The long-term dynamics of many dynamical systems evolves on a low-dimensional, attracting, invariant slow manifold, which can be parameterized by only a few variables (“observables”). The explicit derivation of such a slow manifold (and thus, the reduction of the long-term system dynamics) can be extremely difficult, or practically impossible. For this class of problems, the equation-free compu...

This paper presents a nonlinear solver based on the Newton-Krylov methods, where the Newton equations are solved by Krylov-subspace type approaches. We focus on the solution of unsteady systems, in which the temporal terms are discretized by the backward Euler method using finite difference. To save computational cost, an adaptive time stepping is used to minimize the number of time steps. The ...

Inexact Newton methods for finding a zero of F 1 1 are variations of Newton's method in which each step only approximately satisfies the linear Newton equation but still reduces the norm of the local linear model of F. Here, inexact Newton methods are formulated that incorporate features designed to improve convergence from arbitrary starting points. For each method, a basic global convergence ...

We analyse the elementwise convergence in the Jacobian of the classical Newton method as the solution of a non-linear system f(x) = 0 is approached. It is shown that each Jacobian element approaches a constant value at a rate determined by the nonlinearity of the relevant function with respect to the appropriate scalar variable. For many practical problems, this means that in the last steps of ...

Periodic solutions of certain large scale systems of ODEs can be computed eeciently using a hybrid Newton-Picard scheme, especially in a continuation context. In this paper we describe and analyse a numerical method to accurately compute period doubling bifurcation points using the Newton-Picard approach. The method avoids the computation of the full monodromy matrix, which is present in the de...

This paper proposes a new predictor-corrector interior-point method for a class of semidefinite programs, which numerically traces the central trajectory in a space of Lagrange multipliers. The distinguished features of the method are full use of the BFGS quasi-Newton method in the corrector procedure, and an application of the conjugate gradient method with an effective preconditioning matrix ...

The randomized subspace Newton convex methods for the sensor selection problem are proposed. algorithm is straightforwardly applied to formulation, and customized method in which part of update variables selected be present best candidates also considered. In converged solution, almost same results obtained by original randomized-subspace-Newton methods. As expected, require more computational ...

This paper proposes a new predictor-corrector interior-point method for a class of semidefinite programs, which numerically traces the central trajectory in a space of Lagrange multipliers. The distinguished features of the method are full use of the BFGS quasi-Newton method in the corrector procedure and an application of the conjugate gradient method with an effective preconditioning matrix i...

In this paper we study new preconditioners to be used within the Nonlinear Conjugate Gradient (NCG) method, for large scale unconstrained optimization. The rationale behind our proposal draws inspiration from quasi– Newton updates, and its aim is to possibly approximate in some sense the inverse of the Hessian matrix. In particular, at the current iteration of the NCG we consider some precondit...