Juan Carlos Cortés Lopéz

Polytechnic University of Valencia

Spain

Quantifying uncertainties in differential equations with Jumps

Differential equations are powerful tools to model real-world problems. However, their deterministic formulation neglects epistemic and aleatoric uncertainties coming from the lack of knowledge of the phenomenon under modeling and possible unpredictable facts often inherent to the random nature of the studied problem. Apart from randomness, the dynamics of many phenomena are often subject to abrupt changes such as shocks, harvesting or natural disasters. These short-term perturbations on rapid changes can be mathematically treated as having acted instantaneously using discontinuities or impulses. Combining the rigorous treatment of these two elements, uncertainties and impulses in the setting of differential equations is a significant challenge that is currently attracting the scientific community’s attention. However, more progress has been made using stochastic differential equations where randomness is driven by a specific stochastic process, typically white noise and the Poisson process. This talk will present some recent results about Uncertainty Quantification for differential equations with discontinuities and jumps using the so-called Random Differential Equations (RDEs) approach. In this setting, uncertainties can be directly assumed in all the terms (initial conditions, source term and coefficients) of the RDE using arbitrary distributions. Moreover, we will present a method to calculate the probability density function of the solution, which is a stochastic process, rather than obtaining only the first moments. The theoretical findings will be illustrated using simulations and real-world examples where inverse Uncertainty Quantification techniques will be applied to reasonably fix the distributions of the model parameters from real data.

Short bio

Juan Carlos Cortés López (http://www.upv.es/ficha-personal/jccortes), is a full professor in the Department of Applied Mathematics at the Universitat Politècncia de Valéncia (UPV), Spain. Since 2009, he is the deputy director of the University Institute of Multidisciplinary Mathematics of the UPV, developing his research in the MUNQU group (Modeling and Uncertainty Quantification) https://munqu.webs.upv.es/. His research is oriented to the extension of the classical theory of differential equations (in a broad sense; ODEs, PDEs, equations with delay, fractional ODEs, etc.) to the random scenario under the approach of the so-called Random Differential Equations (RDEs) and their application in modeling with real data, paying attention to the development of inverse techniques for the estimation of the probability distributions of the parameters that appear in the RDEs so that they capture the uncertainty of the phenomenon being modeled.