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Efficient solvers for hybridized three-field mixed finite element coupled poromechanics
We consider a mixed hybrid finite element formulation for coupled poromechanics. A stabilization strategy based on a macro-element approach is advanced to eliminate the spurious pressure modes appearing in undrained/incompressible conditions. The efficient solution of the stabilized mixed hybrid block system is addressed by developing a class of block triangular preconditioners based on a Schur-complement approximation strategy. Robustness, computational efficiency and scalability of the proposed approach are theoretically discussed and tested using challenging benchmark problems on massively parallel architectures.
An aggregation-based nonlinear multigrid solver for two-phase flow and transport in porous media
A nonlinear multigrid solver for two-phase flow and transport in a mixed fractional-flow velocity-pressure-saturation formulation is proposed. The solver, which is under the framework of the full approximation scheme (FAS), extends our previous work on nonlinear multigrid for heterogeneous diffusion problems. The coarse spaces in the multigrid hierarchy are constructed by first aggregating degrees of freedom, and then solving some local flow problems. The mixed formulation and the choice of coarse spaces allow us to assemble the coarse problems without visiting finer levels during the solving phase, which is crucial for the scalability of multigrid methods. Specifically, a natural generalization of the upwind flux can be evaluated directly on coarse levels using the precomputed coarse flux basis vectors. We find that the resulting solver is applicable to problems discretized on general unstructured grids. The performance of the proposed nonlinear multigrid solver in comparison with the standard single level Newton's method is demonstrated through challenging numerical examples. It is observed that the proposed solver is robust for highly nonlinear problems and clearly outperforms Newton's method in the case of high Courant-Friedrichs-Lewy (CFL) numbers.
Multigrid reduction preconditioning framework for coupled processes in porous and fractured media
Not Available
Finite-size effects in periodic coupled cluster calculations
Here, we provide the first rigorous study of the finite-size error in the simplest and representative coupled cluster theory, namely the coupled cluster doubles (CCD) theory, for gapped periodic systems. Given exact Hartree-Fock orbitals and their corresponding orbital energies, we demonstrate that the correlation energy obtained from the approximate CCD method, after a finite number of fixed-point iterations over the amplitude equation, exhibits a finite-size error scaling as $\mathcal{O}(N^{-\frac{1}{3}}_k)$. Here $N_k$ is the number of discretization points in the Brillouin zone and characterizes the system size. Under additional assumptions ensuring the convergence of the fixed-point iterations, we demonstrate that the CCD correlation energy also exhibits a finite-size error scaling as $\mathcal{O}(N^{-\frac{1}{3}}_k)$. Our analysis shows that the dominant error lies in the coupled cluster amplitude calculation, and the convergence of the finite-size error in energy calculation can be boosted to $\mathcal{O}(N^{-1}_k)$ with accurate amplitudes. This also provides the first proof of the scaling of the finite-size error in the third order Møller-Plesset perturbation theory (MP3) for periodic systems.
Tsirelson's problem and asymptotically commuting unitary matrices
In this paper, we consider quantum correlations of bipartite systems having a slight interaction, and reinterpret Tsirelson's problem (and hence Kirchberg's and Connes's conjectures) in terms of finite-dimensional asymptotically commuting positive operator valued measures. We also consider the systems of asymptotically commuting unitary matrices and formulate the Stronger Kirchberg Conjecture.
Combinatorial theory of the semiclassical evaluation of transport moments II: Algorithmic approach for moment generating functions
Electronic transport through chaotic quantum dots exhibits universal behaviour which can be understood through the semiclassical approximation. Within the approximation, calculation of transport moments reduces to codifying classical correlations between scattering trajectories. These can be represented as ribbon graphs and we develop an algorithmic combinatorial method to generate all such graphs with a given genus. This provides an expansion of the linear transport moments for systems both with and without time reversal symmetry. The computational implementation is then able to progress several orders further than previous semiclassical formulae as well as those derived from an asymptotic expansion of random matrix results. The patterns observed also suggest a general form for the higher orders.
An index for two-dimensional SPT states
We define an index for 2D G-invariant invertible states of bosonic lattice systems in the thermodynamic limit for a finite symmetry group G with a unitary action. Furthermore, we show that this index is an invariant of the symmetry protected phase.
Novel superposed kinklike and pulselike solutions for several nonlocal nonlinear equations
In this work, we show that a number of nonlocal nonlinear equations, including the Ablowitz–Musslimani and Yang variant of the nonlocal nonlinear Schrödinger (NLS) equation, the nonlocal modified Korteweg de Vries (mKdV) equation, and the nonlocal Hirota equation, admit novel kinklike and pulselike superposed periodic solutions. Furthermore, we show that the nonlocal mKdV equation also admits the superposed (hyperbolic) kink–antikink solution. In addition, we show that while the nonlocal Ablowitz–Musslimani variant of the NLS admits complex parity-time reversal-invariant kink and pulse solutions, neither the local NLS nor the Yang variant of the nonlocal NLS admits such solutions. Finally, except for the Yang variant of the nonlocal NLS, we show that the other three nonlocal equations admit both the kink and pulse solutions in the same model.
Some properties of three-dimensional Klein polyhedra
We study properties of three-dimensional Klein polyhedra.
