Optimal control problems with PDEs as constraints arise very often in scientific and industrial problems. Due to the difficulties arising in their numerical solution, researchers have put a great effort into devising robust solvers for this class of problems. An example of a highly challenging problem attracting significant attention is the (distributed) control of incompressible viscous fluid flow problems. In this case, the physics may be described, for very viscous flow, by the (linear) incompressible Stokes equations, or, in case the convection of the fluid plays a non-negligible role in the physics, by the (non-linear) incompressible Navier–Stokes equations. In particular, as the PDEs given in the constraints are non-linear, in order to obtain a solution of Navier–Stokes control problems one has to iteratively solve linearizations of the problems until a prescribed tolerance on the non-linear residual is achieved.

In this talk, we present novel, fast, and parameter-robust preconditioned iterative methods for the solution of the distributed time-dependent Navier–Stokes control problems with Crank-Nicolson discretization in time. The key ingredients of the solver are a saddle-point type approximation for the linear systems, an inner iteration for the $(1, 1)$-block accelerated by a generalization of the preconditioner for convection–diffusion control derived in [2], and an approximation to the Schur complement based on a potent commutator argument applied to an appropriate block matrix. The flexibility of the commutator argument, which is a generalization of the technique derived in [1], allows one to alternatively apply a backward Euler scheme in time, as well as to solve the stationary Navier–Stokes control problem. We show the effectiveness and robustness of our approach through a range of numerical experiments.

This talk is based on the work in [3].

[1] D. Kay, D. Loghin, A.J. Wathen. A Preconditioner for the Steady-State Navier–Stokes Equations, SIAM Journal on Scientific Computing 24: 237–256, 2002.

[2] S. Leveque, J.W. Pearson. Fast Iterative Solver for the Optimal Control of Time-Dependent PDEs with Crank–Nicolson Discretization in Time, Numerical Linear Algebra with Applications 29: e2419, 2022.

[3] S. Leveque, J.W. Pearson. Parameter-Robust Preconditioning for Oseen Iteration Applied to Stationary and Instationary Navier–Stokes Control, SIAM Journal on Scientific Computing 44: B694–B722, 2022.

In this talk, we will present a method for Bayesian inference by implicitly defining a posterior distribution as the solution to a regularized linear least-squares problem with randomized data. This method combines Gaussian distributions with the deterministic effects of sparsity-inducing regularization like $l_1$ norms, total variation and/or constraints. The resulting posterior distributions assign positive probability to various low-dimensional subspaces and therefore promote sparsity. Samples from this distribution can be generated by repeatedly solving regularized linear least-squares problems with properly chosen data perturbations, thus, existing tools from optimization theory can be used for sampling. We will discuss some properties of the methodology and discuss an efficient algorithm for sampling from a Bayesian hierarchical model with sparsity structure.

We propose a rapid and robust iterative algorithm to solve inverse acoustic scattering problems formulated as a PDE constrained shape optimization problem. We use a level-set method to represent the obstacle geometry and propose a new scheme for updating the geometry based on an adaptation of accelerated gradient descent methods. The resulting algorithm aims at reducing the number of iterations and improving the accuracy of reconstructions. To cope with regularization issues, we propose a smoothing to the shape gradient using a single layer potential associated with $ik$ where $k$ is the wave number. Numerical experiments are given for several data types (full aperture, backscattering, phaseless, multiple frequencies) and show that our method outperforms a nonaccelerated approach in terms of convergence speed, accuracy, and sensitivity to initial guesses.

We consider a pde-constrained optimization problem and based on the nonlinear primal dual proximal splitting method, a nonconvex generalization of the well-known Chambolle-Pock algorithm, we develop a new iterative algorithmic approach to the problem by splitting the inner problem of solving the pde in each step over the outer iterations. In our work we split our pde-problem in a fashion similar to the classical Gauss-Seidel and Jacobi methods, though other iterative schemes may be fruitful too. We show through numerical experiments that significant speed ups can be attained compared to a naive full pde-solve in each step, and we prove convergence under sufficients second-order growth conditions.