7,293 research outputs found
On an adaptive regularization for ill-posed nonlinear systems and its trust-region implementation
In this paper we address the stable numerical solution of nonlinear ill-posed
systems by a trust-region method. We show that an appropriate choice of the
trust-region radius gives rise to a procedure that has the potential to
approach a solution of the unperturbed system. This regularizing property is
shown theoretically and validated numerically.Comment: arXiv admin note: text overlap with arXiv:1410.278
Local convergence of the Levenberg-Marquardt method under H\"{o}lder metric subregularity
We describe and analyse Levenberg-Marquardt methods for solving systems of
nonlinear equations. More specifically, we propose an adaptive formula for the
Levenberg-Marquardt parameter and analyse the local convergence of the method
under H\"{o}lder metric subregularity of the function defining the equation and
H\"older continuity of its gradient mapping. Further, we analyse the local
convergence of the method under the additional assumption that the
\L{}ojasiewicz gradient inequality holds. We finally report encouraging
numerical results confirming the theoretical findings for the problem of
computing moiety conserved steady states in biochemical reaction networks. This
problem can be cast as finding a solution of a system of nonlinear equations,
where the associated mapping satisfies the \L{}ojasiewicz gradient inequality
assumption.Comment: 30 pages, 10 figure
The geometry of nonlinear least squares with applications to sloppy models and optimization
Parameter estimation by nonlinear least squares minimization is a common
problem with an elegant geometric interpretation: the possible parameter values
of a model induce a manifold in the space of data predictions. The minimization
problem is then to find the point on the manifold closest to the data. We show
that the model manifolds of a large class of models, known as sloppy models,
have many universal features; they are characterized by a geometric series of
widths, extrinsic curvatures, and parameter-effects curvatures. A number of
common difficulties in optimizing least squares problems are due to this common
structure. First, algorithms tend to run into the boundaries of the model
manifold, causing parameters to diverge or become unphysical. We introduce the
model graph as an extension of the model manifold to remedy this problem. We
argue that appropriate priors can remove the boundaries and improve convergence
rates. We show that typical fits will have many evaporated parameters. Second,
bare model parameters are usually ill-suited to describing model behavior; cost
contours in parameter space tend to form hierarchies of plateaus and canyons.
Geometrically, we understand this inconvenient parametrization as an extremely
skewed coordinate basis and show that it induces a large parameter-effects
curvature on the manifold. Using coordinates based on geodesic motion, these
narrow canyons are transformed in many cases into a single quadratic, isotropic
basin. We interpret the modified Gauss-Newton and Levenberg-Marquardt fitting
algorithms as an Euler approximation to geodesic motion in these natural
coordinates on the model manifold and the model graph respectively. By adding a
geodesic acceleration adjustment to these algorithms, we alleviate the
difficulties from parameter-effects curvature, improving both efficiency and
success rates at finding good fits.Comment: 40 pages, 29 Figure
Matrix-Norm Approach of Computing Levenberg-Marquardt Reg- ularization Parameter for Nonlinear Equations
In this paper, we present Levenberg-Marquardt method for solving nonlinear systems of equations. Here, both the objective function and the symmetric Jacobian matrix are assumed to be Lipchitz continuous. The regularization parameter is derived using Matrix-Norm approach. Numerical performance on some benchmark problems that demonstrates the effectiveness and efficiency of our approach are reported and have shown that the proposed algorithm is very promising.Mathematics Subject Classification: 65H10, 65K05, 65F22, 65F35.keywords: Nonlinear system of equations. Levenberg-Marquardt method. Regularization. Matrix-norm. Global convergence
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