pinecone/core-2020-05-10-deduplication huggingface dataset:
Limited memory preconditioners for symmetric indefinite problems with application to structural mechanics

This paper presents a class of limited memory preconditioners (LMP) for solving linear systems of equations with symmetric indefinite matrices and multiple right-hand sides. These preconditioners based on limited memory quasi-Newton formulas require a small number k of linearly independent vectors and may be used to improve an existing first-level preconditioner. The contributions of the paper are threefold. First, we derive a formula to characterize the spectrum of the preconditioned operator. A spectral analysis of the preconditioned matrix shows that the eigenvalues are all real and that the LMP class is able to cluster at least k eigenvalues at 1. Secondly, we show that the eigenvalues of the preconditioned matrix enjoy interlacing properties with respect to the eigenvalues of the original matrix provided that the k linearly independent vectors have been prior projected onto the invariant subspaces associated with the eigenvalues of the original matrix in the open right and left half-plane, respectively. Third, we focus on theoretical properties of the Ritz-LMP variant, where Ritz information is used to determine the k vectors. Finally, we illustrate the numerical behaviour of the Ritz limited memory preconditioners on realistic applications in structural mechanics that require the solution of sequences of large-scale symmetric saddle-point systems. Numerical experiments show the relevance of the proposed preconditioner leading to a significant decrease in terms of computational\ud
operations when solving such sequences of linear systems. A saving of up to 43% in terms of computational effort is obtained on one of these applications
Computing nearest stable matrix pairs

In this paper, we study the nearest stable matrix pair problem: given a
square matrix pair $(E,A)$, minimize the Frobenius norm of
$(\Delta_E,\Delta_A)$ such that $(E+\Delta_E,A+\Delta_A)$ is a stable matrix
pair. We propose a reformulation of the problem with a simpler feasible set by
introducing dissipative Hamiltonian (DH) matrix pairs: A matrix pair $(E,A)$ is
DH if $A=(J-R)Q$ with skew-symmetric $J$, positive semidefinite $R$, and an
invertible $Q$ such that $Q^TE$ is positive semidefinite. This reformulation
has a convex feasible domain onto which it is easy to project. This allows us
to employ a fast gradient method to obtain a nearby stable approximation of a
given matrix pair.Comment: 19 pages, 4 figure
A Riemannian trust-region method for low-rank tensor completion

The goal of tensor completion is to fill in missing entries of a partially
known tensor (possibly including some noise) under a low-rank constraint. This
may be formulated as a least-squares problem. The set of tensors of a given
multilinear rank is known to admit a Riemannian manifold structure, thus
methods of Riemannian optimization are applicable. In our work, we derive the
Riemannian Hessian of an objective function on the low-rank tensor manifolds
using the Weingarten map, a concept from differential geometry. We discuss the
convergence properties of Riemannian trust-region methods based on the exact
Hessian and standard approximations, both theoretically and numerically. We
compare our approach to Riemannian tensor completion methods from recent
literature, both in terms of convergence behaviour and computational
complexity. Our examples include the completion of randomly generated data with
and without noise and recovery of multilinear data from survey statistics
On the Singular Neumann Problem in Linear Elasticity

The Neumann problem of linear elasticity is singular with a kernel formed by
the rigid motions of the body. There are several tricks that are commonly used
to obtain a non-singular linear system. However, they often cause reduced
accuracy or lead to poor convergence of the iterative solvers. In this paper,
different well-posed formulations of the problem are studied through
discretization by the finite element method, and preconditioning strategies
based on operator preconditioning are discussed. For each formulation we derive
preconditioners that are independent of the discretization parameter.
Preconditioners that are robust with respect to the first Lam\'e constant are
constructed for the pure displacement formulations, while a preconditioner that
is robust in both Lam\'e constants is constructed for the mixed formulation. It
is shown that, for convergence in the first Sobolev norm, it is crucial to
respect the orthogonality constraint derived from the continuous problem. Based
on this observation a modification to the conjugate gradient method is proposed
that achieves optimal error convergence of the computed solution
Objective acceleration for unconstrained optimization

Acceleration schemes can dramatically improve existing optimization
procedures. In most of the work on these schemes, such as nonlinear Generalized
Minimal Residual (N-GMRES), acceleration is based on minimizing the $\ell_2$
norm of some target on subspaces of $\mathbb{R}^n$. There are many numerical
examples that show how accelerating general purpose and domain-specific
optimizers with N-GMRES results in large improvements. We propose a natural
modification to N-GMRES, which significantly improves the performance in a
testing environment originally used to advocate N-GMRES. Our proposed approach,
which we refer to as O-ACCEL (Objective Acceleration), is novel in that it
minimizes an approximation to the \emph{objective function} on subspaces of
$\mathbb{R}^n$. We prove that O-ACCEL reduces to the Full Orthogonalization
Method for linear systems when the objective is quadratic, which differentiates
our proposed approach from existing acceleration methods. Comparisons with
L-BFGS and N-CG indicate the competitiveness of O-ACCEL. As it can be combined
with domain-specific optimizers, it may also be beneficial in areas where
L-BFGS or N-CG are not suitable.Comment: 18 pages, 6 figures, 5 table
Low-complexity minimization algorithms

Structured matrix algebras L and a generalized BFGS-type iterative scheme have been recently investigated to introduce low-complexity quasi-Newton methods, named LQN, for solving general (non-structured) minimization problems. In this paper we introduce the LkQN methods, which exploit ad hoc algebras at each step. Since the structure of the updated matrices can be modified at each iteration, the new methods can better fit the Hessian matrix, thereby improving the rate of convergence of the algorithm
Computation of multiple eigenvalues and generalized eigenvectors for
  matrices dependent on parameters

The paper develops Newton's method of finding multiple eigenvalues with one
Jordan block and corresponding generalized eigenvectors for matrices dependent
on parameters. It computes the nearest value of a parameter vector with a
matrix having a multiple eigenvalue of given multiplicity. The method also
works in the whole matrix space (in the absence of parameters). The approach is
based on the versal deformation theory for matrices. Numerical examples are
given. The implementation of the method in MATLAB code is available.Comment: 19 pages, 3 figure
A fast algorithm for computing the smallest eigenvalue of a symmetric positive-definite Toeplitz matrix

Recent progress in signal processing and estimation has generated considerable interest in the problem of computing the smallest eigenvalue of a symmetric positive-definite (SPD) Toeplitz matrix.
An algorithm for computing upper and lower bounds to the smallest eigenvalue of a SPD Toeplitz matrix has been recently derived (Linear Algebra Appl. 2007; DOI: 10.1016/j.laa.2007.05.008). The algorithm relies on the computation of the R factor of the QR factorization of the Toeplitz matrix and the inverse of R. The simultaneous computation of R and R−1 is efficiently accomplished by the generalized Schur algorithm. In this paper, exploiting the properties of the latter algorithm, a numerical method to compute the smallest eigenvalue and the corresponding eigenvector of SPD Toeplitz matrices in an accurate way is proposed.status: publishe
On solving periodic Riccati equations

Numerically reliable algorithms to compute the periodic non-negative definite stabilizing solutions of the periodic differential Riccati equation (PRDE) and discrete-time periodic Riccati equation (DPRE) are proposed. For the numerical solution of PRDEs, a new multiple shooting-type algorithm is developed to compute the periodic solutions in an arbitrary number of time moments within one period by employing suitable discretizations of the continuous-time problems. In contrast to single shooting periodic generator methods, the multiple shooting-type methods have the main advantage of being able to address problems with larger periods. Three methods are discussed to solve DPREs. Two of the methods represetn extensions of a quotient-product swapping and collapsing "fast" algorithm. All proposed approches are completely general, being applicable to periodic Riccati equations with time-varying dimensions as well as with singular weighting matrices
Fast truncation of mode ranks for bilinear tensor operations

We propose a fast algorithm for mode rank truncation of the result of a
bilinear operation on 3-tensors given in the Tucker or canonical form. If the
arguments and the result have mode sizes n and mode ranks r, the computation
costs $O(nr^3 + r^4)$. The algorithm is based on the cross approximation of
Gram matrices, and the accuracy of the resulted Tucker approximation is limited
by square root of machine precision.Comment: 9 pages, 2 tables. Submitted to Numerical Linear Algebra and
  Applications, special edition for ICSMT conference, Hong Kong, January 201
Bounding the equilibrium distribution of Markov population models

Cataloged from PDF version of article.We propose a bounding technique for the equilibrium probability distribution of continuous-time Markov
chains with population structure and infinite state space. We use Lyapunov functions to determine a finite set
of states that contains most of the equilibrium probability mass. Then we apply a refinement scheme based
on stochastic complementation to derive lower and upper bounds on the equilibrium probability for each
state within that set. To show the usefulness of our approach, we present experimental results for several
examples from biology. Copyright © 2011 John Wiley & Sons, Ltd
On the Moore-Penrose inverse in solving saddle-point systems with singular diagonal blocks

This paper deals with the role of the generalized inverses in solving saddle-point systems arising naturally in the solution of many scientific and engineering problems when finite-element tearing and interconnecting based domain decomposition methods are used to the numerical solution. It was shown that the Moore–Penrose inverse may be obtained in this case at negligible cost by projecting an arbitrary generalized inverse using orthogonal projectors. Applying an eigenvalue analysis based on the Moore–Penrose inverse, we proved that for simple model problems, the number of conjugate gradient iterations required for the solution of associate dual systems does not depend on discretization norms. The theoretical results were confirmed by numerical experiments with linear elasticity problems.Web of Science19469967
On the solution of a quadratic vector equation arising in Markovian
  Binary Trees

We present some advances, both from a theoretical and from a computational
point of view, on a quadratic vector equation (QVE) arising in Markovian Binary
Trees. Concerning the theoretical advances, some irreducibility assumptions are
relaxed, and the minimality of the solution of the QVE is expressed in terms of
properties of the Jacobian of a suitable function. From the computational point
of view, we elaborate on the Perron vector-based iteration proposed in
[http://arxiv.org/abs/1006.0577]. In particular we provide a condition which
ensures that the Perron iteration converges to the sought solution of the QVE.
Moreover we introduce a variant of the algorithm which consists in applying the
Newton method instead of a fixed-point iteration. This method has the same
convergence behaviour as the Perron iteration, since its convergence speed
increases for close-to-critical problems. Moreover, unlike the Perron
iteration, the method has a quadratic convergence. Finally, we show that it is
possible to alter the bilinear form defining the QVE in several ways without
changing the solution. This modification has an impact on convergence speed of
the algorithms
Rayleigh wave correction for the BEM analysis of two-dimensional elastodynamic problems in a half-space

This is the pre-peer reviewed version of the following article: Arias, I.; Achenbach, J. Rayleigh wave correction for the BEM analysis of two-dimensional elastodynamic problems in a half-space. "International journal for numerical methods in engineering", Agost 2004, vol. 60, núm. 13, p. 2131-2146, which has been published in final form at http://www3.interscience.wiley.com/journal/109060861/abstractA simple, elegant approach is proposed to correct the error introduced by the truncation of the infinite boundary in the BEM modelling of two-dimensional wave propagation problems in elastic half-spaces. The proposed method exploits the knowledge of the far-field asymptotic behaviour of the solution to adequately correct the BEM displacement system matrix for the truncated problem to account for the contribution of the omitted part of the boundary. The reciprocal theorem of elastodynamics is used for a convenient computation of this contribution involving the same boundary integrals that form the original BEM system. The method is formulated for a two-dimensional homogeneous, isotropic, linearly elastic half-space and its implementation in a frequency domain boundary element scheme is discussed in some detail. The formulation is then validated for a free Rayleigh pulse travelling on a half-space and successfully tested for a benchmark problem with a known approximation to the analytical solution.Peer ReviewedPostprint (author’s final draft
A posteriori finite element bounds to linear functional outputs of the three-dimensional Navier-Stokes equations

An implicit a posteriori finite element error estimation method is presented to inexpensively calculate lower and upper bounds for a linear functional output of the numerical solutions to the three-dimensional Navier-Stokes (N-S) equations. The novelty of this research is to utilize an augmented Lagrangian based on a coarse mesh linearization of the N-S equations and the finite element tearing and interconnecting (FETI) procedure. The latter approach extends the a posteriori bound method to the three-dimensional Crouzeix-Raviart space for N-S problems. The computational advantage of the bound procedure is that a single coupled non-symmetric large problem can be decomposed into several uncoupled symmetric small problems. A simple model problem, which is selected here to illustrate the procedure, is to find the lower and upper bounds of the average velocity of a pressure driven, incompressible, steady Newtonian fluid flow moving at low Reynolds numbers through an endless square channel which has an array of rectangular obstacles. Numerical results show that the bounds for this output are rigorous, i.e. always in the asymptotic certainty regime, that they are sharp and that the required computational resources decrease significantly. Parallel implementation on a Beowulf cluster is also reported. \u
Stress analysis around crack tips in finite strain problems using the eXtended finite element method

International audienceFracture of rubber-like materials is still an open problem. Indeed, it deals with modelling issues (crack growth law, bulk behaviour) and computational issues (robust crack growth in 2D and 3D, incompressibility). The present study focuses on the application of the eXtended Finite Element Method (X-FEM) to large strain fracture mechanics for plane stress problems. Two important issues are investigated: the choice of the formulation used to solve the problem and the determination of suitable enrichment functions. It is demonstrated that the results obtained with the method are in good agreement with previously published works
Solving high-order partial differential equations with indirect radial basis function networks

This paper reports a new numerical method based on radial basis function networks (RBFNs) for solving high-order partial differential equations (PDEs). The variables and their derivatives in the governing equations are represented by integrated RBFNs. The use of integration in constructing neural networks allows the straightforward implementation of multiple boundary conditions and the accurate approximation of high-order derivatives. The proposed RBFN method is verified successfully through the solution of thin-plate bending and viscous flow problems which are governed by biharmonic equations. For thermally driven cavity flows, the solutions are obtained up to a high Rayleigh number
A combined rh-adaptive scheme based on domain subdivision. Formulation and linear examples

An adaptive scheme is proposed in which the domain is split into two subdomains. One subdomain consists of regions where the discretization is refined with an h-adaptive approach, whereas in the other subdomain node relocation or r-adaptivity is used. Through this subdivision the advantageous properties of both remeshing strategies (accuracy and low computer costs, respectively) can be exploited in greater depth. The subdivision of the domain is based on the formulation of a desired element size, which renders the approach suitable for coupling with various error assessment tools. Two-dimensional linear examples where the analytical solution is known illustrate the approach. It is shown that the combined rh-adaptive approach is superior to its components r- and h-adaptivity, in that higher accuracies can be obtained compared to a purely r-adaptive approach, while the computational costs are lower than that of a purely h-adaptive approach. As such, a more flexible formulation of adaptive strategies is given, in which the relative importance of attaining a pre-set accuracy and speeding-up the computational process can be set by the user.Peer ReviewedPostprint (author’s final draft
A high-level programming-language implementation of topology
  optimization applied to steady-state Navier-Stokes flow

We present a versatile high-level programming-language implementation of
nonlinear topology optimization. Our implementation is based on the commercial
software package Femlab, and it allows a wide range of optimization objectives
to be dealt with easily. We exemplify our method by studies of steady-state
Navier-Stokes flow problems, thus extending the work by Borrvall and Petersson
on topology optimization of fluids in Stokes flow [Int. J. Num. Meth. Fluids
2003; 41:77--107]. We analyze the physical aspects of the solutions and how
they are affected by different parameters of the optimization algorithm. A
complete example of our implementation is included as Femlab code in an
appendix.Comment: RevTex 4, 19 pages including 10 figure
Computer-aided simulation and testing of spatial linkages with joint mechanical errors

Tolerance allocation influences production costs in a big way. For this reason it is very important to have an accurate study about the effects of manufacturing errors on the functioning and performances of linkages. In this paper, the authors present a computer-aided methodology based on a 3D geometrical approach using the dual-algebra fundamentals. The purpose is to give ail useful tool which can be integrated into CAD software in order to evaluate the performances of spatial mechanisms with mechanical errors. The proposed methodology has been validated by means of experimental tests on a Cardan joint mechanism with clearances, misalignments and dimensional errors. Copyright (c) 2005 John Wiley & Sons, Ltd
Semi-deterministic and genetic algorithms for global optimization of microfluidic protein folding devices

In this paper we reformulate global optimization problems in terms of boundary value problems (BVP). This allows us to introduce a new class of optimization algorithms. Indeed, current optimization methods, including non-deterministic ones, can be seen as discretizations of initial value problems for differential equations, or systems of differential equations. Furthermore, in order to reduce computational time approximate state and sensitivity evaluations are introduced during optimization. Lastly, we demonstrated the efficacy of two algorithms, included in the former class, on two academic test cases and on the design of a fast microfluidic protein folding device. The aim of the latter design is to reduce mixing times of proteins to microsecond timescales. Results are compared with those obtained with a classical genetic algorithm
Non-conservative and conservative formulations of characteristics-based numerical reconstructions for incompressible flows.

An investigation of characteristics-based (CB) schemes for solving the incompressible Navier-Stokes equations in conjunction with the artificial-compressibility approach, is presented. Both non-conservative and conservative CB numerical reconstructions are derived and their accuracy and convergence properties are assessed analytically and numerically. We demonstrate by means of eigenvalue analysis that there are differences in the spectral characteristics of these formulations that result in different convergence properties. Numerical tests for two- and three-dimensional flows reveal that the two formulations provide similar accuracy but the non-conservative formulation converges faster
Boundary element analysis of an integral equation for three-dmensional crack problems

International audienceIn another paper, the authors proposed  an  integral equation for arbitrary shaped three-dimensional cracks. In  the present paper, a discretization of this equation using a tensor formalism  is  formulated. This approach has  the advantage of providing the displacement discontinuity vector  in  the local basis which varies  as  a function of the point of the crack surface. This also facilitates the computation of the stress intensity factors along the crack  edge.  Numerical examples reported  for  a circular crack and a semi-elliptical surface crack  in  a cylindrical bar show that one can obtain good results, using  few  Gaussian  points  and no singular elements
Arbitrary lagrangian-eulerian finite element analysis of strain localization in transient problems

Non-local models guaranty that finite element computations on strain softening materials remain sound up to failure from a theoretical and computational viewpoint. The non-locality prevents strain localization with zero global dissipation of energy, and consequently finite element calculations converge upon mesh refinements to non-zero width localization zones. One of the major drawbacks of these models is that the element size needed in order to capture the localization zone must be smaller than the intemallength. Hence, the total number of degrees of freedom becomes rapidly prohibitive for most engineering applications and there is an obvious need for mesh adaptivity. This paper deals with the application of the arbitrary Lagrangian-Eulerian (ALE) formulation, well known in hydrodynamics and fluid-structure interaction problems, to transient strain localization in a non-local damageable material. It is shown that the ALE formulation which is employed in large boundary motion problems can also be well suited for non-linear transient analysis of softening materials where localization bands appear. The remeshing strategy is based on the equidistribution of an indicator that quantifies the interelement jump of a selected state variable. Two well known one-dimensional examples illustrate the capabilities of this technique: the first one deals with localization due to a propagating wave in a bar, and the second one studies the wave propagation in a hollow sphere.Peer ReviewedPostprint (author’s final draft