Projects
Projects:
If you are a student and interested in a PhD, master or bachelor project, we encourage you contact anyone in the group for suggestion and more information.
Project: Fluid flow in porous media and geothermal energy
This project concerns modelling and understanding industrial scale heat-plants running on geothermal energy from a mathematical perspective. The general description of the process, where geothermal heat is extracted, is fairly simple. The process begins when two 7-km-deep holes are drilled into the crust of the Earth. Water is fed down to the bedrock where its temperature will rise due to geothermal heat. The hot water will rise up, and the heat will be entered into a district heating network. A complete plant should produce as much as 40 megawatts of energy. To model such a system we need to model how water flows through seemingly "solid" rock. The idea is that the granite in the nordics have due to the absence of earthquakes formed a crystalline structure, i.e. the water that was trapped in the cracks of the rock has mineralized into a type of cement. This cement is weaker than the surrounding rock, so if we with high pressure try to push water into the rock, this will open up these mineralized cracks and allow creeping water-flow. The goal of this project is to find good mathematical models to describe this type of flow and find numerically efficient ways to solve them.
St1 Deep Heat
Project: Optimization of neural networks
This project focuses on the concept of supervised learning, sometimes called `learning with a teacher'. That is, the learning is done on the basis of direct comparison of model output and the known correct answers. The models we are interested in are multilayer feed forward neural networks. Specifically they can be written as a recursive composition of linear functions with non-linear activations.
The history of neural networks started after the development of the Perceptron algorithm by Rosenblatt in 1958. Since then, neural networks have seen many interesting applications, but the computational complexity and slow learning has always been the limiting factor. The theory of neural networks has since been developed in an on and off fashion. And was ultimately revived in 2013, when Krizhevsky et.al. developed very efficient algorithms for using GPUs, Graphical Processing Units, together with a neural network with a special topology called AlexNet. With this network they won the ImageNet, image classification contest. This led to a string of improvements by many authors and has in some cases exceeded human performance on tasks like, image classification, chess and video games.
From a mathematical perspective, the training of these networks is very interesting. In principal it is because these models have several orders of magnitude more parameters than data-points used to estimate said parameters. This stands in direct contrast with standard parameter estimation, where we often have the reverse situation. This project focuses on understanding how we can understand the usual training procedures from a continuous perspective using the theory of stochastic differential equations and partial differential equations.
Project: Boundary value problems and free boundary problems for PDEs
The study of the Dirichlet problem, obstacle problems, and the boundary behaviour and Harnack inequalities, in the interior as well at the boundary, for solutions of elliptic and parabolic differential equations in bounded and unbounded domains play a central role in the theory of elliptic and parabolic differential equations as well as in many applications where these equations occur. In particular, interior and boundary Harnack inequalities are routinely used to study interior and boundary regularity of solutions for large classes of linear, quasi-linear, and fully non-linear equations. The inequalities are fundamental in understanding the notion of degeneracy in degenerate and/or singular elliptic-parabolic equations, and the boundary behaviour of the corresponding solutions. Focusing on boundary Harnack inequalities applications of such inequalities are fundamental, to give examples, to the regularity theory of free boundaries occurring in obstacle problems, to the regularity theory of free boundaries in one- and two-phase free boundary problems of Bernoulli type and in certain free boundary-inverse type problems below the continuous threshold.
This project consists of several independent subprojects targeting certain sets of partial differential operators. The purpose of the subprojects is to advance the mathematical analysis for these operators with a particular focus on boundary estimates in rough geometries, singular integrals and obstacle problems, and with subsequent applications to machine learning and finance.
Examples of subprojects include Obstacle problems for Kolmogorov type operators with applications to American versions of Asian type financial derivatives and Quasi-linear pdes and low-dimensional sets.
Project: Data-driven Discovery with Partial Differential Equations
Partial differential equations (PDEs) serve as important models and tools in many areas of applied mathematics, physics, chemistry, finance, material science, computer science, and engineering. PDEs are usually derived based on constituent physical laws or empirical observations. However, the governing equations for many complex systems in modern applications are still not completely known. Due to the rapid development of sensors, computational power, and the capacity for data storage, huge quantities of data can now be easily collected and efficiently stored. Such vast quantity of data offers new opportunities for data-driven discovery of hidden laws of nature and physics.
Inspired by the recent development of neural network designs in deep learning, uncertainty quantification using Bayesian statistics and Gaussian processes, and techniques from machine learning in general, the purpose of this project is to develop methods which can accurately predict the dynamics of complex systems and which can automatically detect underlying hidden PDE models.
The project can be decomposed into a number of independent subprojects, all connected but each with a unique flavor.
Develop methods based on deep learning and other techniques for machine learning to solve boundary value problems and free boundary problems for PDEs. Such techniques have to be compared to more traditional mesh based methods such as finite elements, finite differences, Wavelets and Monte Carlo methods.
Develop methods based on deep learning and related techniques for large ill-posed inverse problems involving PDEs. Many inverse problems consist in determining the coefficient(s) of a PDE given more or less noisy measurements of its solution. A typical example is the heat distribution in a material with unknown thermal conductivity. Given measurements of the temperature at certain locations, we are to estimate the thermal conductivity of the material by solving the inverse problem for the heat equation. The problem is of both practical and theoretical interest.
