My group's research concerns numerical methods for differential equations, mainly evolution equations. My focus is on error analysis, but I am also interested in the development of new methods. I am particularly interested in splitting schemes, which decompose the problem into easier or cheaper subproblems. Then they iterate between approximately solving these, and combine the subproblem solution approximations into an approximation to the solution of the full problem. This often greatly reduces the computational resource requirements.

The three main application areas of my research have been general nonlinear parabolic problems, stochastic optimization and differential Riccati equations. Nonlinear parabolic problems are exemplified by nonlinear diffusion reaction equations, where both the diffusion operator and the reaction term are nonlinear. These can model e.g. flows in porous media.

Stochastic optimization involves finding the minimum of a given cost function F that could arise from, e.g., training a neural network. This is usually done through methods related to the stochastic gradient descent (SGD), which iteratively moves in the direction of the negative gradient of a random approximation of F. The randomization is mainly because it is too costly to evaluate F completely, but it also provides other benefits. By rephrasing the minimization problem as a gradient flow, a differential equation, we can see that SGD is merely randomized Euler method. From this viewpoint, we can now apply other numerical methods for differential equations, and see them as optimization methods. That has led to several new stochastic optimization methods that are robust to parameter choices.

A differential Riccati equation is a matrix- or operator-valued equation with a quadratic nonlinearity. They are prevalent in control theory, but also appear in many other areas. In the large-scale case, they cannot be solved using normal methods, because simply storing the approximative solution would require too much memory. But by using the fact that the solution has low rank, we can formulate low-rank methods that vastly decrease both the storage and computational requirements. Due to its structure, the differential Riccati equation is very suitable to splitting, and I have developed and implemented a low-rank splitting schemes for several variations of  Riccati equations.