25,706 research outputs found

    Continuation-conjugate gradient methods for the least squares solution of nonlinear boundary value problems

    Get PDF
    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

    Full text link
    This manuscript proposes a probabilistic framework for algorithms that iteratively solve unconstrained linear problems Bx=bBx = b with positive definite BB for xx. The goal is to replace the point estimates returned by existing methods with a Gaussian posterior belief over the elements of the inverse of BB, 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

    Get PDF
    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
    • …
    corecore