What is the subject of your research at Inria?
I am working on the development of numerical methods for computer simulations of partial differential equations (PDEs). In the POMDAPI project team, we are traditionally concentrating on equations describing environmental phenomena such as fluid flow and contaminant transport in subsurface porous media. These equations are quite scary: unsteady, nonlinear, degenerate, coupled... The applications are then, e.g., predictions of subsurface pollution after chemical leakage, security calculations for deep underground nuclear waste repositories, or verifications of intended projects of geological sequestration of carbon dioxide.
What is the ERC grant about?
Although numerical simulations of porous media have been an important research subject for decades, in computational practice, the accuracy of the final outcome is not guaranteed and, surprisingly, often as much as such 90% of the CPU time is literally wasted. The reason is that the issue is complex (it links several rather disconnected domains like mathematical modeling, analysis of PDEs, numerical analysis, numerical linear algebra, and scientific computing) and addressing it rigorously is extremely challenging. In the GATIPOR project, we shall design novel inexact algebraic and linearization solvers, where each step is viewed in the context of the simulated PDE and adaptively steered by an a posteriori error estimate. Kind of interconnecting all parts of the numerical simulation (algebraic solver, linearization solver, spatial discretization, time stepping), analyzing at each moment what is going on, and trying to take the next step in a good direction. Some of the deep numerical concepts are problem- and discretization-dependent multilevel algebraic solvers and local adaptive stopping criteria. We shall theoretically prove the convergence of the new algorithms and justify their optimality. We shall also test how they work for challenging real-life problems such as those described above. The final goal: certify the total simulation error and cut by orders of magnitude the current computational burden.
What is an a posteriori error estimate?
In the last decades, a beautiful mathematical theory was developed, which allows for a miracle: you do not know the exact solution of your PDE, yet you can estimate the error in a numerical simulation. A posteriori error estimates are at the very heart of the GATIPOR project.
What does this grant mean for you?
Appreciation of the whole community. Good material resources to go on. And also a reward for my whole family, for the many white nights...
What attracted you to this field of research?
I have always loved mathematics, but I have also always been attracted to do something applicable, so that hopefully the results directly serve in practice. This is how I came to mathematical engineering and to numerical analysis of PDEs more specifically.
Born in 1977 in the Czech Republic, Martin Vohralik did his Ph.D. under double supervision between the University Paris-Sud (Laboratory of Mathematics) and the Czech Technical University in Prague (Faculty of Nuclear Sciences and Physical Engineering). He started his research career as a postdoctoral fellow at the French National Center for Scientific Research in Orsay, in collaboration with the HydroExpert company. In 2006, Martin Vohralik joined the Pierre and Marie Curie (Paris 6) University (Jacques-Louis Lions Laboratory) as an associate professor. In 2012, he then became a research director at INRIA in the project-team POMDAPI. He received the SIEMENS “Prize for Research” for his diploma thesis in 2000, benefited from the J. T. Oden Faculty fellowship for his stays at the University of Texas at Austin between 2008 et 2011, and became member of the editorial board of the SIAM Journal on Numerical Analysis in 2013.