OCCAM Wednesday Morning Event (past)

********** PLEASE NOTE THE SPECIAL TIME **********

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.

Complex systems emerging from many biochemical applications often exhibit multiscale and multiphysics (MSMP) features: The systems incorporate a variety of physical processes or subsystems across a broad range of scales. A typical MSMP system may come across scales with macroscopic, mesoscopic and microscopic kinetics,
deterministic and stochastic dynamics, continuous and discrete state space, fastscale and slow-scale reactions, and species of both large and small populations. These complex features present great challenges in the modeling and simulation practice. The goal of our research is to develop innovative computational methods and rigorous fundamental theories to answer these challenges. In this talk we will start with introduction of basic stochastic simulation algorithms for biochemical systems and multiscale
features in the stochastic cell cycle model of budding yeast. With detailed analysis of these multiscale features, we will introduce recent progress on simulation algorithms and computational theories for multiscale stochastic systems, including tau-leaping methods, slow-scale SSA, and the hybrid method. 

One of the main problems occurring in the aorta is the development of aneurysms, in which case the artery wall thickens and its diameter increases. Suffice to say that many other factors may be involved in this process. These include, amongst others, geometry, non-homogeneous material, anisotropy, growth, remodeling, age, etc. In this talk, we examine the bifurcation of inflated thick-walled cylindrical shells under axial loading and its interpretation in terms of the mechanical response of arterial tissue and the formation and propagation of aneurysms. We will show that this mechanical approach is able to capture features of the mechanisms involved during the formation and propagation of aneurysms.

Although the importance of hydrologic uncertainty is widely recognized it is rarely considered in control problems, especially real-time control. One of the reasons is that stochastic control is computationally expensive, especially when control decisions are derived from spatially distributed models. This talk reviews relevant control concepts and describes how reduced order models can make stochastic control feasible for computationally demanding applications. The ideas are illustrated with a classic problem -- hydraulic control of a moving contaminant plume.

Telomeres, non-coding terminal structures of DNA strands, consist of repetitive long tandem repeats of a specific length. An absence of an enzyme, telomerase, in certain cellular structures requires an alternative telomerase-independent pathway for telomeric sequence length regulation. Besides linear telomeres other configurations such as telomeric circles and telomeric loops were experimentally observed. They are suspected to play an important role in a universal mechanism for stabilization of the ends of linear DNA that possibly dates back to pre-telomerase ages. We propose a mathematical model that captures biophysical interactions of various telomeric structures on a short time scale and that is able to reproduce experimental measurements in mtDNA of yeast. Moreover, the model opens up a couple of interesting mathematical problems such as validity of a quasi-steady state approximation and dynamic properties of discrete coagulation-fragmentation systems. We also identify and estimate key factors influencing the length distribution of telomeric circles, loops and strand invasions using numerical simulations.