An extension of the Gauss-Newton algorithm is proposed to find local minimizers of penalized nonlinear least squares problems, under generalized Lipschitz assumptions. Convergence results of local type are obtained, as well as an estimate of the radius of the convergence ball. Some applications for solving constrained nonlinear equations are discussed and the numerical performance of the method is assessed on some significant test problems.
An alternative approach is the classical Tikhonov method @cite_35 replacing by the minimization of the associated Tikhonov functional. A problem of the same type of @math arises, with @math weighted by a properly chosen parameter. Up to our knowledge, besides our method, the papers by Ramlau and Teschke, e.g. @cite_17 @cite_27 , are the only ones providing algorithms for the minimization of such type of functionals. In addition, the scheme they propose is different from ours and in general it converges only weakly. On the other hand, our algorithm assumes the derivative @math to be injective with closed range --- a hypothesis not suitable for handling the ill-posed case.
@cite_35 @cite_27 @cite_17
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We shall be concerned with the construction of Tikhonov-based iteration schemes for solving nonlinear operator equations. In particular, we are interested in algorithms for the computation of a minimizer of the Tikhonov functional. To this end, we introduce a replacement functional, that has much better properties than the classical Tikhonov functional with nonlinear operator. Namely, the replacement functional is globally convex and can effectively be minimized by a fixed point iteration. On the basis of the minimizers of the replacement functional, we introduce an iterative algorithm that converges towards a critical point of the Tikhonov functional, and under additional assumptions for the nonlinear operator F, to a global minimizer. Combining our iterative strategy with an appropriate parameter selection rule, we obtain convergence and convergence rates. The performance of the resulting numerical scheme is demonstrated by solving the nonlinear inverse SPECT (single photon emission computerized tomography) problem. In this paper, we consider nonlinear inverse problems where the solution is assumed to have a sparse expansion with respect to a preassigned basis or frame. We develop a scheme which allows to minimize a Tikhonov functional where the usual quadratic regularization term is replaced by a one-homogeneous (typically weighted l p ) penalty on the coefficients (or isometrically transformed coefficients) of such expansions. For (p < 2), the regularized solution will have a sparser expansion with respect to the basis or frame under consideration. The computation of the regularized solution amounts in our setting to a Landweber-fixed-point iteration with a projection applied in each fixed-point iteration step. The performance of the resulting numerical scheme is demonstrated by solving the nonlinear inverse single photon emission computerized tomography (SPECT) problem. For variational problems of the form Infv∈V f(Av)+g(v) , we propose a dual method which decouples the difficulties relative to the functionals f and g from the possible ill-conditioning effects of the linear operator A. The approach is based on the use of an Augmented Lagrangian functional and leads to an efficient and simply implementable algorithm. We study also the finite element approximation of such problems, compatible with the use of our algorithm. The method is finally applied to solve several problems of continuum mechanics. We present a new efficient algorithm for the search version of the approximate Closest Vector Problem with Preprocessing (CVPP). Our algorithm achieves an approximation factor of O(n sqrt log n ), improving on the previous best of O(n^ 1.5 ) due to Lag arias, Lenstra, and Schnorr hkzbabai . We also show, somewhat surprisingly, that only O(n) vectors of preprocessing advice are sufficient to solve the problem (with the slightly worse approximation factor of O(n)). We remark that this still leaves a large gap with respect to the decisional version of CVPP, where the best known approximation factor is O(sqrt n log n ) due to Aharonov and Regev AharonovR04 . To achieve these results, we show a reduction to the same problem restricted to target points that are close to the lattice and a more efficient reduction to a harder problem, Bounded Distance Decoding with preprocessing (BDDP). Combining either reduction with the previous best-known algorithm for BDDP by Liu, Lyubashevsky, and Micciancio LiuLM06 gives our main result. In the setting of CVP without preprocessing, we also give a reduction from (1+eps)gamma approximate CVP to gamma approximate CVP where the target is at distance at most 1+1 eps times the minimum distance (the length of the shortest non-zero vector) which relies on the lattice sparsification techniques of Dadush and Kun DadushK13 . As our final and most technical contribution, we present a substantially more efficient variant of the LLM algorithm (both in terms of run-time and amount of preprocessing advice), and via an improved analysis, show that it can decode up to a distance proportional to the reciprocal of the smoothing parameter of the dual lattice MR04 . We show that this is never smaller than the LLM decoding radius, and that it can be up to an wide tilde Omega (sqrt n ) factor larger.