Dynamically orthogonal tensor methods for high-dimensional nonlinear PDEs

Tensor sparsity of solutions of high dimensional PDEs

We introduce a class of Bernstein-Durrmeyer operators with respect to an arbitrary measure on a multi-dimensional simplex. These operators generalize the well-known Bernstein-Durrmeyer operators with Jacobi weights. A motivation for this generalization comes from learning theory. In the talk, we discuss the question which properties of the measure are important for convergence of the operators....

On the dynamically orthogonal approximation of time dependent random PDEs

In this work we discuss the Dynamically Orthogonal (DO) approximation of time dependent partial differential equations with random data. The approximate solution is expanded at each time instant on a time dependent orthonormal basis in the physical domain with fixed and small number of terms. Dynamic equations are written for the evolution of the basis as well as the evolution of the stochastic...

Infinite dimensional backstepping for nonlinear parabolic PDEs

Numerical Methods for Nonlinear PDEs in Finance

Many problems in finance can be posed in terms of an optimal stochastic control. Some well-known examples include transaction cost/uncertain volatility models [17, 2, 25], passport options [1, 26], unequal borrowing/lending costs in option pricing [9], risk control in reinsurance [23], optimal withdrawals in variable annuities[13], optimal execution of trades [20, 19], and asset allocation [28,...

Adaptive tensor product wavelet methods for solving PDEs