# Academic collaboration

**Easily implemented iterative solution methods for a class of finite dimensional constrained saddle point problems**

**Financial Supporting:** Finnish Academy.

**Kazan Federal University, head of project**: Prof. Alexander Lapin

**Partner University and collaborator**: University of Oulu, Department of Mathematical Sciences, Finland, Dr. Erkki Laitinen.

**Project time**: 01.01.2014 - 31.12.2015.

**Objectives and methods**

The aim of the project is the development of fast convergent and easily implemented iterative solution methods for the finite dimensional approximations of the following problems:

1) 2-nd order variational inequalities with constraints for the gradient of the solutions;

2) control- and state-constrained optimal control problems for parabolic equations.

For variational inequalities we will construct and investigate theoretically and numerically the iterative methods which implementation is reduced to the solution of the system of low-dimensional minimization problems and system of linear equations. For control- and state-constrained optimal control problems for parabolic equations we will construct the effective methods by using the explicit approximations of the parabolic equations with time variable steps.

**Project summary**

Large scale finite dimensional variational inequalities, inclusions and constrained saddle point problems arise from the mesh approximations of the different problems containing partial differential operators. These problems are, for example, variational inequalities with linear or nonlinear differential operators and with constraints for the gradient of the solutions, control- and state-constrained optimal control problems for the elliptic and parabolic equations. By mesh approximations we mean the approximations by using finite difference, finite element or finite volume methods.

Many iterative solution methods are constructed and investigated for different classes of the aforementioned problems. Among them: iterative algorithms based on the augmented Lagrangian approaches for the variational inequalities, regularization and penalty

methods, active set and interior point methods for the constrained optimal control problems and many others.

Some new iterative methods were proposed and investigated by the applicants of this project for the elliptic variational inequalities with linear and nonlinear differential operators and for the control- and state-constrained optimal control problems governed by linear elliptic partial differential equations.

The development of the efficient iterative methods for different classes of the constrained saddle point problems is still an actual problem.

The aim of the project is the development of the fast convergent and easily implemented iterative solution methods.

We call the iterative methods for the mesh variational inequalities with nonlinear differential operators as easily implemented if their implementation is reduced to the solution of the system of low-dimensional minimization problems and system of linear equations. Well-known methods are not easily implemented in the case of non-potential nonlinear operators.

For the parabolic optimal control problems, we plan to construct and study the iterative methods with special explicit in time approximations of the state parabolic equation. These explicit schemes with variable time steps are easily implemented and stable under the reasonable constraints for these steps. We expect to construct new iterative algorithms which will an advantage in time of calculations in relation with well-known methods.

The project is expected to result in:

**Published articles**

**Topological and geometrical properties of Banach spaces and operator algebras**

**Financial Supporting:** Grant Agency of the Czech Republic

**Project time**: 2012 - 2017

**Partner Universities**: Czech Technical University in Prague, Karlov University

**Published articles**