Robust Preconditioners for the High-Contrast Elliptic Partial Differential Equations

In this thesis, we discuss a robust preconditioner (the AGKS preconditioner) for solving linear systems arising from approximations of partial differential equations (PDEs) with high-contrast coefficients. The problems considered here include the standard second and higher order elliptic PDEs such as high-contrast diffusion equation, Stokes\u27 equation and biharmonic-plate equation. The goal of this study is the development of robust and parallelizable preconditioners that can easily be integrated to treat large configurations. The construction of the preconditioner consists of two phases. The first one is an algebraic phase which partitions the degrees of freedom into high and low permeability regions which may be of arbitrary geometry. This yields a corresponding block partitioning of the stiffness matrix allowing us to use a formula for the action of its inverse involving the inverses of both the high permeability block and its Schur complement in the original matrix. Singular perturbation analysis plays a big role to analyze the structure of the required subblock inverses in the high contrast case which shows that for high enough contrast each of the subblock inverses can be approximated well by solving only systems with constant coefficients. The second phase involves an efficient multigrid approximation of this exact inverse. After applying singular perturbation theory to each of the sub-blocks, we obtain that inverses of each of the subblocks with high contrast entries can be approximated efficiently using geometric multigrid methods, and that this approximation is robust with respect to both the contrast and the mesh size. The result is a multigrid method for high contrast problems which is provably optimal to both contrast and mesh size. We demonstrate the advantageous properties of the AGKS preconditioner using experiments on model high-contrast problems. We examine its performance against multigrid method under varying discretizations of diffusion equation, Stokes equation and biharmonic-plate equation. Thus, we show that we accomplished a desirable preconditioning design goal by using the same family of preconditioners to solve the elliptic family of PDEs with varying discretizations

A Preconditioned Inexact Active-Set Method for Large-Scale Nonlinear Optimal Control Problems

We provide a global convergence proof of the recently proposed sequential homotopy method with an inexact Krylov--semismooth-Newton method employed as a local solver. The resulting method constitutes an active-set method in function space. After discretization, it allows for efficient application of Krylov-subspace methods. For a certain class of optimal control problems with PDE constraints, in which the control enters the Lagrangian only linearly, we propose and analyze an efficient, parallelizable, symmetric positive definite preconditioner based on a double Schur complement approach. We conclude with numerical results for a badly conditioned and highly nonlinear benchmark optimization problem with elliptic partial differential equations and control bounds. The resulting method is faster than using direct linear algebra for the 2D benchmark and allows for the parallel solution of large 3D problems.Comment: 26 page

Composable code generation for high order, compatible finite element methods

It has been widely recognised in the HPC communities across the world, that exploiting modern computer architectures, including exascale machines, to a full extent requires software commu- nities to adapt their algorithms. Computational methods with a high ratio of floating point op- erations to bandwidth are favorable. For solving partial differential equations, which can model many physical problems, high order finite element methods can calculate approximations with a high efficiency when a good solver is employed. Matrix-free algorithms solve the corresponding equations with a high arithmetic intensity. Vectorisation speeds up the operations by calculating one instruction on multiple data elements. Another recent development for solving partial differential are compatible (mimetic) finite ele- ment methods. In particular with application to geophysical flows, compatible discretisations ex- hibit desired numerical properties required for accurate approximations. Among others, this has been recognised by the UK Met office and their new dynamical core for weather and climate fore- casting is built on a compatible discretisation. Hybridisation has been proven to be an efficient solver for the corresponding equation systems, because it removes some inter-elemental coupling and localises expensive operations. This thesis combines the recent advances on vectorised, matrix-free, high order finite element methods in the HPC community on the one hand and hybridised, compatible discretisations in the geophysical community on the other. In previous work, a code generation framework has been developed to support the localised linear algebra required for hybridisation. First, the framework is adapted to support vectorisation and further, extended so that the equations can be solved fully matrix-free. Promising performance results are completing the thesis.Open Acces

Preconditioned iterative methods for optimal control problems with time-dependent PDEs as constraints

