A new approach to nonlinear constrained Tikhonov regularization

We present a novel approach to nonlinear constrained Tikhonov regularization from the viewpoint of optimization theory. A second-order sufficient optimality condition is suggested as a nonlinearity condition to handle the nonlinearity of the forward operator. The approach is exploited to derive convergence rates results for a priori as well as a posteriori choice rules, e.g., discrepancy principle and balancing principle, for selecting the regularization parameter. The idea is further illustrated on a general class of parameter identification problems, for which (new) source and nonlinearity conditions are derived and the structural property of the nonlinearity term is revealed. A number of examples including identifying distributed parameters in elliptic differential equations are presented.Comment: 21 pages, to appear in Inverse Problem

Discrete localized modes supported by an inhomogeneous defocusing nonlinearity

We report that infinite and semi-infinite lattices with spatially inhomogeneous self-defocusing (SDF)\ onsite nonlinearity, whose strength increases rapidly enough toward the lattice periphery, support stable unstaggered (UnST) discrete bright solitons, which do not exist in lattices with the spatially uniform SDF nonlinearity. The UnST solitons coexist with stable staggered (ST) localized modes, which are always possible under the defocusing onsite nonlinearity. The results are obtained in a numerical form, and also by means of variational approximation (VA). In the semi-infinite (truncated) system, some solutions for the UnST surface solitons are produced in an exact form. On the contrary to surface discrete solitons in uniform truncated lattices, the threshold value of the norm vanishes for the UnST solitons in the present system. Stability regions for the novel UnST solitons are identified. The same results imply the existence of ST discrete solitons in lattices with the spatially growing self-focusing nonlinearity, where such solitons cannot exist either if the nonlinearity is homogeneous. In addition, a lattice with the uniform onsite SDF nonlinearity and exponentially decaying inter-site coupling is introduced and briefly considered too. Via a similar mechanism, it may also support UnST discrete solitons, under the action of the SDF nonlinearity. The results may be realized in arrayed optical waveguides and collisionally inhomogeneous Bose-Einstein condensates trapped in deep optical lattices. A generalization for a two-dimensional system is briefly considered too.Comment: 14 pages, 7 figures, accepted for publication in PR

Third-order nonlinear optical properties of bismuth-borate glasses measured by conventional and thermally managed eclipse Z scan

Third-order nonlinearity one order of magnitude larger than silica is measured in bismuth-borate glasses presenting a fast response (<200 fs). The results for the sign and magnitude of the nonlinearity were obtained using a combination of the eclipse Z scan with thermal nonlinearity managed Z scan, whereas the Kerr shutter technique was employed to obtain the electronic time response of the nonlinearity, all performed with 76 MHz repetition rate 150 fs pulses at 800 nm. Conventional Z scans in the picosecond regime at 532 and 1064 nm were also independently performed, yielding the values of the third-order nonlinear susceptibilities at those wavelengths. The results obtained for the femtosecond response, enhanced third-order nonlinearity of this glass (with respect to silica), place this glass system as an important tool in the development of photonics devices. Electro-optical modulators, optical switches, and frequency converters are some of the applications using second-order nonlinear properties of the Bi-glass based on the rectification model