To follow
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Total generalised variation (TGV) was introduced by Bredies et al. as a high quality regulariser for variational problems arising in mathematical image processing like denoising and deblurring. The main advantage over the classical total variation regularisation is the elimination of the undesirable stairscasing effect. In this talk we will give a small introduction to TGV and provide some properties of the exact solutions to the L^{2}-TGV model in the one dimensional case.
******************** PLEASE NOTE THIS SEMINAR WILL TAKE PLACE ON TUESDAY ********************
Well-known iterative schemes for the solution of ill-posed linear equations are the Landweber iteration, the cg-iteration and semi-iterative algorithms like the $\nu$-methods. After introducing these methods, we show that for ill-posed problems a slight modification of the underlying three-term recurrence relation of the $\nu$-methods provides accelerated Landweber algorithms with better performance properties than the $\nu$-methods. The new semi-iterative methods are based on the family of co-dilated ultraspherical polynomials. Compared to the standard $\nu$-methods, the residual polynomials of the modified methods have a faster decay at the origin. This results in an earlier termination of the iteration if the spectrum of the involved operator is clustered around the origin. The convergence order of the modified methods turns out to be the same as for the original $\nu$-methods. The new algorithms are tested numerically and a simple adaptive scheme is developed in which an optimal dilation parameter is determined. At the end, the new semi-iterative methods are used to solve a parameter identification problem obtained from a model in elastography.
In this talk aimed at a general audience I will discuss the ways in which we have used mathematical models of the regulation of haematopoiesis (blood cell production) to understand haematological diseases, and suggest successful treatment strategies for these diseases. At the end I will talk about our current work on tailoring chemotherapy so that it has less damaging effects on the haematopoietic system and, consequently, improve the quality of life for patients being treated for a variety of tumours.
***** PLEASE NOTE THIS SEMINAR WILL TAKE PLACE ON TUESDAY 19TH FEBRUARY *****
With "fully anisotropic" I describe diffusion models of the form u_t=\nabla \nabla (D(x) u), where the diffusion tensor appears inside both derivatives. This model arises naturally in the modeling of brain tumor spread and wolf movement and other applications. Since this model does not satisfy a maximum principle, it can lead to interesting spatial pattern formation, even in the linear case. I will present a detailed derivation of this model and discuss its application to brain tumors and wolf movement. Furthermore, I will present an example where, in the linear case, the solution blows-up in infinite time; a quite surprising result for a linear parabolic equation (joint work with K.J. Painter and M. Winkler).
When two drops come into contact they will rapidly merge and form a single drop. Here we address the coalescence of drops on a substrate, focussing on the initial dynamics just after contact. For very viscous drops we present similarity solutions for the bridge that connects the two drops, the size of which grows linearly with time. Both the dynamics and the self-similar bridge profiles are verified quantitatively by experiments. We then consider the coalescence of water drops, for which viscosity can be neglected and liquid inertia takes over. Once again, we find that experiments display a self-similar dynamics, but now the bridge size grows with a power-law $t^{2/3}$. We provide a scaling theory for this behavior, based on geometric arguments. The main result for both viscous and inertial drops is that the contact angle is important as it determines the geometry of coalescence -- yet, the contact line dynamics appears irrelevant for the early stages of coalescence.
***** PLEASE NOTE THIS SEMINAR WILL TAKE PLACE ON THURSDAY 31ST JANUARY *****