Inverse Problems for Anomalous Diffusion Processes (22w5043)

The Banff International Research Station will host the "Inverse Problems for Anomalous Diffusion Processes" workshop in Banff from May 8 - 13, 2022.

In his ``annus mirabalis'' of 1905 Einstein published four far reaching papers. Two of these have been idolized in the popular literature: the equivalence of mass and energy $(E=mc^2$) and special relativity. A third won a Nobel Prize for the photoelectric effect, but the fourth is by far the most cited. This is his explanation of Brownian motion, the seemingly chaotic behaviour of small particles when seen under a microscope, which had been discovered in the early nineteenth century. In the Einstein formulation the change in the direction of motion of a particle is random and that its average long term displacement is proportional to the time taken. It is easily shown that this ``random walk'' model gives rise to one of the classical equations of mathematical physics the so-called heat equation. This is an example of a partial differential equation and such creatures which are an extension of familiar ideas from calculus are the lingua franca of the physical sciences and engineering. In such language one interprets the first derivative of the displacement as a velocity, the second as an acceleration and so forth.

However, this tidy picture makes fundamental assumptions on the underlying materials that more recent research has shown simply don't hold. The correction for this leads to so-called ``anomalous diffusion'' where the random walk fails the Einstein formula. The outcome is that instead of the partial differential equation with normal derivatives one obtains an equation with fractional derivatives; these animals have quite different properties from ones of integer order.

A standard problem is to assume a physical model and its attendant parameters such as the value of the conductivity in a heat conduction experiment. This leads, using the differential equation, to an ability to predict the value of the temperature at any given point is space or time. This direct or forwards problem makes the assumption that all the ingredients are known. The inverse problem in some sense reverses this; We may not know the value of this conductivity which may in fact vary from point to point, but ask the question: ``If we are able to measure, say, the temperature at some later time, can we use this information to reconstruct the conductivity?'' Questions as these (and their positive answers) form the basis of modern medical imaging. Such inverse problems are mathematically difficult.

Again using the above example, we want to know if our measured data allows a recovery of a unique conductivity or a whole family of possibilities. We also want to know if small errors in our data measurement correspond to small errors in our reconstruction of the conductivity. Unfortunately the answer to the latter question is often negative; it is prohibited by the accumulated randomness in the underlying model. Quantifying this latter effect is an important question and has been intensively studied over the last few decades. This all leads to answering similar questions under the now more complex model where our integer order equation is replaced by one with fractional derivatives. This adds considerable mathematical complexity but has important physical applications; for example a negative answer means that under ``anomalous materials'' time cannot be effectively reversed or if so, with a fundamentally different outcome.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).