We introduce a numerical methodology which applies to a broad class of partial differential equations and discrete models, and is referred to here as the transport-based mesh-free method. It led us to several numerical algorithms which are now implemented in a Python library, called CodPy. We develop a mesh-free discretization technique based on the (so-called RKHS) theory of reproducing kernels and the theory of transport mappings, in a way that is reminiscent of Lagrangian methods in computational fluid dynamics. The strategy is relevant when a large number of dimensions or degrees of freedom are present, as is the case in mathematical finance and machine learning, but is also applicable in fluid dynamics. We present our algorithms primarily for the Fokker-Planck-Kolmogorov system of mathematical finance and for neural networks based on support vector machines. The proposed algorithms are nonlinear in nature and enjoy quantitative error estimates based on the notion of discrepancy error, which allow one to evaluate the relevance and accuracy of given data and numerical solutions.
Philippe G. LeFloch holds a permanent position at Sorbonne University, as a Research Professor of the Centre National de la Recherche Scientifique (CNRS). He graduated from the Ecole Normale Superieure (Saint- Cloud) and, in 1988, obtained a Ph.D. in Applied Mathematics from the Ecole Polytechnique (Palaiseau). In 1995, he received a Faculty Early Career Development award from the National Science Foundation. He worked at the Courant Institute of Mathematical Sciences (NY) and at the University of Southern California. P.G. LeFloch published about 250 research papers with more than 100 different co-authors, and several textbooks including graduate courses. For the past ten years he developed a new methodology for mathematical finance and data science, which led to the development of the CodPy algorithms.
Jean-Marc Mercier is the head of the research and development team at MPG-Partners, a consulting firm for the financial services industry. He graduated from the University of Bordeaux (France) with a Ph.D. in applied mathematics obtained in 1996. He started his career as an Academic researcher, then moved to engineering in the finance industry. He now devotes his time to solve a variety of challenging industrial problems which he tackles with fundamental research tools and which led to the development of the CodPy algorithms.