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Person# Tommaso Vanzan

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Partial differential equation

In mathematics, a partial differential equation (PDE) is an equation which computes a function between various partial derivatives of a multivariable function.
The function is often thought of as

Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a d

Domain decomposition methods

In mathematics, numerical analysis, and numerical partial differential equations, domain decomposition methods solve a boundary value problem by splitting it into smaller boundary value problems on

Courses taught by this person (1)

MATH-616: Numerical methods for random PDEs and uncertainty

The course focuses on mathematical models based on PDEs with random parameters, and presents numerical techniques for forward uncertainty propagation, inverse uncertainty analysis in a Bayesian framework and optimal control under uncertainty.

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This work presents a new computational optimization framework for the robust control of parks of Wave Energy Converters (WEC) in irregular waves. The power of WEC parks is maximized with respect to the individual control damping and stiffness coefficients of each device. The results are robust with respect to the incident wave direction, which is treated as a random variable. Hydrodynamic properties are computed using the linear potential model, and the dynamics of the system is computed in the frequency domain. A slamming constraint is enforced to ensure that the results are physically realistic. We show that the stochastic optimization problem is well posed. Two optimization approaches for dealing with stochasticity are then considered: stochastic approximation and sample average approximation. The outcomes of the above mentioned methods in terms of accuracy and computational time are presented. The results of the optimization for complex and realistic array configurations of possible engineering interest are then discussed. Results of extensive numerical experiments demonstrate the efficiency of the proposed computational framework.

We present a multigrid algorithm to solve efficiently the large saddle-point systems of equations that typically arise in PDE-constrained optimization under uncertainty. The algorithm is based on a collective smoother that at each iteration sweeps over the nodes of the computational mesh, and solves a reduced saddle-point system whose size depends on the number of samples used to discretize the probability space. We show that this reduced system can be solved with optimal complexity. We test the multigrid method on three problems: a linear-quadratic problem for which the multigrid method is used to solve directly the linear optimality system; a nonsmooth problem with box constraints and L^1 norm penalization on the control, in which the multigrid scheme is used within a semismooth Newton iteration; a risk-adverse problem with the smoothed CVaR risk measure where the multigrid method is called within a preconditioned Newton iteration. In all cases, the multigrid algorithm exhibits very good performances and robustness with respect to all parameters of interest.

2023We present a combination technique based on mixed differences of both spatial approximations and quadrature formulae for the stochastic variables to solve efficiently a class of Optimal Control Problems (OCPs) constrained by random partial differential equations. The method requires to solve the OCP for several low-fidelity spatial grids and quadrature formulae for the objective functional. All the computed solutions are then linearly combined to get a final approximation which, under suitable regularity assumptions, preserves the same accuracy of fine tensor product approximations, while drastically reducing the computational cost. The combination technique involves only tensor product quadrature formulae, thus the discretized OCPs preserve the convexity of the continuous OCP. Hence, the combination technique avoids the inconveniences of Multilevel Monte Carlo and/or sparse grids approaches, but remains suitable for high dimensional problems. The manuscript presents an a-priori procedure to choose the most important mixed differences and an asymptotic complexity analysis, which states that the asymptotic complexity is exclusively determined by the spatial solver. Numerical experiments validate the results.