In this work, we study fast and robust solvers for optimal control problems with Partial Differential Equations (PDEs) as constraints. Speci cally, we devise preconditioned iterative methods for time-dependent PDE-constrained optimization problems, usually when a higher-order discretization method in time is employed as opposed to most previous solvers. We also consider the control of stationary problems arising in uid dynamics, as well as that of unsteady Fractional Differential Equations (FDEs). The preconditioners we derive are employed within an appropriate Krylov subspace method. The fi rst key contribution of this thesis involves the study of fast and robust preconditioned iterative solution strategies for the all-at-once solution of optimal control problems with time-dependent PDEs as constraints, when a higher-order discretization method in time is employed. In fact, as opposed to most work in preconditioning this class of problems, where a ( first-order accurate) backward Euler method is used for the discretization of the time derivative, we employ a (second-order accurate) Crank-Nicolson method in time. By applying a carefully tailored invertible transformation, we symmetrize the system obtained, and then derive a preconditioner for the resulting matrix. We prove optimality of the preconditioner through bounds on the eigenvalues, and test our solver against a widely-used preconditioner for the linear system arising from a backward Euler discretization. These theoretical and numerical results demonstrate the effectiveness and robustness of our solver with respect to mesh-sizes and regularization parameter. Then, the optimal preconditioner so derived is generalized from the heat control problem to time-dependent convection{diffusion control with Crank- Nicolson discretization in time. Again, we prove optimality of the approximations of the main blocks of the preconditioner through bounds on the eigenvalues, and, through a range of numerical experiments, show the effectiveness and robustness of our approach with respect to all the parameters involved in the problem. For the next substantial contribution of this work, we focus our attention on the control of problems arising in fluid dynamics, speci fically, the Stokes and the Navier-Stokes equations. We fi rstly derive fast and effective preconditioned iterative methods for the stationary and time-dependent Stokes control problems, then generalize those methods to the case of the corresponding Navier-Stokes control problems when employing an Oseen approximation to the non-linear term. The key ingredients of the solvers are a saddle-point type approximation for the linear systems, an inner iteration for the (1,1)-block accelerated by a preconditioner for convection-diffusion control problems, and an approximation to the Schur complement based on a potent commutator argument applied to an appropriate block matrix. Through a range of numerical experiments, we show the effectiveness of our approximations, and observe their considerable parameter-robustness. The fi nal chapter of this work is devoted to the derivation of efficient and robust solvers for convex quadratic FDE-constrained optimization problems, with box constraints on the state and/or control variables. By employing an Alternating Direction Method of Multipliers for solving the non-linear problem, one can separate the equality from the inequality constraints, solving the equality constraints and then updating the current approximation of the solutions. In order to solve the equality constraints, a preconditioner based on multilevel circulant matrices is derived, and then employed within an appropriate preconditioned Krylov subspace method. Numerical results show the e ciency and scalability of the strategy, with the cost of the overall process being proportional to N log N, where N is the dimension of the problem under examination. Moreover, the strategy presented allows the storage of a highly dense system, due to the memory required being proportional to N

Hybrid mimetic finite-difference and virtual element formulation for coupled poromechanics

We present a hybrid mimetic finite-difference and virtual element formulation for coupled single-phase poromechanics on unstructured meshes. The key advantage of the scheme is that it is convergent on complex meshes containing highly distorted cells with arbitrary shapes. We use a local pressure-jump stabilization method based on unstructured macro-elements to prevent the development of spurious pressure modes in incompressible problems approaching undrained conditions. A scalable linear solution strategy is obtained using a block-triangular preconditioner designed specifically for the saddle-point systems arising from the proposed discretization. The accuracy and efficiency of our approach are demonstrated numerically on two-dimensional benchmark problems.Comment: 25 pages, 17 figure

