Linearizing non-linear inverse problems and an application to inverse backscattering

We propose an abstract approach to prove local uniqueness and conditional Hölder stability to non-linear inverse problems by linearization. The main condition is that, in addition to the injectivity of the linearization A, we need a stability estimate for A as well. That condition is satisfied in particular, if A^∗A is an elliptic pseudo-differential operator. We apply this scheme to show uniqueness and Hölder stability for the inverse backscattering problem for the acoustic equation near a constant sound speed.     

On travelling-wave solutions for a moving boundary problem of Hele-Shaw type

We discuss a 2D moving boundary problem for the Laplacian withRobin boundary conditions in an exterior domain. It arises asa model for Hele–Shaw flow of a bubble with kinetic undercoolingregularization and is also discussed in the context of modelsfor electrical streamer discharges. The corresponding evolutionequation is given by a degenerate, non-linear transport problemwith non-local lower-order dependence. We identify the localstructure of the set of travelling-wave solutions in the vicinityof trivial (circular) ones. We find that there is a unique non-trivialtravelling wave for each velocity near the trivial one. Therefore,the trivial solutions are unstable in a comoving frame. Thedegeneracy of our problem is reflected in a loss of regularityin the estimates for the linearization. Moreover, there is anupper bound for the regularity of its solutions. To prove ourresults, we use a quasi-linearization by differentiation, indexresults for degenerate ordinary differential operators on thecircle and perturbation arguments for unbounded Fredholm operators.     

Trapping of a Thiolate→Dibromine Charge‐Transfer Adduct by a Macrocyclic Dinickel Complex and Its Conversion into an Arenesulfenyl Bromide Derivative

Stuck on sulfur : The first transition‐metal complexes with S? Br units are surprisingly stable. Solid 3 is stable for at least six months and under vacuum solid 2 does not lose Br₂. The formation of the first structurally characterized transition‐metal arenesulfenyl bromide complex 3 occurs with a change of the spin ground state from S=2 to S=0.

     

Efficient calculation of the worst-case error and (fast) component-by-component construction of higher order polynomial lattice rules

We show how to obtain a fast component-by-component construction algorithm for higher order polynomial lattice rules. Such rules are useful for multivariate quadrature of high-dimensional smooth functions over the unit cube as they achieve the near optimal order of convergence. The main problem addressed in this paper is to find an efficient way of computing the worst-case error. A general algorithm is presented and explicit expressions for base 2 are given. To obtain an efficient component-by-component construction algorithm we exploit the structure of the underlying cyclic group. We compare our new higher order multivariate quadrature rules to existing quadrature rules based on higher order digital nets by computing their worst-case error. These numerical results show that the higher order polynomial lattice rules improve upon the known constructions of quasi-Monte Carlo rules based on higher order digital nets.     

High-efficiency ultrasmall polarization converter in InP membrane

An ultrasmall (<10 μm length) polarization converter in InP membrane is fabricated and characterized. The device relies on the beating between the two eigenmodes of chemically etched triangular waveguides. Measurements show a very high polarization conversion efficiency of >99% with insertion losses of <-1.2 dB at a wavelength of 1.53?μm. Furthermore, our design is found to be broadband and tolerant to dimension variations.