Third-cycle (doctoral) programme in mathematics

Contact: Christian Engström

PDEs are key tools in the modeling of processes in science and engineering. Many state-of-the-art PDE models in physics, finance, and biology include fractional derivatives to model long-range structural stochasticity and terms that represent a time delay. The project concerns breakthrough development in analysis and numerical analysis of (fractional) partial differential equations (PDEs) with memory. The aim of the project is to develop mathematical and numerical analysis that results in robust and very fast numerical methods that can be used to build digital twins of our world. The mathematical theory and new numerical methods provide a cornerstone in the development of efficient machine learning algorithms for PDEs and optimization with PDE constraints.

It is within the project possible to focus on (i) efficient numerical methods for the time-evolution problem, with applications to scientific machine learning and optimization (ii) new spectral theory and functional analysis for operator functions that is related to PDEs with memory.

The project will provide training in modern analysis or numerical analysis with applications in contemporary science and engineering, thus equipping the student with highly desirable skills for working in both industry and academia.

The project is part of a very active research area the Ph.D. student will benefit from existing collaborations in the research area with Durham University (UK), University of Zagreb (Croatia), and University of Bern (Switzerland).

We are looking for a highly motivated, enthusiastic, and curiosity-driven researcher. You have, or will shortly acquire, an M.Sc. degree in the field of applied mathematics, mathematical analysis, or a closely related study program. You have good team spirit and like to work in an internationally oriented environment. You are proficient in English.

The project is related to several ongoing national and international projects and the groups

Scientific Computing and Partial Differential Equations

Deterministic and Stochastic Modelling

Contact: Joachim Toft

Pseudo-differential operators emerged in the 1930s in quantum physics for quantization (i.e. finding suitable transitions between observables in classical mechanics and quantum mechanics). A pseudo-differential calculus is a certain rule that transfer a function, defined in the phase space, to an operator. This function is called the symbol of the operator. The function corresponds to an observable classical mechanics and the operator is the corresponding observable in quantum mechanics.

Pseudo-differential operators fell somewhat into oblivion, but return to scene during 1960s in mathematics, initially to find convenient ways to invert certain differential operators. Later on, they became important in several other fields. For example, in time frequency analysis and signal processing, pseudo-differential operators are used to model non-stationary filters and to sort out noise.

The Bargmann transform transfers functions and (ultra) distributions on R^n to analytical functions, or more generally, to power series expansions on C^n. Correspondingly, operators can be transferred to so-called analytical pseudo-differential operators (also called Wick operators). Ideas for this appeared in the 1970s, but some of the fundamental continuity investigations were performed first during the past recent years. The advantage of this approach is that the objects are analytical functions, which possess several special structures, convenient in calculations and applications.

Similarly as for regular differential operators, one picks a function (still called a symbol) and arrange an operator. For analytical pseudo-differential operators, however, both the symbol and the functions that the operators act on, are analytical functions.

The project aims at, among others:

1. Finding new conditions on the symbols to the analytic pseudo-differential operators, to guarantee suitable continuity properties
2. Investigations of propagations of singularities (certain types of so-called wave-front sets).
3. Finding characterizations of certain types of spaces of analytic functions.

We are looking for a person with enthusiasm and good knowledge in mathematical analysis. It is extra meritorious for the applicant to possess good knowledge in English.

Contact: Nacira Agram, Astrid Hilbert, Torsten Lindström, and Roger Pettersson

Formulating models using systems of differential equations is a standard procedure for taking into account the total effect of several different interacting processes. The solutions that arise are generally very complicated. Qualitative analysis involves analyzing well-chosen properties of the solutions instead. The long-term behaviour of the solutions becomes more important than the solutions themselves.

Systems of differential equations arise in many different applications and we concentrate primarily on ecological, epidemiological and economical systems. In these systems, species or market players can, for example, interact by competing with each other or parasitizing on each other. In some cases one can observe large variations over time and in others not. The need to be able to regulate and optimize the systems considered can be substantial.

Reasons for the variations are important for the possibilities of making predictions and controlling the observed systems. Sometimes there are mechanisms that cause purely random outbreaks and variations as in epidemiology, while systems that describe food chains can give rise to deterministic fluctuations with much larger predictability. In economic applications, predictions can be subjected to large uncertainty.

There is a variety of things to keep in mind when beginning the analysis of a system that potentially gives rise to variation over time. First, each model is a simplification of reality and one must ask to what extent features excluded from the model can affect the results. Within qualitative analysis, the concept of structural stability has been developed to deal with this type of omission. If the model is structurally stable, the results are not affected by small disturbances that result from what you have chosen not to include. Another question that often arises is how the model is able to handle parts of the system that can be isolated and studied separately. Various ways to ensure that a suitable model is analyzed are included in the project.

A qualitative analysis of a model's long-term behavior provides answers to a wide range of questions, but there is often reason to supplement such an analysis with numerical simulations. Such simulations must be validated by requiring that they coincide with other analysis. Any discrepancies must be possible to explain and often give rise to the need for fairly in-depth analyzes of the algorithms used.

When you build a system of differential equations, you start from the mechanisms you choose to include in a system. But everyone knows that a child can learn to ride a bike without knowing anything about relevant balance equations. If we start from data, can we then build up another type of model with help of statistical models? What qualitative characteristics do models that are built on the basis of adaptation to data have in relation to models that are built on the basis of mechanisms and what limitations do new areas such as machine learning have?

Knowledge of differential equations, stochastic processes and dynamic systems is important for the project.

Contact: Karl-Olof Lindahl

The main goal of the project is to develop robust algorithms in reinforcement learning with applications in optimization and control of dynamical systems. Mathematically, the project lies in the intersection between dynamical systems, optimization and mathematical statistics. As a student, you need to have a good foundation in mathematics at the basic and advanced level, as well as good knowledge and skills in programming.

Reinforcement learning is a hot area in machine learning and artificial intelligence. It is used to solve complex optimization problems that form the basis for advanced decision-making in optimal planning and operation of energy systems, resource management, robotics, image recognition, self-driving cars, stock portfolio selection, to name a few examples. It can also be said that the area is about maximizing future accumulated rewards over time for an intelligent system operating in an environment.

The project specifically concerns distribution-based reinforcement learning (DRL). The total reward over time is characterized in these cases by a stochastic variable whose distribution function is then to be estimated. In more classical approaches to reinforcement learning, the focus is on the expected value of this distribution rather than the distribution itself. The project is based on Bellemere, Dabney and Muno's work "A Distributional Perspective on Reinforcement Learning" where it is justified why distribution-based reinforcement learning can lead to better approximate learning.

Development, experiments, and analysis of algorithms in this field include functional analysis (metric spaces, projections, transforms), probability theory and statistics (measure theory, stochastic convergence), optimization and theory of neural networks. Computer experiments, implementation and testing are important elements in the research and involve a lot of programming work.

In the project we develop:

• Specialized algorithms for machine learning and its applications
• Code for implementation
• Theory and analysis methods for robustness

The project will be performed within the research theme "AI and machine learning for optimization and operations research" at LNU.