## Contributed Session Mon.2.MA 415

#### Monday, 13:15 - 14:45 h, Room: MA 415

**Cluster 19: PDE-constrained optimization & multi-level/multi-grid methods** [...]

### Iterative solution of PDE constrained optimization and subproblems

**Chair: Lutz Lehmann**

**Monday, 13:15 - 13:40 h, Room: MA 415, Talk 1**

**Pavel Zhlobich**

Multilevel quasiseparable matrices in PDE-constrained optimization

**Coauthor: Jacek Gondzio**

**Abstract:**

Discretization of PDE-constrained optimization problems leads

to linear systems of saddle-point type. Numerical solution of such systems is often a challenging task due to their large size and poor spectral properties. In this work we propose and develop the novel approach to solving saddle-point systems, which is based on the exploitation of low-rank structure of discretized differential operators and their inverses. This structure is known in the scientific literature as "quasiseparable''. One may think of a usual quasiseparable matrix as of a discrete analog of Green's function of a one-dimensional differential operator. The remarkable feature of such matrices is that almost all of the algorithms with them have linear complexity. We extend the range of applicability of quasiseparable matrices to problems in higher dimensions. In particular, we construct a class of preconditioners that can be computed and applied at a linear computational cost. Their use with appropriate Krylov subspace methods leads to algorithms for solving saddle-point systems mentioned above of asymptotically linear complexity.

**Monday, 13:45 - 14:10 h, Room: MA 415, Talk 2**

**Gregor Kriwet**

Covariance matrix computation for parameter estimation in nonlinear models solved by iterative linear algebra methods

**Abstract:**

For solving parameter estimation (PE) and optimum experimental design (OED) problems we need covariance matrix of the parameter estimates. So far numerical methods for PE and OED in dynamic processes have been based on direct linear algebra methods which involve explicit matrix factorizations. They are originally developed for systems of non-linear DAE where direct linear algebra methods are more effective for forward model problems than iterative methods. On the other hand for large scale constrained problems with sparse matrices of special structure, e.g., originating from discretization of PDE, direct linear algebra methods are not competitive with iterative linear algebra methodseven for forward models. Hence, for PDE models, generalizations of iterative linear algebra methods to the computation of covariance matrices are crucial for practical applications. One of the intriguing results is that solving nonlinear constrained least squares problems by Krylov type methods we get as a by-product the covariance matrix and confidence intervals.

The talk is based on joint work with H. G. Bock, E. Kostina, O. Kostyukova, I. Schierle and M. Saunders.

**Monday, 14:15 - 14:40 h, Room: MA 415, Talk 3**

**Lutz Lehmann**

Optimal sequencing of primal, adjoint and design steps.

**Coauthors: Torsten Bosse, Andreas Griewank**

**Abstract:**

Many researchers have used design optimization methods based on a user specified primal state iteration, corresponding adjoint iterations and appropriately preconditioned design steps. Our goal is to develop heuristics for the sequencing of these three subtasks in order to optimize the convergence rate of the resulting coupled iteration. We present numerical results that confirm the theoretical predictions.