Augmented Lagrangian preconditioners for the Oseen-Frank model of nematic and cholesteric liquid crystals

We propose a robust and efficient augmented Lagrangian-type preconditioner for solving linearizations of the Oseen-Frank model arising in cholesteric liquid crystals. By applying the augmented Lagrangian method, the Schur complement of the director block can be better approximated by the weighted mass matrix of the Lagrange multiplier, at the cost of making the augmented director block harder to solve. In order to solve the augmented director block, we develop a robust multigrid algorithm which includes an additive Schwarz relaxation that captures a pointwise version of the kernel of the semi-definite term. Furthermore, we prove that the augmented Lagrangian term improves the discrete enforcement of the unit-length constraint. Numerical experiments verify the efficiency of the algorithm and its robustness with respect to problem-related parameters (Frank constants and cholesteric pitch) and the mesh size

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

On the design of block preconditioners for maritime engineering

The iterative error can be an important part of the total numerical error of any Com- putational Fluid Dynamics simulation when the iterative convergence stagnates or when loose convergence criteria are used. In the quest for better iterative convergence of CFD simulations, we consider the design of iterative methods for the Reynolds-averaged Navier-Stokes equations, discretized by finite-volume methods with cell-centered, co-located variables. The central point is the approximation of the Schur complement (pressure matrix) in the block factorization of the discrete system of mass and momentum equations. We show particular approximations of these blocks that yield either segregated solvers or block preconditioners for fully coupled solvers. The performance of these solvers are then demonstrated by computing the flow over a flat plate and around a tanker on both structured and unstructured grids. We find that iterative convergence to machine precision is attainable despite the high Reynolds numbers and mesh aspect ratio’s. Improved approximations of the Schur complement do result in improved convergence rates, but do not seem to pay-off in terms of total cost compared to the basic SIMPLE-type approximation

A DECOMPOSITION PROCEDURE BASED ON APPROXIMATE NEWTON DIRECTIONS

The efficient solution of large-scale linear and nonlinear optimization problems may require exploiting any special structure in them in an efficient manner. We describe and analyze some cases in which this special structure can be used with very little cost to obtain search directions from decomposed subproblems. We also study how to correct these directions using (decomposable) preconditioned conjugate gradient methods to ensure local convergence in all cases. The choice of appropriate preconditioners results in a natural manner from the structure in the problem. Finally, we conduct computational experiments to compare the resulting procedures with direct methods, as well as to study the impact of different preconditioner choices.

A decomposition procedure based on approximate newton directions

The efficient solution of large-scale linear and nonlinear optimization problems may require exploiting any special structure in them in an efficient manner. We describe and analyze some cases in which this special structure can be used with very little cost to obtain search directions from decomposed subproblems. We also study how to correct these directions using (decomposable) preconditioned conjugate gradient methods to ensure local convergence in all cases. The choice of appropriate preconditioners results in a natural manner from the structure in the problem. Finally, we conduct computational experiments to compare the resulting procedures with direct methods, as well as to study the impact of different preconditioner choices

Discretisations and Preconditioners for Magnetohydrodynamics Models

The magnetohydrodynamics (MHD) equations are generally known to be difficult to solve numerically, due to their highly nonlinear structure and the strong coupling between the electromagnetic and hydrodynamic variables, especially for high Reynolds and coupling numbers. In the first part of this work, we present a scalable augmented Lagrangian preconditioner for a finite element discretisation of the $\mathbf{B}$-$\mathbf{E}$ formulation of the incompressible viscoresistive MHD equations. For stationary problems, our solver achieves robust performance with respect to the Reynolds and coupling numbers in two dimensions and good results in three dimensions. Our approach relies on specialised parameter-robust multigrid methods for the hydrodynamic and electromagnetic blocks. The scheme ensures exactly divergence-free approximations of both the velocity and the magnetic field up to solver tolerances. In the second part, we focus on incompressible, resistive Hall MHD models and derive structure-preserving finite element methods for these equations. We present a variational formulation of Hall MHD that enforces the magnetic Gauss's law precisely (up to solver tolerances) and prove the well-posedness of a Picard linearisation. For the transient problem, we present time discretisations that preserve the energy and magnetic and hybrid helicity precisely in the ideal limit for two types of boundary conditions. In the third part, we investigate anisothermal MHD models. We start by performing a bifurcation analysis for a magnetic Rayleigh--B\'enard problem at a high coupling number $S=1{,}000$ by choosing the Rayleigh number in the range between 0 and $100{,}000$ as the bifurcation parameter. We study the effect of the coupling number on the bifurcation diagram and outline how we create initial guesses to obtain complex solution patterns and disconnected branches for high coupling numbers.Comment: Doctoral thesis, Mathematical Institute, University of Oxford. 174 page

Hessian Matrix-Free Lagrange-Newton-Krylov-Schur-Schwarz Methods for Elliptic Inverse Problems

This study focuses on the solution of inverse problems for elliptic systems. The inverse problem is constructed as a PDE-constrained optimization, where the cost function is the L2 norm of the difference between the measured data and the predicted state variable, and the constraint is an elliptic PDE. Particular examples of the system considered in this stud, are groundwater flow and radiation transport. The inverse problems are typically ill-posed due to error in measurements of the data. Regularization methods are employed to partially alleviate this problem. The PDE-constrained optimization is formulated as the minimization of a Lagrangian functional, formed from the regularized cost function and the discretized PDE, with respect to the parameters, the state variables, and the Lagrange multipliers. Our approach is known as an all at once method. An algorithm is proposed for an inverse problem that is capable of being extended to large scales. To overcome storage limitations, we develop a parallel preconditioned Newton-Krylov method employed in a Hessian-free manner. The preconditioners have an inner-outer structure, taking the form of a Schur complement (block factorization) at the outer level and Schwarz projections at the inner level. However, building an exact Schur complement is prohibitively expensive. Thus, we use Schur complement approximations, including the identity, probing, the Laplacian, the J operator, and a BFGS operator. For exact data the exact Schur complements are superior to the inexact approximations. However, for data with noise the inexact methods are competitive to or even better than the exact in every computational aspect. We also find that nousymmetric forms of the Karush-Kuhn-Tucker matrices and preconditioners are competitive to or better than the symmetric forms that are commonly used in the optimization community. In this study, iterative Tikhonov and Total Variation regularizations are proposed and compared to the standard regularizations and each other. For exact data with jump discontinuities the standard and iterative Total Variation regulations are superior to the standard and iterative Tikhonov regularizations. However, in the case of noisy data the proposed iterative Tikhonov regularizations are superior to the standard and iterative Total Variation methods. We also show that in some cases the iterative regularizations are better than the noniterative. To demonstrate the performance of the algorithm, including the effectiveness of the preconditioners and regularizations, synthetic one- and two-dimensional elliptic inverse problems are solved, and we also compare with other methodologies that are available in the literature. The proposed algorithm performs well with regard to robustness, reconstructs the parameter models effectively, and is easily implemented in the framework of the available parallel PDE software PETSc and the automatic differentiation software ADIC. The algorithm is also extendable to three-dimensional problems