Particle-hole symmetry and composite fermions in fractional quantum Hall states
We study fractional quantum Hall states at filling fractions in the Jain sequences using the framework of composite Dirac fermions. Synthesizing previous work, we write down an effective field theory consistent with all symmetry requirements, including Galilean invariance and particle-hole symmetry. Employing a Fermi liquid description, we demonstrate the appearance of the Girvin-Macdonlald-Platzman algebra and compute the dispersion relation of neutral excitations and various response functions. Our results satisfy requirements of particle-hole symmetry. We show that while the dispersion relation obtained from the HLR theory is particle-hole symmetric, correlation functions obtained from HLR are not. The results of the Dirac theory are shown to be consistent with the Haldane bound on the projected structure factor, while those of the HLR theory violate it.
Spectral Upscaling for Graph Laplacian Problems with Application to Reservoir Simulation
Here, we consider coarsening procedures for graph Laplacian problems written in a mixed saddle-point form. In that form, in addition to the original (vertex) degrees of freedom (dofs), we also have edge degrees of freedom. We extend previously developed aggregation-based coarsening procedures applied to both sets of dofs to now allow more than one coarse vertex dof per aggregate. Those dofs are selected as certain eigenvectors of local graph Laplacians associated with each aggregate. Additionally, we coarsen the edge dofs by using traces of the discrete gradients of the already constructed coarse vertex dofs. These traces are defined on the interface edges that connect any two adjacent aggregates. The overall procedure is a modification of the spectral upscaling procedure developed in for the mixed finite element discretization of diffusion type PDEs which has the important property of maintaining inf-sup stability on coarse levels and having provable approximation properties. We consider applications to partitioning a general graph and to a finite volume discretization interpreted as a graph Laplacian, developing consistent and accurate coarse-scale models of a fine-scale problem.
Auxiliary Space Preconditioning of Finite Element Equations Using a Nonconforming Interior Penalty Reformulation and Static Condensation
We modify the well-known interior penalty finite element discretization method so that it allows for element-by-element assembly. This is possible due to the introduction of additional unknowns associated with the interfaces between neighboring elements. The resulting bilinear form, and a Schur complement (reduced) version of it, are utilized in a number of auxiliary space preconditioners for the original conforming finite element discretization problem. Furthermore, these preconditioners are analyzed on the fine scale, and their performance is illustrated on model second order scalar elliptic problems discretized with high order elements.
Multilevel Spectral Coarsening for Graph Laplacian Problems with Application to Reservoir Simulation
We extend previously developed two-level coarsening procedures for graph Laplacian problems written in a mixed saddle point form to the fully recursive multilevel case. The resulting hierarchy of discretizations gives rise to a hierarchy of upscaled models, in the sense that they provide approximation in the natural norms (in the mixed setting). This property enables us to utilize them in three applications: (i) as an accurate reduced model, (ii) as a tool in multilevel Monte Carlo simulations (in application to finite volume discretizations), and (iii) for providing a sequence of nonlinear operators in a full approximation scheme for solving nonlinear pressure equations discretized by the conservative two-point flux approximation. Finally, we illustrate the potential of the proposed multilevel technique in all three applications on a number of popular benchmark problems used in reservoir simulation.
A Condensed Constrained Nonconforming Mortar-Based Approach for Preconditioning Finite Element Discretization Problems
This paper presents and studies an approach for constructing auxiliary space preconditioners for finite element problems using a constrained nonconforming reformulation that is based on a proposed modified version of the mortar method. The well-known mortar finite element discretization method is modified to admit a local structure, providing an element-by-element or subdomain-by-subdomain assembly property. This is achieved via the introduction of additional trace finite element spaces and degrees of freedom (unknowns) associated with the interfaces between adjacent elements or subdomains. The resulting nonconforming formulation and a reduced-via-static-condensation Schur complement form on the interfaces are used in the construction of auxiliary space preconditioners for a given conforming finite element discretization problem. Overall, the properties of these preconditioners are studied and their performance is illustrated on model second order scalar elliptic problems utilizing high order elements.
Final report for Texas A&M University Group Contribution to DE-FG02-09ER25949/DE-SC0002505: Topology for Statistical Modeling of Petascale Data (and ASCR-funded collaboration between Sandia National Labs, Texas A&M University and University of Utah)
We summarize the contributions of the Texas A\&M University Group to the project (DE-FG02-09ER25949/DE-SC0002505: Topology for Statistical Modeling of Petascale Data - an ASCR-funded collaboration between Sandia National Labs, Texas A\&M U, and U Utah) during 6/9/2011 -- 2/27/2013.
Extending PETSc's Composable Hierarchical Solvers (Final Technical Report)
This report documents research activities conducted at CU Boulder as part of Extending PETSc’s Composable Hierarchical Solvers, which has been part of a collaboration with Argonne National Laboratory (separate award). Our work has focused on performance-portable end-to-end GPU solvers demonstrated via exemplary applications in nonlinear fluid and structural mechanics. We describe advances in algorithmic composition and analysis in the context of these applications, but the implementations are fully documented and decoupled, and in use by other projects. We believe the vertical integration achieved through collaboration with ECP’s CEED and the PSAAP center at CU was necessary to take risks with data structures and algorithms.
Research study to develop methods for solving problems associated with space vehicles Monthly progress report /Final/
Trigonometric function algorithms for algebraic simplification in computer programs