113,173 research outputs found
Conjugate gradient type methods for linear systems with complex symmetric coefficient matrices
We consider conjugate gradient type methods for the solution of large sparse linear system Ax equals b with complex symmetric coefficient matrices A equals A(T). Such linear systems arise in important applications, such as the numerical solution of the complex Helmholtz equation. Furthermore, most complex non-Hermitian linear systems which occur in practice are actually complex symmetric. We investigate conjugate gradient type iterations which are based on a variant of the nonsymmetric Lanczos algorithm for complex symmetric matrices. We propose a new approach with iterates defined by a quasi-minimal residual property. The resulting algorithm presents several advantages over the standard biconjugate gradient method. We also include some remarks on the obvious approach to general complex linear systems by solving equivalent real linear systems for the real and imaginary parts of x. Finally, numerical experiments for linear systems arising from the complex Helmholtz equation are reported
Do optimization methods in deep learning applications matter?
With advances in deep learning, exponential data growth and increasing model
complexity, developing efficient optimization methods are attracting much
research attention. Several implementations favor the use of Conjugate Gradient
(CG) and Stochastic Gradient Descent (SGD) as being practical and elegant
solutions to achieve quick convergence, however, these optimization processes
also present many limitations in learning across deep learning applications.
Recent research is exploring higher-order optimization functions as better
approaches, but these present very complex computational challenges for
practical use. Comparing first and higher-order optimization functions, in this
paper, our experiments reveal that Levemberg-Marquardt (LM) significantly
supersedes optimal convergence but suffers from very large processing time
increasing the training complexity of both, classification and reinforcement
learning problems. Our experiments compare off-the-shelf optimization
functions(CG, SGD, LM and L-BFGS) in standard CIFAR, MNIST, CartPole and
FlappyBird experiments.The paper presents arguments on which optimization
functions to use and further, which functions would benefit from
parallelization efforts to improve pretraining time and learning rate
convergence
Computation of Ground States of the Gross-Pitaevskii Functional via Riemannian Optimization
In this paper we combine concepts from Riemannian Optimization and the theory
of Sobolev gradients to derive a new conjugate gradient method for direct
minimization of the Gross-Pitaevskii energy functional with rotation. The
conservation of the number of particles constrains the minimizers to lie on a
manifold corresponding to the unit norm. The idea developed here is to
transform the original constrained optimization problem to an unconstrained
problem on this (spherical) Riemannian manifold, so that fast minimization
algorithms can be applied as alternatives to more standard constrained
formulations. First, we obtain Sobolev gradients using an equivalent definition
of an inner product which takes into account rotation. Then, the
Riemannian gradient (RG) steepest descent method is derived based on projected
gradients and retraction of an intermediate solution back to the constraint
manifold. Finally, we use the concept of the Riemannian vector transport to
propose a Riemannian conjugate gradient (RCG) method for this problem. It is
derived at the continuous level based on the "optimize-then-discretize"
paradigm instead of the usual "discretize-then-optimize" approach, as this
ensures robustness of the method when adaptive mesh refinement is performed in
computations. We evaluate various design choices inherent in the formulation of
the method and conclude with recommendations concerning selection of the best
options. Numerical tests demonstrate that the proposed RCG method outperforms
the simple gradient descent (RG) method in terms of rate of convergence. While
on simple problems a Newton-type method implemented in the {\tt Ipopt} library
exhibits a faster convergence than the (RCG) approach, the two methods perform
similarly on more complex problems requiring the use of mesh adaptation. At the
same time the (RCG) approach has far fewer tunable parameters.Comment: 28 pages, 13 figure
On Algorithms Based on Joint Estimation of Currents and Contrast in Microwave Tomography
This paper deals with improvements to the contrast source inversion method
which is widely used in microwave tomography. First, the method is reviewed and
weaknesses of both the criterion form and the optimization strategy are
underlined. Then, two new algorithms are proposed. Both of them are based on
the same criterion, similar but more robust than the one used in contrast
source inversion. The first technique keeps the main characteristics of the
contrast source inversion optimization scheme but is based on a better
exploitation of the conjugate gradient algorithm. The second technique is based
on a preconditioned conjugate gradient algorithm and performs simultaneous
updates of sets of unknowns that are normally processed sequentially. Both
techniques are shown to be more efficient than original contrast source
inversion.Comment: 12 pages, 12 figures, 5 table
A COMPARISON OF PRECONDITIONING TECHNIQUES FOR PARALLELIZED PCG SOLVERS FOR THE CELL-CENTERED FINITE-DIFFERENCE PROBLEM
Abstract This paper reports on a parallelization of the preconditioned conjugate gradient algorithm for sparse, symmetric matrices. Parallelization is based in domain partitioning into non-overlapping subdomains; the resulting parallelized algorithm is briefly described. Comparisons are made between three block preconditioners commonly used in the parallelization of the preconditoned conjugate gradient methods: Jacobi, incomplete Choleski, and Gauss Seidel. Basic timing and iteration results for these preconditioners are presented; these results tentatively indicate that the simpler block Jacobi algorithm is as efficient as the more complex block incomplete Cholesky and block Gauss Seidel
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