Develop methods to assess uncertainty and noisy sensors. For both boundary value problems, free boundary problems, and inverse problems for PDEs, it is important to assess the uncertainty/errors entering through uncertainties in data and measurements.
The ultimate and most visionary step is to develop robust methods, based on deep learning and other techniques for machine learning, using which one can, given a stream of data, automatically produce a compact PDE model describing the quantities of interest. Naturally this problem is very versatile and while the vision may seem clear, intuitive and exciting, it is far from clear how to do this and how to understand the PDEs produced.
Project: Regularity for non-local and singular p-Laplace equations
A very active area within the PDE community right now, is equations involving the fractional p-Laplacian:
This operator arises naturally in the calculus of variations in fractional Sobolev spaces and serves as a prototype for nonlocal nonlinear operators. It has been used in image processing and machine learning. A first step of this project, will be to extend the higher integrability result from [1] for this type of equations, to the sublinear case p < 2. As in the classical estimate for the p-Laplacian, we expect the estimate for p < 2 to be qualitatively different from the one for p > 2. A second step could be to extend the explicit Hölder regularity from [2] to the sublinear case.
[1] Brasco, Lindgren: Higher Sobolev regularity for the fractional p-Laplace equation in the superquadratic case, Advances in Mathematics, Volume 304, 2017, Pages 300-354
[2] Brasco, Lindgren, Schikorra: Higher Hölder regularity for the fractional p-Laplacian in the superquadratic case, Advances in Mathematics, Volume 338, 2018, Pages 782-846,
Project: Doubly nonlinear flows, inverse iterations and nonlinear eigenvalue problems
In [1-4] we study the connection between evolution equations, inverse iterations and nonlinear eigenvalue problems. Our results apply to various eigenvalue-type problems: Poincaré inequalities in classical and fractional Sobolev spaces, Morrey-type inequalities, the operator norm of the trace, the Steklov problem.
Many of our results are proved under the assumption that the ground state, or the first eigenvalue, is simple. This is true for many problems of interest but also unknown in some interesting problems. We believe that our results are true under a weaker assumption. Perhaps it is enough if the set of linearly independent ground states is finite or discrete. This requires further investigation. It would require the development of a good notion of distance between ground states.
In addition, our results produce numerical methods but they do not come with convergence rates. This would require further studies. In particular, it would require a deeper understanding of the impact of the gap between the first and the second eigenvalue for these problems. Furthermore, there is a need to practically implement our methods.
[1] Hynd, Ryan; Lindgren, Erik: A doubly nonlinear evolution for the optimal Poincaré inequality, Calc. Var. Partial Differential Equations 55 (2016), no. 4, 55:100.
[2] Hynd, Ryan; Lindgren, Erik: Inverse iteration for p-ground states, Proc. Amer. Math. Soc. 144 (2016), no. 5, 2121–2131.
[3] Hynd, Ryan; Lindgren, Erik: Hölder estimates and large time behavior for a nonlocal doubly nonlinear evolution, Anal. PDE 9 (2016), no. 6, 1447–1482.
[4] Hynd, Ryan; Lindgren, Erik: Approximation of the least Rayleigh quotient for degree p homogeneous functionals, J. Funct. Anal. 272 (2017), no. 12, 4873–4918.
Project: Analytical and numerical studies of anyons
Anyons are quantum mechanical particles with unusual statistical properties, which can appear in physical systems that are confined to one or two spatial dimensions. We currently have a rather limited understanding of the behavior of such particles in large numbers, because they always behave as interacting and not as free particles. There are however a few analytical and/or numerical approaches to the problem by means of models of fewer particles and which are suitable for a Bachelor's thesis project or in greater depth in a Master's thesis project.
See for example https://people.kth.se/~dogge/research.html and contact Douglas for more information.
Project: Clifford algebras in mathematics and physics
Geometric algebra, also known as the Clifford algebra generated over a vector (quadratic) space together with a set of geometric operations, is a powerful tool with applications in numerous areas of mathematics and physics. For example, not only does one obtain intuitive formulations of classical mechanics, relativity theory and electromagnetism, but also discrete geometry including simplicial complexes and graph theory, as well as purely algebraic questions such as factorization identities, can be treated using these tools.
See for example http://www.mathematik.uni-muenchen.de/~lundholm/clifford.php and contact Douglas for more information.
Project: Zeros and recurrence coefficients for multiple orthogonal polynomials
Given a measure on the real line, one can perform the standard Gram-Schmidt procedure to the sequence of monomials x^j to create a sequence of orthonormal polynomials. These polynomials satisfy the three-term recurrence relation with coefficients that can be formed into a tridiagonal (Jacobi) matrix. Spectral properties of this Jacobi matrix is intimately related to the properties of the orthogonality measure and the associated orthogonal polynomials.
Similar theory can be developed for the case when the orthogonality measure is supported on the unit circle of the complex plane.
If one requires simultaneous orthogonality with respect to several measures then one obtains the so called multiple orthogonal polynomials. The project is to investigate in detail existence, recurrence relations and zero distribution of these polynomials in certain cases (Angelesco or Nikishin).
Contact Rostyslav for more information.