At the present time I’m mostly interested in collaboration in terms of the post doctorate research.
Mathematics
The collaboration subjects include, but are not limited to, the following

Numerical methods for evolutionary, nonlocal or inverse problems:
 Parallel numerical methods for the abstract Schrödinger equation.
 Numerical methods to approximate the solution of final value problem for abstract parabolic equation.
 Fully discrete numerical method for the Cauchy problem with elliptic matrix partial differential operator coefficient based on spectral approximation in time domain and FEM for space discretization.

Numerical linear algebra:
 Development of new iterative numerical methods for shifted linear systems. These methods ought to be deliberately adapted to solve a sequence of discretized resolvent equations arising from the DunfordCauchy formula. Overview of the problem is given in [1], where authors highlight its importance and propose a way how to improve the Richardson iteration and CG methods by the use of tunedup preconditioner. Considering the specific structure of this sequence one may expect to improve convergence results given in [1] by adapting the GMRES method. There are several techniques to achieve this. Efficient restart, deflation by eigenvectors and Krylov subspase recycling are among such. Although used disjointly there neither analysis nor implementation of these techniques when combined together in one method designed for the mentioned sequence of linear systems. (If you know such methods to exist let me know please!)

Research oriented software development:
 PamPeRO based software development for mathematical modelling in a specific application area. I am interested in the physical, biological, financial or other models which can be interpreted mathematically in terms of abstract Cauchy or nonlocal evolutionary problem with elliptic differential operator. Once some interesting application problem is identified and found suitable it should be worthwhile to collaborate on it. And develop an implementation of numerical method for solving such problem using a fairy general numerical technique PamPeRO project is based upon (see [2], [3]). Heat dissipation, Black–Scholes model and atmospheric dispersion of pollutants are model examples where some initial implementation have been performed already (These models are discussed in the Examples section of PamPeRO project).
 Software development within PamPeRO project. Any collaboration and code or ideas contribution which may result in the extension or improvement of PamPeRO project are greatly appreciated.
References
[1] McLean, W. & Thomée, V. Iterative methods for shifted positive definite linear systems and time discretization of the heat equation ArXiv eprints, 2011
[2] Gavrilyuk, I. & Makarov, V. Exponentially convergent algorithms for the operator exponential with applications to inhomogeneous problems in Banach spaces SIAM Journal on Numerical Analysis, 2005, 43, 21442171
[3] Gavrilyuk, I. & Makarov, V. An Exponential Convergent Algorithm For Nonlinear Differential Equations in Banach Spaces Math. Comp., 2007, 76, 18951923