175 research outputs found
GMRES-Accelerated ADMM for Quadratic Objectives
We consider the sequence acceleration problem for the alternating direction
method-of-multipliers (ADMM) applied to a class of equality-constrained
problems with strongly convex quadratic objectives, which frequently arise as
the Newton subproblem of interior-point methods. Within this context, the ADMM
update equations are linear, the iterates are confined within a Krylov
subspace, and the General Minimum RESidual (GMRES) algorithm is optimal in its
ability to accelerate convergence. The basic ADMM method solves a
-conditioned problem in iterations. We give
theoretical justification and numerical evidence that the GMRES-accelerated
variant consistently solves the same problem in iterations
for an order-of-magnitude reduction in iterations, despite a worst-case bound
of iterations. The method is shown to be competitive against
standard preconditioned Krylov subspace methods for saddle-point problems. The
method is embedded within SeDuMi, a popular open-source solver for conic
optimization written in MATLAB, and used to solve many large-scale semidefinite
programs with error that decreases like , instead of ,
where is the iteration index.Comment: 31 pages, 7 figures. Accepted for publication in SIAM Journal on
Optimization (SIOPT
The Anderson model of localization: a challenge for modern eigenvalue methods
We present a comparative study of the application of modern eigenvalue
algorithms to an eigenvalue problem arising in quantum physics, namely, the
computation of a few interior eigenvalues and their associated eigenvectors for
the large, sparse, real, symmetric, and indefinite matrices of the Anderson
model of localization. We compare the Lanczos algorithm in the 1987
implementation of Cullum and Willoughby with the implicitly restarted Arnoldi
method coupled with polynomial and several shift-and-invert convergence
accelerators as well as with a sparse hybrid tridiagonalization method. We
demonstrate that for our problem the Lanczos implementation is faster and more
memory efficient than the other approaches. This seemingly innocuous problem
presents a major challenge for all modern eigenvalue algorithms.Comment: 16 LaTeX pages with 3 figures include
On large-scale diagonalization techniques for the Anderson model of localization
We propose efficient preconditioning algorithms for an eigenvalue problem arising in quantum physics, namely the computation of a few interior eigenvalues and their associated eigenvectors for large-scale sparse real and symmetric indefinite matrices of the Anderson model
of localization. We compare the Lanczos algorithm in the 1987 implementation by Cullum and Willoughby with the shift-and-invert techniques in the implicitly restarted Lanczos method and in the Jacobi–Davidson method. Our preconditioning approaches for the shift-and-invert symmetric indefinite linear system are based on maximum weighted matchings and algebraic multilevel incomplete
LDLT factorizations. These techniques can be seen as a complement to the alternative idea of using more complete pivoting techniques for the highly ill-conditioned symmetric indefinite Anderson matrices. We demonstrate the effectiveness and the numerical accuracy of these algorithms. Our numerical examples reveal that recent algebraic multilevel preconditioning solvers can accelerate the computation of a large-scale eigenvalue problem corresponding to the Anderson model of localization
by several orders of magnitude
A distributed primal-dual interior-point method for loosely coupled problems using ADMM
In this paper we propose an efficient distributed algorithm for solving
loosely coupled convex optimization problems. The algorithm is based on a
primal-dual interior-point method in which we use the alternating direction
method of multipliers (ADMM) to compute the primal-dual directions at each
iteration of the method. This enables us to join the exceptional convergence
properties of primal-dual interior-point methods with the remarkable
parallelizability of ADMM. The resulting algorithm has superior computational
properties with respect to ADMM directly applied to our problem. The amount of
computations that needs to be conducted by each computing agent is far less. In
particular, the updates for all variables can be expressed in closed form,
irrespective of the type of optimization problem. The most expensive
computational burden of the algorithm occur in the updates of the primal
variables and can be precomputed in each iteration of the interior-point
method. We verify and compare our method to ADMM in numerical experiments.Comment: extended version, 50 pages, 9 figure
An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices, part 2
It is shown how the look-ahead Lanczos process (combined with a quasi-minimal residual QMR) approach) can be used to develop a robust black box solver for large sparse non-Hermitian linear systems. Details of an implementation of the resulting QMR algorithm are presented. It is demonstrated that the QMR method is closely related to the biconjugate gradient (BCG) algorithm; however, unlike BCG, the QMR algorithm has smooth convergence curves and good numerical properties. We report numerical experiments with our implementation of the look-ahead Lanczos algorithm, both for eigenvalue problem and linear systems. Also, program listings of FORTRAN implementations of the look-ahead algorithm and the QMR method are included
A STUDY OF MARINE CSEM SURVEY GEOMETRY FOR SEABED LOGGING
In recent years marine Controlled Source Electromagnetic (CSEM) method is
being used for hydrocarbon exploration in deeper water. The marine CSEM is
preferred over seismic due to its ability to differentiate reservoir of resistive
hydrocarbon and conductive saline fluids. In marine CSEM method, however, the
survey data is highly dependent on source-receiver position and orientation.
