Manuel Jesús Castro Díaz

Manuel Jesús Castro Díaz


  • His research areas include: finite volume schemes for hyperbolic nonconservative systems, modeling and simulation of geophysical flows, high eficient algorithms and software for geophysical flows and GPU computing.
  • He has published a wide range of papers in top-level interantional scientific journals and colaborated with interdisciplinary research groups.
  • He develops software for geophysical flows, with application to a wide range of tests, physical and academic.
  • He has participated, as collaborator and director, in different research projects.


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Plenary talk: Multi-level Monte Carlo Finite volume method: an efficient  method for uncertainty quantification in geophysical flows


Existing numerical methods for approximating PDEs appearing in the simulation of geophysical flows require the initial data, boundary conditions, coefficients, and source terms as inputs. However, in most practical situations, it is not possible to measure these inputs precisely. The inputs are uncertain and this uncertainty propagates in the solution. The modeling and approximation of the propagation of uncertainty in the solution due to uncertainty in inputs constitutes the theme of uncertainty quantification (UQ).  UQ for geophysical flows is vitally important for risk evaluation and hazard mitigation.

The modeling and computation of solution statistics is highly non-trivial. Challenges include possibly large number of random variables (fields) to parametrize the uncertainty and the sheer computational challenge of evaluating statistical moments that might necessitate a large number of PDE solves. The challenges are particularly accentuated for hyperbolic and convection-dominated PDEs as the discontinuities in physical space such as shocks can propagate into stochastic space resulting in a loss of regularity of the underlying solution with respect to the random parameters. A very large number of degrees of freedom in stochastic space might be needed to resolve such irregular functions. 
Nevertheless, several numerical methods have been developed for UQ in hyperbolic PDEs. Methods include the stochastic Galerkin methods based on generalized Polynomial Chaos (gPC), stochastic collocation methods and stochastic finite volume methods (SFVM). Some of these methods (particularly stochastic Galerkin) have the huge disadvantage of being highly intrusive: existing codes for computing deterministic solutions of conservation laws need to be completely reconfigured for implementation. Furthermore, none of these methods are currently able to handle even a moderate number of sources of uncertainty (stochastic dimensions).
Another class of methods are the so-called Monte Carlo (MC) methods in which the probability space is sampled, the underlying deterministic PDE is solved for each sample and the samples are combined to determine statistical information about the random field. Although non-intrusive, easy to code and to parallelize, MC methods converge at rate 1/2 as the number M of MC samples increases. 
This slow convergence entails high computational costs for MC type methods and makes them infeasible for computing uncertainty in complex geophysical flows.  This slow convergence has inspired the development of Multi-Level Monte Carlo or MLMC methods. In particular, Mishra, Schwab and Sukys have extended and analyzed the MLMC algorithm for scalar conservation laws and for systems of conservation laws, respectively. The asymptotic analysis for the MLMC method, presented Mishra and Schwab, showed that the method allows the computation of approximate statistical moments with the same accuracy versus cost ratio as a single deterministic solve on the same mesh.
In this talk,  Multi-level Monte Carlo (MLMC) method for approximating the stochastic hyperbolic PDEs appearing in the simulation  of some geophysical flow problems is presented. This entails discretizing space, time as well as the probability space. For spatio-temporal discretization, we will employ a PVM-path-conservative  finite volume scheme and the probabilistic space is discretized by means of MLMC method.
Finally, some comments about the GPU implementation of the method  and some applications to several geophysical problems, like the generation and propagation of a tsunami, or the evolution of a sedimentary basin will be presented.