Adaptive SDE based interpolation for random PDEs

A numerical method for the fully adaptive sampling and interpolation of PDE with random data is presented. It is based on the idea that the solution of the PDE with stochastic data can be represented as conditional expectation of a functional of a corresponding stochastic differential equation (SDE). The physical domain is decomposed subject to a non-uniform grid and a classical Euler scheme is employed to approximately solve the SDE at grid vertices. Interpolation with a conforming finite element basis is employed to reconstruct a global solution of the problem. An a posteriori error estimator is introduced which provides a measure of the different error contributions. This facilitates the formulation of an adaptive algorithm to control the overall error by either reducing the stochastic error by locally evaluating more samples, or the approximation error by locally refining the underlying mesh. Numerical examples illustrate the performance of the presented novel method

SDE based regression for random PDEs

A simulation based method for the numerical solution of PDE with random coefficients is presented. By the Feynman-Kac formula, the solution can be represented as conditional expectation of a functional of a corresponding stochastic differential equation driven by independent noise. A time discretization of the SDE for a set of points in the domain and a subsequent Monte Carlo regression lead to an approximation of the global solution of the random PDE. We provide an initial error and complexity analysis of the proposed method along with numerical examples illustrating its behaviour

Biomaterial based strategies to reconstruct the nigrostriatal pathway in organotypic slice co-cultures

Protection or repair of the nigrostriatal pathway represents a principal disease-modifying therapeutic strategy for Parkinson's disease (PD). Glial cell line-derived neurotrophic factor (GDNF) holds great therapeutic potential for PD, but its efficacious delivery remains difficult. The aim of this study was to evaluate the potential of different biomaterials (hydrogels, microspheres, cryogels and microcontact printed surfaces) for reconstructing the nigrostriatal pathway in organotypic co-culture of ventral mesencephalon and dorsal striatum. The biomaterials (either alone or loaded with GDNF) were locally applied onto the brain co-slices and fiber growth between the co-slices was evaluated after three weeks in culture based on staining for tyrosine hydroxylase (TH). Collagen hydrogels loaded with GDNF slightly promoted the TH+ nerve fiber growth towards the dorsal striatum, while GDNF loaded microspheres embedded within the hydrogels did not provide an improvement. Cryogels alone or loaded with GDNF also enhanced TH+ fiber growth. Lines of GDNF immobilized onto the membrane inserts via microcontact printing also significantly improved TH+ fiber growth. In conclusion, this study shows that various biomaterials and tissue engineering techniques can be employed to regenerate the nigrostriatal pathway in organotypic brain slices. This comparison of techniques highlights the relative merits of different technologies that researchers can use/develop for neuronal regeneration strategies

Geometric methods on low-rank matrix and tensor manifolds

In this chapter we present numerical methods for low-rank matrix and tensor problems that explicitly make use of the geometry of rank constrained matrix and tensor spaces. We focus on two types of problems: The first are optimization problems, like matrix and tensor completion, solving linear systems and eigenvalue problems. Such problems can be solved by numerical optimization for manifolds, called Riemannian optimization methods. We will explain the basic elements of differential geometry in order to apply such methods efficiently to rank constrained matrix and tensor spaces. The second type of problem is ordinary differential equations, defined on matrix and tensor spaces. We show how their solution can be approximated by the dynamical low-rank principle, and discuss several numerical integrators that rely in an essential way on geometric properties that are characteristic to sets of low rank matrices and tensors

Transglutaminase activation in neurodegenerative diseases

The following review examines the role of calcium in promoting the in vitro and in vivo activation of transglutaminases in neurodegenerative disorders. Diseases such as Alzheimer's disease, Parkinson's disease and Huntington's disease exhibit increased transglutaminase activity and rises in intracellular calcium concentrations, which may be related. The aberrant activation of transglutaminase by calcium is thought to give rise to a variety of pathological moieties in these diseases, and the inhibition has been shown to have therapeutic benefit in animal and cellular models of neurodegeneration. Given the potential clinical relevance of transglutaminase inhibitors, we have also reviewed the recent development of such compounds

A mesh-free partition of unity method for diffusion equations on complex domains

We present a numerical method for solving partial differential equations on domains with distinctive complicated geometrical properties. These will be called complex domains. Such domains occur in many real-world applications, for example in geology or engineering. We are, however, particularly interested in applications stemming from the life sciences, especially cell biology. In this area complex domains, such as those retrieved from microscopy images at different scales, are the norm and not the exception. Therefore geometry is expected to directly influence the physiological function of different systems, for example signalling pathways. New numerical methods that are able to tackle such problems in this important area of application are urgently needed. In particular, the mesh generation problem has imposed many restrictions in the past. The approximation approach presented here for such problems is based on a promising mesh-free Galerkin method: the partition of unity method (PUM). We introduce the main approximation features and then focus on the construction of appropriate covers as the basis of discretizations. As a main result we present an extended version of cover construction, ensuring fast convergence rates in the solution process. Parametric patches are introduced as a possible way of approximating complicated boundaries without increasing the overall problem size. Finally, the versatility, accuracy and convergence behaviour of the PUM are demonstrated in several numerical examples