University of Bath

In my talk I will introduce the concept of spectral discrete solitons

(SDSs): solutions of nonlinear Schroedinger type equations, which are localized on a regular grid in frequency space. In time domain such solitons correspond to periodic trains of pulses. SDSs play important role in cascaded four-wave-mixing processes (frequency comb generation) in optical fibres, where initial excitation by a two-frequency pump leads to the generation of multiple side-bands. When free space diffraction is taken into consideration, a non-trivial generalization of 1D SDSs will be discussed, in which every individual harmonic is an optical vortex with its own topological charge. Such excitations correspond to spatio-temporal helical beams.

TBA

We show that data assimilation using four-dimensional variation

(4DVar) can be interpreted as a form of Tikhonov regularisation, a

familiar method for solving ill-posed inverse problems. It is known from

image restoration problems that $L_1$-norm penalty regularisation recovers

sharp edges in the image better than the $L_2$-norm penalty

regularisation. We apply this idea to 4DVar for problems where shocks are

present and give some examples where the $L_1$-norm penalty approach

performs much better than the standard $L_2$-norm regularisation in 4DVar.

As research in topology optimisation has reached a level of maturity, two main classes of methods have emerged and their applications to real engineering design in industry are increasing. It has therefore become important to identify the limitations and challenges in order to ensure that topology optimisation is appropriately employed during the design process whilst research may continue to offer a more reliable and fast design tool to engineers.

The seminar will begin by introducing the topology optimisation problem and the two popular finite element based approaches. A range of numerical methods used in the typical implementations will be outlined. This will form the basis for the discussion on the short-comings and challenges as an easy-to-use design tool for engineers, particularly in the context of reliably providing the consistent optimum solutions to given problems with minimum a priori information. Another industrial requirement is a fast solution time to easy-to-set-up problems. The seminar will present the recent efforts in addressing some of these issues and the remaining challenges for the future.

Multi-well relaxation problem emerges e.g. in characterising effective properties of composites and in phase transformations. This is a nonlinear problem and one approach uses its reformulation in Fourier space, known in the theory of composites as Hashin-Shtrikman approach, adapted to nonlinear composites by Talbot and Willis. Characterisation of admissible mixtures, subjected to appropriate differential constraints, leads to a quasiconvexification problem. The latter is equivalently reformulated in the Fourier space as minimisation with respect to (extremal points of) H-measures of characteristic functions (Kohn), which in a sense separates the microgeometry of mixing from the differential constraints. For three-phase mixtures in 3D we obtain a full characterisation of certain extremal H-measures. This employs Muller's Haar wavelet expansion estimates in terms of Riesz transform to establish via the tools of harmonic analysis weak lower semicontinuity of certain functionals with rank-2 convex integrands. As a by-product, this allows to fully solve the problem of characterisation of quasiconvex hulls for three arbitrary divergence-free wells. We discuss the applicability of the results to problems with other kinematic constraints, and other generalisations. Joint work with Mariapia Palombaro, Leipzig.