22 research outputs found
A Newton-like method and its application
AbstractIn this paper we prove an existence and uniqueness theorem for solving the operator equation F(x)+G(x)=0, where F is a Gateaux differentiable continuous operator while the operator G satisfies a Lipschitz-condition on an open convex subset of a Banach space. As corollaries, a theorem of Tapia on a weak Newton's method and the classical convergence theorem for modified Newton-iterates are deduced. An existence theorem for a generalized Euler–Lagrange equation in the setting of Sobolev space is obtained as a consequence of the main theorem. We also obtain a class of Gateaux differentiable operators which are nowhere Frechet differentiable. Illustrative examples are also provided
Refraction-corrected ray-based inversion for three-dimensional ultrasound tomography of the breast
Ultrasound Tomography has seen a revival of interest in the past decade,
especially for breast imaging, due to improvements in both ultrasound and
computing hardware. In particular, three-dimensional ultrasound tomography, a
fully tomographic method in which the medium to be imaged is surrounded by
ultrasound transducers, has become feasible. In this paper, a comprehensive
derivation and study of a robust framework for large-scale bent-ray ultrasound
tomography in 3D for a hemispherical detector array is presented. Two
ray-tracing approaches are derived and compared. More significantly, the
problem of linking the rays between emitters and receivers, which is
challenging in 3D due to the high number of degrees of freedom for the
trajectory of rays, is analysed both as a minimisation and as a root-finding
problem. The ray-linking problem is parameterised for a convex detection
surface and three robust, accurate, and efficient ray-linking algorithms are
formulated and demonstrated. To stabilise these methods, novel
adaptive-smoothing approaches are proposed that control the conditioning of the
update matrices to ensure accurate linking. The nonlinear UST problem of
estimating the sound speed was recast as a series of linearised subproblems,
each solved using the above algorithms and within a steepest descent scheme.
The whole imaging algorithm was demonstrated to be robust and accurate on
realistic data simulated using a full-wave acoustic model and an anatomical
breast phantom, and incorporating the errors due to time-of-flight picking that
would be present with measured data. This method can used to provide a
low-artefact, quantitatively accurate, 3D sound speed maps. In addition to
being useful in their own right, such 3D sound speed maps can be used to
initialise full-wave inversion methods, or as an input to photoacoustic
tomography reconstructions
An acceleration technique for a conjugate direction algorithm for nonlinear regression
A linear acceleration technique, LAT, is developed which is applied to three conjugate direction algorithms: (1) Fletcher-Reeves algorithm, (2) Davidon-Fletcher-Powell algorithm and (3) Grey\u27s Orthonormal Optimization Procedure (GOOP). Eight problems are solved by the three algorithms mentioned above and the Levenberg-Marquardt algorithm. The addition of the LAT algorithm improves the rate of convergence for the GOOP algorithm in all problems attempted and for some problems using the Fletcher-Reeves algorithm and the Davidon-Fletcher-Powell algorithm. Using the number of operations to perform function and derivative evaluations, the algorithms mentioned above are compared. Although the GOOP algorithm is relatively unknown outside of the optics literature, it was found to be competitive with the other successful algorithms. A proof of convergence of the accelerated GOOP algorithm for nonquadratic problems is also developed --Abstract, page ii
ON THE RELATION BETWEEN 2 LOCAL CONVERGENCE THEORIES OF LEAST-CHANGE SECANT UPDATE METHODS
In this paper, we show that the main results of the local convergence theory for least-change secant update methods of Dennis and Walker (SIAM J. Numer. Anal. 18 (1981), 949-987) can be proved using the theory introduced recently by Martinez (Math. Comp. 55 (1990), 143-167). In addition, we exhibit two generalizations of well-known methods whose local convergence can be easily proved using Martinez's theory.5920045748
Novos resultados sobre formulas secantes e aplicações
Orientador: Jose Mario MartinezTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação CientíficaResumo: Não informado.Abstract: Not informed.DoutoradoDoutor em Matemática Aplicad
A trust region-type normal map-based semismooth Newton method for nonsmooth nonconvex composite optimization
We propose a novel trust region method for solving a class of nonsmooth and
nonconvex composite-type optimization problems. The approach embeds inexact
semismooth Newton steps for finding zeros of a normal map-based stationarity
measure for the problem in a trust region framework. Based on a new merit
function and acceptance mechanism, global convergence and transition to fast
local q-superlinear convergence are established under standard conditions. In
addition, we verify that the proposed trust region globalization is compatible
with the Kurdyka-{\L}ojasiewicz (KL) inequality yielding finer convergence
results. We further derive new normal map-based representations of the
associated second-order optimality conditions that have direct connections to
the local assumptions required for fast convergence. Finally, we study the
behavior of our algorithm when the Hessian matrix of the smooth part of the
objective function is approximated by BFGS updates. We successfully link the KL
theory, properties of the BFGS approximations, and a Dennis-Mor{\'e}-type
condition to show superlinear convergence of the quasi-Newton version of our
method. Numerical experiments on sparse logistic regression and image
compression illustrate the efficiency of the proposed algorithm.Comment: 56 page
Novos resultados sobre formulas secantes e aplicações
Tese (doutorado) - Universidade Estadual de Campinas. Instituto de Matematica Estatistica e Ciencias da Computaçã