes have been also investigated. Qualitatively consistent with experimental observations, our results provide physical insight and helpful guidance for the experimental design of such stimuli-responsive polymer materials.
Discontinuous Galerkin formulation for multi-component multiphase flow
(2010) Ho, Christina; Riviere, Beatrice M.
The understanding of multiphase multi-component transport in capillary porous media plays an important role in scientific and engineering disciplines such as the petroleum and environmental industries. The two most commonly used tools to model multiphase multi-component flow are finite difference and finite volume methods. While these are well-established methods, they either fail to provide stability on unstructured meshes or they yield low order approximation. In this thesis, a presentation of both fully coupled and sequential discontinuous Galerkin (DG) formulations for the multiphase multi-component flow is given. Two physical models are examined: the black oil model and the CO2 sequestration model. The attractive attribute of using DG is that it permits the use of unstructured meshes while maintaining high order accuracy. Furthermore, the method can be structured to ensure mass conservation, which is another appealing feature when one is dealing with fluid dynamic problems.
Discontinuous Galerkin Methods for Elliptic Partial Differential Equations with Random Coefficients
(2011) Liu, Kun; Riviere, Beatrice M.
This thesis proposes and analyses two numerical methods for solving elliptic partial differential equations with random coefficients. The stochastic problem is first transformed into a parametrized one by the use of the Karhunen--Loève expansion. This new problem is then discretized by the discontinuous Galerkin (DG) method. A priori error estimate in the energy norm for the stochastic discontinuous Galerkin solution is derived. In addition, the expected value of the numerical error is theoretically bounded in the energy norm and the L2 norm. In the second approach, the Monte Carlo method is used to generate independent identically distributed realizations of the stochastic coefficients. The resulting deterministic problems are solved by the DG method. Next, estimates are obtained for the error between the average of these approximate solutions and the expected value of the exact solution. The Monte Carlo discontinuous Galerkin method is tested numerically on several examples. Results show that the nonsymmetric DG method is stable independently of meshes and the value of penalty parameter. Symmetric and incomplete DG methods are stable only when the penalty parameter is large enough. Finally, comparisons with the Monte Carlo finite element method and the Monte Carlo discontinuous Galerkin method are presented for several cases.
Discontinuous Galerkin Methods for Parabolic Partial Differential Equations with Random Input Data
(2013-09-16) Liu, Kun; Riviere, Beatrice M.; Heinkenschloss, Matthias; Symes, William W.; Vannucci, Marina
This thesis discusses and develops one approach to solve parabolic partial differential equations with random input data. The stochastic problem is firstly transformed into a parametrized one by using finite dimensional noise assumption and the truncated Karhunen-Loeve expansion. The approach, Monte Carlo discontinuous Galerkin (MCDG) method, randomly generates $M$ realizations of uncertain coefficients and approximates the expected value of the solution by averaging M numerical solutions. This approach is applied to two numerical examples. The first example is a two-dimensional parabolic partial differential equation with random convection term and the second example is a benchmark problem coupling flow and transport equations. I first apply polynomial kernel principal component analysis of second order to generate M realizations of random permeability fields. They are used to obtain M realizations of random convection term computed from solving the flow equation. Using this approach, I solve the transport equation M times corresponding to M velocity realizations. The MCDG solution spreads toward the whole domain from the initial location and the contaminant does not leave the initial location completely as time elapses. The results show that MCDG solution is realistic, because it takes the uncertainty in velocity fields into consideration. Besides, in order to correct overshoot and undershoot solutions caused by the high level of oscillation in random velocity realizations, I solve the transport equation on meshes of finer resolution than of the permeability, and use a slope limiter as well as lower and upper bound constraints to address this difficulty. Finally, future work is proposed.
Discrete Search Optimization for Real-Time Path Planning in Satellites
(2012-09-05) Mays, Millie; Heinkenschloss, Matthias; Symes, William W.; Riviere, Beatrice M.; Bedrossian, Nazareth
This study develops a discrete search-based optimization method for path planning in a highly nonlinear dynamical system. The method enables real-time trajectory improvement and singular configuration avoidance in satellite rotation using Control Moment Gyroscopes. By streamlining a legacy optimization method and combining it with a local singularity management scheme, this optimization method reduces the computational burden and advances the capability of satellites to make autonomous look-ahead decisions in real-time. Current optimization methods plan offline before uploading to the satellite and experience high sensitivity to disturbances. Local methods confer autonomy to the satellite but use only blind decision-making to avoid singularities. This thesis' method seeks near-optimal trajectories which balance between the optimal trajectories found using computationally intensive offline solvers and the minimal computational burden of non-optimal local solvers. The new method enables autonomous guidance capability for satellites using discretization and stage division to minimize the computational burden of real-time optimization.