Furthermore, real geological conditions are extremely varied and it is rarely possible
to turn the survey data into reliable picture of geological structure using virtual
simulation. Consequently, it is important to understand EM field behavior for various
geological models and source-receiver position to improve virtual simulation process.
The aforementioned aspect studied through comprehensive forward modeling is an
alternate to real time geophysical surveys. In this research, a forward modeling
algorithm is employed as a staggered-grid finite difference solution to the totalelectric
field Maxwell’s equations. Solution are achieved through (i) an optimal grid
technique that extends the boundaries of the mesh outward from the region of interest
using a minimal number of nodes, and (ii) a direct matrix solution technique that
allows for simultaneous solution for all sources. The forward modeling algorithm is
applied on 1D and 2D geological models implemented using MATLAB. Results
obtained provide qualitative understanding of electromagnetic signal propagation
through different stratified media with various source positions and orientation. 1D
forward modeling is used to find optimal frequency for specific depth and for
studying effects of speed variation of dipole source. The 2D forward modeling is used
to understand EM field behavior for resistive hydrocarbon and conductive saline fluid
reservoirs. Furthermore, inaccuracy due to improper survey geometry such as dipping
effect is also analyzed and discussed. In essence, forward modeling was applied to
various scenarios and obtained results were accurately matched with previously
published work of real time survey. This is a significant step towards the
improvement in confidence modeling which in turn can potentially help to reduce
surveying cost
Numerical solution of 3-D electromagnetic problems in exploration geophysics and its implementation on massively parallel computers
The growing significance, technical development and employment of electromagnetic (EM) methods in exploration geophysics have led to the increasing need for reliable and fast techniques of interpretation of 3-D EM data sets acquired in complex geological environments. The first and most important step to creating an inversion method is the development of a solver for the forward problem. In order to create an efficient, reliable and practical 3-D EM inversion, it is necessary to have a 3-D EM modelling code that is highly accurate, robust and very fast. This thesis focuses precisely on this crucial and very demanding step to building a 3-D EM interpretation method.
The thesis presents as its main contribution a highly accurate, robust, very fast and extremely scalable numerical method for 3-D EM modelling in geophysics that is based on finite elements (FE) and designed to run on massively parallel computing platforms. Thanks to the fact that the FE approach supports completely unstructured tetrahedral meshes as well as local mesh refinements, the presented solver is able to represent complex geometries of subsurface structures very precisely and thus improve the solution accuracy and avoid misleading artefacts in images. Consequently, it can be successfully used in geological environments of arbitrary geometrical complexities. The parallel implementation of the method, which is based on the domain decomposition and a hybrid MPI-OpenMP scheme, has proved to be highly scalable - the achieved speed-up is close to the linear for more than a thousand processors. Thanks to this, the code is able to deal with extremely large problems, which may have hundreds of millions of degrees of freedom, in a very efficient way. The importance of having this forward-problem solver lies in the fact that it is now possible to create a 3-D EM inversion that can deal with data obtained in extremely complex geological environments in a way that is realistic for practical use in industry. So far, such imaging tool has not been proposed due to a lack of efficient, parallel FE solutions as well as the limitations of efficient solvers based on finite differences.
In addition, the thesis discusses physical, mathematical and numerical aspects and challenges of 3-D EM modelling, which have been studied during my research in order to properly design the presented software for EM field simulations on 3-D areas of the Earth. Through this work, a physical problem formulation based on the secondary Coulomb-gauged EM potentials has been validated, proving that it can be successfully used with the standard nodal FE method to give highly accurate numerical solutions. Also, this work has shown that Krylov subspace iterative methods are the best solution for solving linear systems that arise after FE discretisation of the problem under consideration. More precisely, it has been discovered empirically that the best iterative method for this kind of problems is biconjugate gradient stabilised with an elaborate preconditioner. Since most commonly used preconditioners proved to be either unable to improve the convergence of the implemented solvers to the desired extent, or impractical in the parallel context, I have proposed a preconditioning technique for Krylov methods that is based on algebraic multigrid. Tests for various problems with different conductivity structures and characteristics have shown that the new preconditioner greatly improves the convergence of different Krylov subspace methods, which significantly reduces the total execution time of the program and improves the solution quality. Furthermore, the preconditioner is very practical for parallel implementation. Finally, it has been concluded that there are not any restrictions in employing classical parallel programming models, MPI and OpenMP, for parallelisation of the presented FE solver. Moreover, they have proved to be enough to provide an excellent scalability for it
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