University of Nice Sophia-Antipolis

Whitney elements on simplices are perhaps the most widely used finite elements in computational electromagnetics. They offer the simplest construction of polynomial discrete differential forms on simplicial complexes. Their associated degrees of freedom (dofs) have a very clear physical meaning and give a recipe for discretizing physical balance laws, e.g., Maxwell’s equations. As interest grew for the use of high order schemes, such as hp-finite element or spectral element methods, higher-order extensions of Whitney forms have become an important computational tool, appreciated for their better convergence and accuracy properties. However, it has remained unclear what kind of cochains such elements should be associated with: Can the corresponding dofs be assigned to precise geometrical elements of the mesh, just as, for instance, a degree of freedom for the space of Whitney 1-forms belongs to a specific edge? We address this localization issue. Why is this an issue? The existing constructions of high order extensions of Whitney elements follow the traditional FEM path of using higher and higher “moments” to define the needed dofs. As a result, such high order finite k-elements in d dimensions include dofs associated to q-simplices, with k < q ≤ d, whose physical interpretation is obscure. The present paper offers an approach based on the so-called “small simplices”, a set of subsimplices obtained by homothetic contractions of the original mesh simplices, centered at mesh nodes (or more generally, when going up in degree, at points of the principal lattice of each original simplex). Degrees of freedom of the high-order Whitney k-forms are then associated with small simplices of dimension k only.  We provide an explicit  basis for these elements on simplices and we justify this approach from a geometric point of view (in the spirit of Hassler Whitney's approach, still successful 30 years after his death).   

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please send email to @email.

 We provide in this work a robust solution theory for random rough differential equations of mean field type

$$

dX_t = V\big( X_t,{\mathcal L}(X_t)\big)dt + \textrm{F}\bigl( X_t,{\mathcal L}(X_t)\bigr) dW_t,

$$

where $W$ is a random rough path and ${\mathcal L}(X_t)$ stands for the law of $X_t$, with mean field interaction in both the drift and diffusivity. Propagation of chaos results for large systems of interacting rough differential equations are obtained as a consequence, with explicit convergence rate. The development of these results requires the introduction of a new rough path-like setting and an associated notion of controlled path. We use crucially Lions' approach to differential calculus on Wasserstein space along the way. This is a joint work with I. Bailleul and R. Catellier.

Joint work with I. Bailleul (Rennes) and R. Catellier (Nice)

(Joint work with P.E. Chaudru de Raynal and F. Delarue)

Several problems in financial mathematics, in particular that of contingent claim pricing, may be casted as decoupled Forward Backward Stochastic Differential Equations (FBSDEs). It is then of a practical interest to find efficient algorithms to solve these equations numerically with a reasonable complexity.

An efficient numerical approach using cubature on Wiener spaces was introduced by Crisan and Manoralakis [1]. The algorithm uses an approximation scheme requiring the calculation of conditional expectations, a task achieved through the cubature kernel approximation. This algorithm features several advantages, notably the fact that it is possible to solve with the same cubature tree several decoupled FBSDEs with different boundary conditions. There is, however, a drawback of this method: an exponential growth of the algorithm's complexity.

In this talk, we introduce a modification on the cubature method based on the recombination method of Litterer and Lyons [2] (as an alternative to Tree Branch Based Algorithm proposed in [1]). The main idea of the method is to modify the nodes and edges of the cubature trees in such a way as to preserve, up to a constant, the order of convergence of the expectation and conditional expectation approximations obtained via the cubature method, while at the same time controlling the complexity growth of the algorithm.

We have obtained estimations on the order of convergence and complexity growth of the algorithm under smoothness assumptions on the coefficients of the FBSDE and uniform ellipticity of the forward equation. We discuss that, just as in the case of the plain cubature method, the order of convergence of the algorithm may be degraded as an effect of solving FBSDEs with rougher boundary conditions. Finally, we illustrate the obtained estimations with some numerical tests.

References

[1] Crisan, D., and K. Manolarakis. “Solving Backward Stochastic Differential Equations Using the Cubature Method. Application to Nonlinear Pricing.” In Progress in Analysis and Its Applications, 389–397. World Sci. Publ., Hackensack, NJ, 2010.

[2] Litterer, C., and T. Lyons. “High Order Recombination and an Application to Cubature on Wiener Space.” The Annals of Applied Probability 22, no. 4 (August 2012): 1301–1327. doi:10.1214/11-AAP786.