744 research outputs found
A technique for accelerating iterative convergence in numerical integration, with application in transonic aerodynamics
A technique is described for the efficient numerical solution of nonlinear partial differential equations by rapid iteration. In particular, a special approach is described for applying the Aitken acceleration formula (a simple Pade approximant) for accelerating the iterative convergence. The method finds the most appropriate successive approximations, which are in a most nearly geometric sequence, for use in the Aitken formula. Simple examples are given to illustrate the use of the method. The method is then applied to the mixed elliptic-hyperbolic problem of steady, inviscid, transonic flow over an airfoil in a subsonic free stream
Scalar Levin-Type Sequence Transformations
Sequence transformations are important tools for the convergence acceleration
of slowly convergent scalar sequences or series and for the summation of
divergent series. Transformations that depend not only on the sequence elements
or partial sums but also on an auxiliary sequence of so-called remainder
estimates are of Levin-type if they are linear in the , and
nonlinear in the . Known Levin-type sequence transformations are
reviewed and put into a common theoretical framework. It is discussed how such
transformations may be constructed by either a model sequence approach or by
iteration of simple transformations. As illustration, two new sequence
transformations are derived. Common properties and results on convergence
acceleration and stability are given. For important special cases, extensions
of the general results are presented. Also, guidelines for the application of
Levin-type sequence transformations are discussed, and a few numerical examples
are given.Comment: 59 pages, LaTeX, invited review for J. Comput. Applied Math.,
abstract shortene
Brezinski Inverse and Geometric Product-Based Steffensen's Methods for Image Reverse Filtering
This work develops extensions of Steffensen's method to provide new tools for
solving the semi-blind image reverse filtering problem. Two extensions are
presented: a parametric Steffensen's method for accelerating the Mann
iteration, and a family of 12 Steffensen's methods for vector variables. The
development is based on Brezinski inverse and geometric product vector inverse.
Variants of these methods are presented with adaptive parameter setting and
first-order method acceleration. Implementation details, complexity, and
convergence are discussed, and the proposed methods are shown to generalize
existing algorithms. A comprehensive study of 108 variants of the vector
Steffensen's methods is presented in the Supplementary Material. Representative
results and comparison with current state-of-the-art methods demonstrate that
the vector Steffensen's methods are efficient and effective tools in reversing
the effects of commonly used filters in image processing
Recycling Krylov Subspaces for Efficient Partitioned Solution of Aerostructural Adjoint Systems
Robust and efficient solvers for coupled-adjoint linear systems are crucial
to successful aerostructural optimization. Monolithic and partitioned
strategies can be applied. The monolithic approach is expected to offer better
robustness and efficiency for strong fluid-structure interactions. However, it
requires a high implementation cost and convergence may depend on appropriate
scaling and initialization strategies. On the other hand, the modularity of the
partitioned method enables a straightforward implementation while its
convergence may require relaxation. In addition, a partitioned solver leads to
a higher number of iterations to get the same level of convergence as the
monolithic one.
The objective of this paper is to accelerate the fluid-structure
coupled-adjoint partitioned solver by considering techniques borrowed from
approximate invariant subspace recycling strategies adapted to sequences of
linear systems with varying right-hand sides. Indeed, in a partitioned
framework, the structural source term attached to the fluid block of equations
affects the right-hand side with the nice property of quickly converging to a
constant value. We also consider deflation of approximate eigenvectors in
conjunction with advanced inner-outer Krylov solvers for the fluid block
equations. We demonstrate the benefit of these techniques by computing the
coupled derivatives of an aeroelastic configuration of the ONERA-M6 fixed wing
in transonic flow. For this exercise the fluid grid was coupled to a structural
model specifically designed to exhibit a high flexibility. All computations are
performed using RANS flow modeling and a fully linearized one-equation
Spalart-Allmaras turbulence model. Numerical simulations show up to 39%
reduction in matrix-vector products for GCRO-DR and up to 19% for the nested
FGCRO-DR solver.Comment: 42 pages, 21 figure
Shanks sequence transformations and Anderson acceleration
This paper presents a general framework for Shanks transformations of sequences of elements in a vector space. It is shown that the Minimal Polynomial Extrapolation (MPE), the
Modified Minimal Polynomial Extrapolation (MMPE), the Reduced Rank Extrapolation (RRE), the Vector Epsilon Algorithm (VEA), the Topological Epsilon Algorithm (TEA), and Anderson Acceleration (AA), which are standard general techniques designed for accelerating arbitrary sequences and/or solving nonlinear equations, all fall into this framework. Their properties and their connections with quasi-Newton and Broyden methods are studied. The paper then exploits this framework to compare these methods. In the linear case, it is known that AA and GMRES are \u2018essentially\u2019
equivalent in a certain sense while GMRES and RRE are mathematically equivalent. This paper
discusses the connection between AA, the RRE, the MPE, and other methods in the nonlinear case
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