## Abstract

The Matheon project D-SE9 focuses on the analysis and efficient numerical solution of optimal control problems for nonlinear evolution equations with Maxwell’s evolution equations as a challenging benchmark example. In particular, we aim at developing low rank approximation techniques for the solution of forward-backward optimality systems that arise whenever optimal control problems for evolution equations are considered. In this project, we thus merge existing expertise in optimal control and low rank matrix and tensor approximation.

This research is carried out in the framework of Matheon supported by the Einstein Foundation Berlin.

## Background

### Optimal control of PDEs

One of the main trends in the optimal control of partial differential equations (PDEs) is the investigation of problems that are modeled by nonlinear systems of equations. We refer e.g. to cardiac medicine, magneto-hydrodynamics, heat treatment of steel, or semiconductor crystal growth to name only a few applications. Related to Maxwell equations, we mention exemplary [Kolmbauer,Langer,2012], [Yousept,Troltzsch,2012], [Hoemberg,Petzold,Rocca,2013], and [Nicaise,2014].

Such fairly complex problems share one main obstacle – the curse of dimension. For this reason, the application of fast numerical methods of optimal control such as semi-smooth Newton or sequential quadratic optimization (SQP) methods, is more or less limited to academic problems in low spatial dimension.

For evolution equations, these techniques require the solution of forward-backward optimality systems consisting of the (forward) state equation and the (backward) adjoint equation. As of today, there is no practicable method to tackle them in higher spatial dimension $$n \ge 2$$.

### Low rank approximation

Low rank approximation of matrices has been used in various fields of applications.
While the best approximation in the $$L_2$$-norm is obtained by (variants of) the well known SVD decomposition, the best rank $$r$$ approximation w.r.t. to other norms is in general an NP hard problem. Good approximations can be obtained by alternating directional search (ALS) or by local optimization methods based on Riemannian gradient techniques (RG) [Koch,Lubich,2012], [Absil,Mahony,Sepulchre,2008].

The former approach is commonly used in model reduction settings and is then known as (full) POD. The latter on the other hand is novel to this setting. In this project we aim to remedy this and apply these methods to the model reduction setting.

## Highlights

Figure 1 comparison of repeated POD to improve bases and a steepest descent using the ALS.

Our implementation show numerical evidence that our local optimization methods like ALS and RG clearly outperform well established standard POD based methods in terms of accuracy. This marks a real breakthrough for our project since, at the beginning, it was an open question whether tensor network techniques can be helpful in practice.
Figure 1 shows the quality of solutions for both mentioned algorithms for an example of the time dependent heat equation in which the source term is controlled.

We did not restrict ourselves to a specific basis in the gradient iteration. Instead the solution of the state equation as well as the adjoint equation were obtained e.g. by the ALS algorithm, that modifies the basis as well. Thus the usual gradient or conjugated gradient methods can be used to simultaneously improve the bases. The convergence and monotonicity results that already exist for the ALS and RG can be expanded to this algorithm as well.

Figure 2 several iterates of the fixed-point iteration for $$P$$.

In optimal control problems it is important to allow for pointwise inequality constraints on the control.
To include such constraints, we merged a semi-smooth Newton method with a fixed point strategy. In the semismooth Newton method, a nonlinear (and non-differentiable) projection like $$P(x) = \max(x,0)$$ appears. In its exact form the complexity of this projection is exponential in the order of the tensor. Instead we have modified a fix-point iteration to preserve the low-rank structure of its iterates (see Figure 2).
This iteration can be used to obtain an approximation of the pointwise application of $$P$$ and its complexity is only linear in the order and polynomial in the rank of the TT tensor.

By expanding the numerical library Xerus to allow sparse component tensors in hierarchical tensor representations and its algorithms, this library now offers the first openly available source code that combines the advantages of low rank structure and sparsity.

## People

 Prof. Dr. Reinhold Schneider – project head Prof. Dr. Fredi Tröltzsch – project head Benjamin Huber – research staff Sebastian Wolf – research staff

## Publications

### Software

 Large parts of the open-source, general purpose tensor and tensor-decomposition library Xerus have been developed in the course of this project. Documentation – Git Repository – Github

### Refereed publications

1. Eduardo Casas and Fredi Tröltzsch.
Second order optimality conditions and their role in pde control.
Jahresbericht der Deutschen Mathematiker-Vereinigung, 117(1):3–44, 2015.
2. Wolfgang Hackbusch and Reinhold Schneider.
Tensor spaces and hierarchical tensor representations.
In Extraction of Quantifiable Information from Complex Systems, pages 237–261. Springer, 2014.
3. Pedro Merino, Ira Neitzel, and Fredi Tröltzsch.
An adaptive numerical method for semi-infinite elliptic control problems based on error estimates.
Optimization Methods and Software, 30(3):492–515, 2015.
4. Serge Nicaise, Simon Stingelin, and Fredi Tröltzsch.
On two optimal control problems for magnetic fields.
Computational Methods in Applied Mathematics, 14(4):555–573, 2014.
5. Serge Nicaise, Simon Stingelin, and Fredi Tröltzsch.
Optimal control of magnetic fields in flow measurement.
Discrete and Continuous Dynamical Systems-S, 8:579–605, 2015.
6. Serge Nicaise and Fredi Tröltzsch.
A coupled Maxwell integrodifferential model for magnetization processes.
Mathematische Nachrichten, 287(4):432–452, 2014.
7. Reinhold Schneider and André Uschmajew.
Approximation rates for the hierarchical tensor format in periodic sobolev spaces.
Journal of Complexity, 30(2):56–71, 2014.
8. Reinhold Schneider and André Uschmajew.
Convergence results for projected line-search methods on varieties of low-rank matrices via Lojasiewicz inequality.
SIAM Journal on Optimization, 25(1):622–646, 2015.
9. Fredi Tröltzsch and Alberto Valli.
Optimal control of low-frequency electromagnetic fields in multiply connected conductors.
To appear in “Optimization”. urn:nbn:de:0296-matheon-13328.

### Submitted articles

1. Markus Bachmayr and Reinhold Schneider.
Iterative methods based on soft thresholding of hierarchical tensors.
Submitted to “FOCM”, 2015. arXiv:1501.07714.
2. Markus Bachmayr, Reinhold Schneider, and André Uschmajew.
Tensor networks and hierarchical tensors for the solution of high-dimensional partial differential equations.
Submitted to “FOCM”. preprint.
3. Fredi Tröltzsch and Albert Valli.
Modeling and control of low-frequency electromagnetic fields in multiply connected conductors.
Submitted to the Springer series: “IFIP Advances in Information and Communication Technology (IFIP AICT Series) System Modeling and Optimization (CSMO 2015)”. preprint.