Research

As a mathematician and computer scientist, I try to understand the world by

Modelling real-life problems using differential equations.
Building efficient solvers for large linear systems.
Developing numerical software.
Researching machine learning approaches for uncertainty.

My research interests are focused on the design and analysis of coupling methods for partial differential equations. These methods arise because of considering multi-physics problems, non-conforming methods or using domain decomposition methods. The applications that I have studied include such fields of science and engineering as fluid dynamics, continuum mechanics, wave propagation and biomolecular physics.

My theoretical investigations include the design of well-posed discrete problems and a priori error analysis for Galerkin finite element methods using continuous and discontinuous approximation spaces.

My computational studies include the design of efficient iterative solvers using domain decomposition methods that allow taking advantage of modern parallel architectures.

Below are more details about my projects

Projects

Virus simulations

In biological settings virus interaction with their solvent is described as implicit solvent. It is mathematically characterised by the Poisson-Boltzmann equation, which is widely used to compute salvation-free energies and mean-field potentials. Unfortunately, the main difficulty of this equation is its non-linearity. Hence, the linear Poisson-Boltzmann equation is in the interest of computational chemists.

Many numerical approaches such as the finite difference, the boundary element, and the finite element methods have been considered to solve the linear Poisson-Boltzmann equation. Since each of them has its pros and cons, coupling different methods allow taking advantage of their strengths while overcoming some of their limitations.

The main objective of this project is to apply my recently developed framework allowing coupling of the finite element and boundary element methods to the linear and later non-linear Poisson-Boltzmann equation. It should address challenges in terms of efficiency and accuracy, the two major concerns. Furthermore, despite the many advantages provided by these two methods, it will be their first coupling for this kind of problem.

Coupling of finite element and boundary element methods.

Many applications require solving multi-physical problems, thus coupling methods are increasingly important in numerical analysis. In my current study, I am interested in problems on an unbounded domain, hence I focus on the coupling of finite element methods with boundary element methods. Such an approach is associated with imposing coupling conditions on the interface. As in the case of domain decomposition methods, weakly imposing interface conditions appear as an appropriate strategy.

The main result of my work is an abstract framework that allows me to solve various scalar and vector coupling problems. The applications that we are considering include continuum mechanics, wave propagation and biomolecular physics. In addition, since each of the coupling sub-problems is solved independently, each of them can be solved by using various local solvers. The work was submitted to a renowned journal.

An implementation is made in FEniCS, a software develop by, among others, the University of Cambridge and in BEMpp, a software developed at the University College London. In the current stage, together with the University of Cambridge, we are developing a parallel implementation of coupling methods.

Efficient discretisation and parallel solution techniques
for Stokes equations.

Nowadays most of the problems arise with linear systems that are too big for direct solvers. Thus, to take advantage of modern parallel architectures, domain decomposition methods are becoming increasingly important in scientific computing. However, solving the Stokes equation by an optimal domain decomposition method involves the use of non-standard interface conditions whose discretisation is not trivial. Together with my advisors, we found that hybrid discontinuous Galerkin discretisation naturally allows us to define interface conditions at the boundary between elements. Thus, my project was a combination of the appropriate discretisation and the associated domain decomposition methods.

The development of domain decomposition implementation brought a collaboration with Laboratorie Jacques-Louis Lions at Sorbonne University. It resulted in a publication in Computational Methods in Applied Mathematics. After experience gained in Paris, I developed two-level domain decomposition methods which allow to increase the number of subdomains (and decrease the size of the local problems) without increasing the computational cost. This result was published in a special volume dedicated to Olof B. Widlund. The theoretical investigation resulted in a stabilised discontinuous method for the Stokes equation with nonstandard boundary conditions, published in the chapter of Lecture Notes in Computational Science and Engineering.

Inexact domain decomposition methods for isogeometric problems.

The isogeometric method is an extension of the classical finite element method and is based on the idea of using splines or other functions constructed from splines. Unfortunately, high-regularity functions increase the computational cost significantly. My collaborators have developed a fast solver that is based on the tensor-product structure of multivariate splines and it is suitable only for a single isogeometric patch.

My contribution was the development of the combination of a non-overlapping domain decomposition method with a fast local solver. Such a solver is robust with respect to the spline degree and in practice has a computational cost that is independent of this degree. The numerical results prove a significant reduction in computational time. This research was presented at the 25th International Domain Decomposition Conference and published in Computers & Mathematics with Applications.

Maximum likelihood estimation for discrete exponential families

Exponential families are of paramount importance in probability and statistics. The study of discrete exponential families, that is exponential families on finite sets allowed us to give a new characterization of the existence of the maximum likelihood estimator for exponential family and the data at hand. The condition can be expressed as a linear programming problem. We obtain the result by a straightforward approach, which does not depend on the delicate convex analysis. Applications of the main result included the model of ferromagnetism in statistical mechanics and the random graphs theory. For this work, I was awarded third prize in the national competition organised by the Polish Mathematical Society.