SlabLU: A Two-Level Sparse Direct Solver for Elliptic PDEs

The paper describes a sparse direct solver for the linear systems that arise from the discretization of an elliptic PDE on a two dimensional domain. The solver is designed to reduce communication costs and perform well on GPUs; it uses a two-level framework, which is easier to implement and optimize than traditional multi-frontal schemes based on hierarchical nested dissection orderings. The scheme decomposes the domain into thin subdomains, or "slabs". Within each slab, a local factorization is executed that exploits the geometry of the local domain. A global factorization is then obtained through the LU factorization of a block-tridiagonal reduced coefficient matrix. The solver has complexity $O(N^{5/3})$ for the factorization step, and $O(N^{7/6})$ for each solve once the factorization is completed. The solver described is compatible with a range of different local discretizations, and numerical experiments demonstrate its performance for regular discretizations of rectangular and curved geometries. The technique becomes particularly efficient when combined with very high-order convergent multi-domain spectral collocation schemes. With this discretization, a Helmholtz problem on a domain of size $1000 \lambda \times 1000 \lambda$ (for which N=100 \mbox{M}) is solved in 15 minutes to 6 correct digits on a high-powered desktop with GPU acceleration

Reduced Order Modeling based Inexact FETI-DP solver for lattice structures

This paper addresses the overwhelming computational resources needed with standard numerical approaches to simulate architected materials. Those multiscale heterogeneous lattice structures gain intensive interest in conjunction with the improvement of additive manufacturing as they offer, among many others, excellent stiffness-to-weight ratios. We develop here a dedicated HPC solver that benefits from the specific nature of the underlying problem in order to drastically reduce the computational costs (memory and time) for the full fine-scale analysis of lattice structures. Our purpose is to take advantage of the natural domain decomposition into cells and, even more importantly, of the geometrical and mechanical similarities among cells. Our solver consists in a so-called inexact FETI-DP method where the local, cell-wise operators and solutions are approximated with reduced order modeling techniques. Instead of considering independently every cell, we end up with only few principal local problems to solve and make use of the corresponding principal cell-wise operators to approximate all the others. It results in a scalable algorithm that saves numerous local factorizations. Our solver is applied for the isogeometric analysis of lattices built by spline composition, which offers the opportunity to compute the reduced basis with macro-scale data, thereby making our method also multiscale and matrix-free. The solver is tested against various 2D and 3D analyses. It shows major gains with respect to black-box solvers; in particular, problems of several millions of degrees of freedom can be solved with a simple computer within few minutes.Comment: 30 pages, 12 figures, 2 table

Lectures on Computational Numerical Analysis of Partial Differential Equations

From Chapter 1: The purpose of these lectures is to present a set of straightforward numerical methods with applicability to essentially any problem associated with a partial differential equation (PDE) or system of PDEs independent of type, spatial dimension or form of nonlinearity.https://uknowledge.uky.edu/me_textbooks/1002/thumbnail.jp

Hybridizable compatible finite element discretizations for numerical weather prediction: implementation and analysis

There is a current explosion of interest in new numerical methods for atmospheric modeling. A driving force behind this is the need to be able to simulate, with high efficiency, large-scale geophysical flows on increasingly more parallel computer systems. Many current operational models, including that of the UK Met Office, depend on orthogonal meshes, such as the latitude-longitude grid. This facilitates the development of finite difference discretizations with favorable numerical properties. However, such methods suffer from the ``pole problem," which prohibits the model to make efficient use of a large number of computing processors due to excessive concentration of grid-points at the poles. Recently developed finite element discretizations, known as ``compatible" finite elements, avoid this issue while maintaining the key numerical properties essential for accurate geophysical simulations. Moreover, these properties can be obtained on arbitrary, non-orthogonal meshes. However, the efficient solution of the resulting discrete systems depend on transforming the mixed velocity-pressure (or velocity-pressure-buoyancy) system into an elliptic problem for the pressure. This is not so straightforward within the compatible finite element framework due to inter-element coupling. This thesis supports the proposition that systems arising from compatible finite element discretizations can be solved efficiently using a technique known as ``hybridization." Hybridization removes inter-element coupling while maintaining the desired numerical properties. This permits the construction of sparse, elliptic problems, for which fast solver algorithms are known, using localized algebra. We first introduce the technique for compatible finite element discretizations of simplified atmospheric models. We then develop a general software abstraction for the rapid implementation and composition of hybridization methods, with an emphasis on preconditioning. Finally, we extend the technique for a new compatible method for the full, compressible atmospheric equations used in operational models.Open Acces