gNek: A GPU Accelerated Incompressible Navier Stokes Solver
(2013-09-16) Stilwell, Nichole; Warburton, Timothy; Riviere, Beatrice M.; Embree, Mark
This thesis presents a GPU accelerated implementation of a high order splitting scheme with a spectral element discretization for the incompressible Navier Stokes (INS) equations. While others have implemented this scheme on clusters of processors using the Nek5000 code, to my knowledge this thesis is the first to explore its performance on the GPU. This work implements several of the Nek5000 algorithms using OpenCL kernels that efficiently utilize the GPU memory architecture, and achieve massively parallel on chip computations. These rapid computations have the potential to significantly enhance computational fluid dynamics (CFD) simulations that arise in areas such as weather modeling or aircraft design procedures. I present convergence results for several test cases including channel, shear, Kovasznay, and lid-driven cavity flow problems, which achieve the proven convergence results.
High order discontinuous Galerkin methods for simulating miscible displacement process in porous media with a focus on minimal regularity
(2015-04-20) Li, Jizhou; Riviere, Beatrice M.; Symes, William; Hirasaki, George; Warburton, Timothy; Heinkenschloss, Matthias
In my thesis, I formulate, analyze and implement high order discontinuous Galerkin methods for simulating miscible displacement in porous media. The analysis concerning the stability and convergence under the minimal regularity assumption is established to provide theoretical foundations for using discontinuous Galerkin discretization to solve miscible displacement problems. The numerical experiments demonstrate the robustness and accuracy of the proposed methods. The performance study for large scale simulations with highly heterogeneous porous media suggests strong scalability which indicates the efficiency of the numerical algorithm. The simulations performed using the algorithms for physically unstable flow show that higher order methods proposed in thesis are more suitable for simulating such phenomenon than the commonly used cell-center finite volume method.
Hybridizable Discontinuous Galerkin Methods for Flow and Transport: Applications, Solvers, and High Performance Computing
(2019-04-16) Fabien, Maurice S; Riviere, Beatrice M.; Knepley, Matthew G.
This thesis proposal explores e cient computational methods for the approximation of solutions to partial di erential equations that model ow and transport phenomena in porous media. These problems can be challenging to solve as the governing equations are coupled, nonlinear, and material properties are often highly varying and discontinuous. The high-order implicit hybridizable discontinuous method (HDG) is utilized for the discretization, which signi cantly reduces the computational cost. To our knowledge, HDG methods have not been previously applied to this class of complex problems in porous media. The HDG method is high-order accurate, locally mass-conservative, allows us to use unstructured complicated meshes, and enables the use of static condensation. We demonstrate that the HDG method is able to e ciently generate high- delity simulations of ow and transport phenomena in porous media. Several challenging benchmarks are used to verify and validate the method in heterogeneous porous media. High-order methods give rise to less sparse discretization matrices, which is problematic for linear solvers. To address the issue of less sparse discretization matrices (compared to low-order methods), we develop and deployed a novel nested multigrid method. It is based on a combination of p-multgrid, h-multigrid and algebraic multigrid. The method is demonstrated to be algorithmically e cient, achieving convergences rates of at most 0:2. We also show how to implement the multigrid technique in many-core parallel architectures. Parallel computing is a critical step in the simulation process, as it allows us to consider larger problems, and potentially generate simulations faster. Traditional performance measures like FLOPs or run-time are not entirely appropriate for nite element problems, as they ignore solution accuracy. A new accuracy-inclusive performance measure has been investigated as a part of my research. This performance measure, called the Time-Accuracy-Size spectrum (TAS), allows us to have a more complete assessment of how e cient our algorithms are. Utilizing TAS also enables a systematic way of determining which discretization is best suited for a given application.
