Numerical Fractional Diffusion: a Pde Approach, Integral Formulation, Extensions and Applications

By now it is clear that computational science is one of the bases, together with theory and experiments, of scientific inquiry. A carefully crafted computational model/experiment can replace a very expensive or unrealizable experimental setting, and it can give new insight into the theoretical developments of a specific discipline. Discussions about computer aided simulation and high performance computing are recurring themes in science. But, are these computational models a faithful representation of the underlying physical processes? How can we make sure that the results obtained by the computer are meaningful? The answer to these questions is, evidently, not a simple one. Even if we, for the time being, assume that the computer is always right, we still might obtain meaningless results simply because the model that we are using does not describe the process of our interest. How to detect this kind of errors is a question for the specialists in the physical sciences and so I will discuss it no more. In the discussion above, a big assumption was made: the computer is right. But, how can we guarantee that? Even more, we may ask, How can we get the best possible answer with the least possible amount of effort? A quest for the answer to these and many related questions is the underlying reason for numerical analysis, which is the analysis of computational schemes. It offers us, in short, a solid mathematical background that describes to what extent the computer’s output approximates the process of interest. In general, my research concentrates on the numerical analysis of partial differential equations (PDE), in particular on fractional diffusion. Diffusion is the tendency of a substance to evenly spread into an available space, and is one of the most common physical processes. The classical models of diffusion lead to well known models and even better studied equations. However, in recent times, it has become evident that many of the assumptions that lead to these models are not always satisfactory or not even realistic at all. Consequently, different models of diffusion have been proposed, fractional diffusion being one of them. Inspired by a novel result from PDE [4], in our research we have made a decisive advance in the numerical solution and analysis of fractional diffusion, a relatively new but rapidly growing area of research. Although its mathematical analysis is highly nontrivial, the computational implementation of such method is done using standard components of numerical analysis; see [7, 8, 11, 13]. This is the main advantage of our scheme, since the approaches advocated in the literature require special attention due to the mathematical difficulties inherent to fractional diffusion; see [15] for a discussion. The mathematical structure of fractional diffusion is shared by a wide class of mathematical objects: nonlocal operators. These objects have a strong connection with realworld problems, since they constitute a fundamental part of the modeling and simulation of complex phenomena. Their applications are vast, and span control theory, finance, electromagnetic fluids, image processing, materials science, optimization, porous media flow, turbulence, continuum field theories and others. We will be concerned with applications in image processing and optimal control of PDE; see [1] for preliminary results in control theory. From this it is evident that the particular type of mathematical object appearing in applications can widely vary and that a unified analysis might be well beyond our reach. A more modest, but nevertheless quite ambitious, goal is to develop a computational analysis that is representative of a particular class: fractional diffusion. In mathematical terms, exploiting the cylindrical extension proposed and investigated by X. Cabré and J. Tan [3] and Capella, Dávila, Dupaigne and Sire [6], in turn inspired in a work of L. Caffarelli and L. Silvestre [4], we have replaced the (intricate) integral