50 research outputs found
Preconditioned fast solvers for large linear systems with specific sparse and/or Toeplitz-like structures and applications
In this thesis, the design of the preconditioners we propose starts from applications instead of treating the problem in a completely general way. The reason is that not all types of linear systems can be addressed with the same tools. In this sense, the techniques for designing efficient iterative solvers depends mostly on properties inherited from the continuous problem, that has originated the discretized sequence of matrices. Classical examples are locality, isotropy in the PDE context, whose discrete counterparts are sparsity and matrices constant along the diagonals, respectively.
Therefore, it is often important to take into account the properties of the originating continuous model for obtaining better performances and for providing an accurate convergence analysis. We consider linear systems that arise in the solution of both linear and nonlinear partial differential equation of both integer and fractional type. For the latter case, an introduction to both the theory and the numerical treatment is given.
All the algorithms and the strategies presented in this thesis are developed having in mind their parallel implementation. In particular, we consider the processor-co-processor framework, in which the main part of the computation is performed on a Graphics Processing Unit (GPU) accelerator.
In Part I we introduce our proposal for sparse approximate inverse preconditioners for either the solution of time-dependent Partial Differential Equations (PDEs), Chapter 3, and Fractional Differential Equations (FDEs), containing both classical and fractional terms, Chapter 5. More precisely, we propose a new technique for updating preconditioners for dealing with sequences of linear systems for PDEs and FDEs, that can be used also to compute matrix functions of large matrices via quadrature formula in Chapter 4 and for optimal control of FDEs in Chapter 6. At last, in Part II, we consider structured preconditioners for quasi-Toeplitz systems. The focus is towards the numerical treatment of discretized convection-diffusion equations in Chapter 7 and on the solution of FDEs with linear multistep formula in boundary value form in Chapter 8
Preconditioned fast solvers for large linear systems with specific sparse and/or Toeplitz-like structures and applications
In this thesis, the design of the preconditioners we propose starts from applications instead of treating the problem in a completely general way. The reason is that not all types of linear systems can be addressed with the same tools. In this sense, the techniques for designing efficient iterative solvers depends mostly on properties inherited from the continuous problem, that has originated the discretized sequence of matrices. Classical examples are locality, isotropy in the PDE context, whose discrete counterparts are sparsity and matrices constant along the diagonals, respectively.
Therefore, it is often important to take into account the properties of the originating continuous model for obtaining better performances and for providing an accurate convergence analysis. We consider linear systems that arise in the solution of both linear and nonlinear partial differential equation of both integer and fractional type. For the latter case, an introduction to both the theory and the numerical treatment is given.
All the algorithms and the strategies presented in this thesis are developed having in mind their parallel implementation. In particular, we consider the processor-co-processor framework, in which the main part of the computation is performed on a Graphics Processing Unit (GPU) accelerator.
In Part I we introduce our proposal for sparse approximate inverse preconditioners for either the solution of time-dependent Partial Differential Equations (PDEs), Chapter 3, and Fractional Differential Equations (FDEs), containing both classical and fractional terms, Chapter 5. More precisely, we propose a new technique for updating preconditioners for dealing with sequences of linear systems for PDEs and FDEs, that can be used also to compute matrix functions of large matrices via quadrature formula in Chapter 4 and for optimal control of FDEs in Chapter 6. At last, in Part II, we consider structured preconditioners for quasi-Toeplitz systems. The focus is towards the numerical treatment of discretized convection-diffusion equations in Chapter 7 and on the solution of FDEs with linear multistep formula in boundary value form in Chapter 8
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
Space-time block preconditioning for incompressible flow
Parallel-in-time methods have become increasingly popular in the simulation
of time-dependent numerical PDEs, allowing for the efficient use of additional
MPI processes when spatial parallelism saturates. Most methods treat the
solution and parallelism in space and time separately. In contrast, all-at-once
methods solve the full space-time system directly, largely treating time as
simply another spatial dimension. All-at-once methods offer a number of
benefits over separate treatment of space and time, most notably significantly
increased parallelism and faster time-to-solution (when applicable). However,
the development of fast, scalable all-at-once methods has largely been limited
to time-dependent (advection-)diffusion problems. This paper introduces the
concept of space-time block preconditioning for the all-at-once solution of
incompressible flow. By extending well-known concepts of spatial block
preconditioning to the space-time setting, we develop a block preconditioner
whose application requires the solution of a space-time (advection-)diffusion
equation in the velocity block, coupled with a pressure Schur complement
approximation consisting of independent spatial solves at each time-step, and a
space-time matrix-vector multiplication. The new method is tested on four
classical models in incompressible flow. Results indicate perfect scalability
in refinement of spatial and temporal mesh spacing, perfect scalability in
nonlinear Picard iterations count when applied to a nonlinear Navier-Stokes
problem, and minimal overhead in terms of number of preconditioner applications
compared with sequential time-stepping.Comment: 28 pages, 7 figures, 4 table
Fast iterative methods for solving large systems arising from variational models in image processing
Seventh Copper Mountain Conference on Multigrid Methods
The Seventh Copper Mountain Conference on Multigrid Methods was held on April 2-7, 1995 at Copper Mountain, Colorado. This book is a collection of many of the papers presented at the conference and so represents the conference proceedings. NASA Langley graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The vibrancy and diversity in this field are amply expressed in these important papers, and the collection clearly shows the continuing rapid growth of the use of multigrid acceleration techniques
HP-multigrid as smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows. Part I. Multilevel Analysis
The hp-Multigrid as Smoother algorithm (hp-MGS) for the solution of higher order accurate space-(time) discontinuous Galerkin discretizations of advection dominated flows is presented. This algorithm combines p-multigrid with h-multigrid at all p-levels, where the h-multigrid acts as smoother in the p-multigrid. The performance of the hp-MGS algorithm is further improved using semi-coarsening in combination with a new semi-implicit Runge-Kutta method as smoother. A detailed multilevel analysis of the hp-MGS algorithm is presented to obtain more insight into the theoretical performance of the algorithm. As model problem a fourth order accurate space-time discontinuous Galerkin discretization of the advection-diffusion equation is considered. The multilevel analysis shows that the hp-MGS algorithm has excellent convergence rates, both for low and high cell Reynolds numbers and on highly stretched meshes
Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018
This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions