25,706 research outputs found
Continuation-conjugate gradient methods for the least squares solution of nonlinear boundary value problems
We discuss in this paper a new combination of methods for solving nonlinear boundary value problems containing a parameter. Methods of the continuation type are combined with least squares formulations, preconditioned conjugate gradient algorithms and finite element approximations.
We can compute branches of solutions with limit points, bifurcation points, etc.
Several numerical tests illustrate the possibilities of the methods discussed in the present paper; these include the Bratu problem in one and two dimensions, one-dimensional bifurcation and perturbed bifurcation problems, the driven cavity problem for the Navier–Stokes equations
Probabilistic Interpretation of Linear Solvers
This manuscript proposes a probabilistic framework for algorithms that
iteratively solve unconstrained linear problems with positive definite
for . The goal is to replace the point estimates returned by existing
methods with a Gaussian posterior belief over the elements of the inverse of
, which can be used to estimate errors. Recent probabilistic interpretations
of the secant family of quasi-Newton optimization algorithms are extended.
Combined with properties of the conjugate gradient algorithm, this leads to
uncertainty-calibrated methods with very limited cost overhead over conjugate
gradients, a self-contained novel interpretation of the quasi-Newton and
conjugate gradient algorithms, and a foundation for new nonlinear optimization
methods.Comment: final version, in press at SIAM J Optimizatio
Bridging Proper Orthogonal Decomposition methods and augmented Newton-Krylov algorithms: an adaptive model order reduction for highly nonlinear mechanical problems
This article describes a bridge between POD-based model order reduction
techniques and the classical Newton/Krylov solvers. This bridge is used to
derive an efficient algorithm to correct, "on-the-fly", the reduced order
modelling of highly nonlinear problems undergoing strong topological changes.
Damage initiation problems are addressed and tackle via a corrected
hyperreduction method. It is shown that the relevancy of reduced order model
can be significantly improved with reasonable additional costs when using this
algorithm, even when strong topological changes are involved
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