Identifying ECG Clusters in Congenital Heart Disease
(2015-04-23) Hendryx, Emily; Riviere, Beatrice M.; Rusin, Craig; Cox, Steven; Dabaghian, Yuri
This thesis presents a method of clustering ECG morphologies for the identification of individual ECG features in congenital heart disease. Clustering is performed on the computed heart dipole moment magnitude using k-medoids clustering with variants of dynamic time warping. The method is applied to both synthetic data and patient data with different parameter values for classic and derivative dynamic time warping. A deterministic k-medoids algorithm demonstrates poor clustering results on both data sets, but an iterative approach with random initialization shows marked improvement. The synthetic data clusters are generally well-defined with the expected number of clusters. Though the patient data derivative results are inconclusive, upon closer examination, the clustering results from classic dynamic time warping with a small warping window seem sensible. Through this project, the groundwork is laid for the future classification of ECG recordings and the development of predictive models in patients with congenital heart disease.
Inverse source problems for time-dependent radiative transport
(2014-03-20) Acosta Valenzuela, Sebastian; Borcea, Liliana; Riviere, Beatrice M.; Symes, William W.; Hardt, Robert M.; Alonso, Ricardo J
In the first part of this thesis, I develop a time reversal method for the radiative transport equation to solve two problems: an inverse problem for the recovery of an initial condition from boundary measurements, and the exact boundary controllability of the transport field with finite steering time. Absorbing and scattering effects, modeled by coefficients with low regularity, are incorporated in the formulation of these problems. This time reversal approach leads to a convergent iterative procedure to reconstruct the initial condition provided that the scattering coefficient is sufficiently small. Then, using duality arguments, I show that the solvability of the inverse problem leads to exact controllability of the transport field. The solution approach to both of these problems may have applications in areas such as optical imaging and optimization of radiation delivery. The second portion of the work is dedicated to the simultaneous recovery of a source of the form "s(t,x,d) f(x)" (with "s" known) and an isotropic initial condition "u0(x)", using the single measurement induced by these data. This result is part of an effort to reconstruct optical properties using unknown illumination embedded in the unknown medium. More precisely, based on exact boundary controllability, I derive a system of equations for the unknown terms "f" and "u0". The system is shown to be of Fredholm type if "s" satisfies a certain positivity condition. This condition requires that the radiation visits the region over which "f" is to be recovered. I show that for generic term "s" and weakly absorbing media, the inverse problem is well-posed.
Locally Mass-Conservative Method With Discontinuous Galerkin In Time For Solving Miscible Displacement Equations Under Low Regularity
(2013-09-16) Li, Jizhou; Riviere, Beatrice M.; Heinkenschloss, Matthias; Symes, William W.; Warburton, Tim
The miscible displacement equations provide the mathematical model for simulating the displacement of a mixture of oil and miscible fluid in underground reservoirs during the Enhance Oil Recovery(EOR) process. In this thesis, I propose a stable numerical scheme combining a mixed finite element method and space-time discontinuous Galerkin method for solving miscible displacement equations under low regularity assumption. Convergence of the discrete solution is investigated using a compactness theorem for functions that are discontinuous in space and time. Numerical experiments illustrate that the rate of convergence is improved by using a high order time stepping method. For petroleum engineers, it is essential to compute finely detailed fluid profiles in order to design efficient recovery procedure thereby increase production in the EOR process. The method I propose takes advantage of both high order time approximation and discontinuous Galerkin method in space and is capable of providing accurate numerical solutions to assist in increasing the production rate of the miscible displacement oil recovery process.
Modeling Laser-Induced Thermotherapy in Biological Tissue
(2015-05-01) Wang, Zheng; Warburton, Timothy; Riviere, Beatrice M.; Symes, William W.
This thesis studies simulations of laser-induced thermotherapy (LITT), a minimally-invasive procedure which ablates cancerous tissue using laser heating. In order to predict this procedure, mathematical models are used to assist in treatment. By simulating the laser heating of tissue, surgeons may estimate regions of tissue exceeding a thermal damage threshold. One important component of the LITT model is laser simulation, which is typically characterized by the radiative transfer equation (RTE). The RTE is a time-dependent integro-differential equation with variables in both angular and physical spaces. In this thesis, we conduct numerical experiments using both discrete ordinate and Galerkin methods. The former discretizes a finite number of directions using finite difference methods, while the latter employs continuous functions for both angular and spatial discretizations. Numerical results indicate that the numerical errors in both methods are dominated by the error in the less restricted space. In addition, the discrete ordinate method suffers from the ray effect, in which isotropic scattering is violated, whereas in the Galerkin method, the ray effect is not observed.

Browsing by Author "Riviere, Beatrice M."

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