Projekt: NA-KEX VT2026
| Handledare | Projektförslag | Beskrivning |
| Gustaf Lorentzon/Elias Jarlebring | Projektförslag: Data-analys med Matrisfunktioner | I detta projekt utforskar vi hur matrisfunktioner, såsom matrisexponetialfunktionen, kan användas för att analysera egenskaper i grafer. Genom att utnyttja tekniker från grafteori och linjär algebra fokuserar vi på viktiga aspekter som centralitet och kommunikabilitet i graferna. Som utgångspunkt har vi artikeln "Network Properties Revealed through Matrix Functions". Projektets mål är att implementera tekniker för att beräkna matrisfunktioner och tillämpa dem på analys av olika nätverk, med fokus på att undersöka deras tillämpbarhet och effektivitet. |
| Jiayu Bian/Elias Jarlebring | Algorithms for image segmentation | Image segmentation, a classical problem in data analysis, aims to divide an image into meaningful regions and is often formulated as a clustering task. This project will focus on the paper Normalized cuts and image segmentation by Shi and Malik (2000), which approaches segmentation through spectral graph partitioning. The work will include a review of the mathematical foundations, implementation of the algorithm, and experiments on both synthetic clustering data and image segmentation. The objective is to compare normalized cuts with alternative partitioning methods and to assess their effectiveness and limitations. |
| Mattias Sandberg | Simulering av trafikflöden | Beskrivning: När hastighetsbegränsningen på ringleden runt Paris sänktes från 80 km/h till 70 km/h ökade medelhastigheten där från 32.6 km/h till 38.4 km/h i morgonrusningen och från 30.3 km/h till 33.9 km/h på eftermiddagen (Le Monde 19 januari 2015). I detta KEX-jobb kommer vi att modellera trafikflöden och se om det är möjligt att förstå varför sänkningen av hastighetsbegränsningen i Paris fick nämnda effekt. Trafikflöden kan modelleras både med hjälp av partiella differentialekvationer såväl som med partikel-modeller i vilka varje bils framfart tas i beaktande. Vi kommer i första skedet att använda partikelmodeller, för att eventuellt även jämföra med numeriska lösningar av partiella differentialekvationer. https://www.lemonde.fr/paris/article/2015/01/19/le-bilan-tres-positif-de-la-reduction-de-la-vitesse-sur-le-peripherique-parisien_4559095_1779311.html |
| Vilhelm P. Lithell /Elias Jarlebring | Robust Rayleigh-quotient minimization in data-science, via nonlinear eigenvalue problems | This project will be concerned with Rayleigh Quotient (RQ) minimization. RQ minimization has numerous applications, from vibrations in mechanical structures to data science, where the main idea is to relate important properties of, e.g., classification problems to the spectrum (eigenvalues) of a related problem. In this project we will study the latter of these two applications, and see how RQ-minimization arises in problems from data-science. The project will mainly be based on this paper by Bai, Lu and Vandereycken (https://epubs.siam.org/doi/10.1137/18M1167681). We will combine aspects from optimization, numerical linear algebra, and statistics, and see how incorporating uncertainties into our analysis of RQ-minimization leads to a formulation as a nonlinear eigenvalue problem. The eigenproblem is of a particularly difficult character, involving a nonlinear dependance on the eigenvectors. This eigenvalue problem will be solved with state-of-the-art techniques, namely a variant of the self-consistent-field (SCF) iteration. The paper contains several example applications, including classifcation problems, and pattern recognition, both incorporating uncertainty. The starting point of the project will be to read the paper and discuss the analysis aspects, as well as re-creating some or all of the numerical experiments. In particular we will verify the convergence properties of the algorithms proposed in the paper, and verify the results from the experiments. If time permits, the project can also be extended to other data-sets or patter-recognition problems, or alternatively the students can investigate some of the other references to different application-areas provided in the paper. |
| Sebastian Myrbäck/Sara Zahedi | Flow around obstacles | Fluid flow around obstacles is an important problem that appears in many applications, such as modelling the wind flow around an airplane wing, or water flowing around a ship. In these applications, the geometry can be complicated, or even change in time. Standard finite element methods require meshes that fit the obstacle exactly, which can be difficult and time-consuming to create. The cut finite element method (CutFEM) avoids this by allowing the geometry to cut through a simple background mesh. In this project, you will use CutFEM to simulate incompressible flow around obstacles of varying complexity of the shape. You may start with the simpler Stokes equations (slow, viscous flow) and, if time allows, extend to the more complicated and non-linear Navier–Stokes equations. It is also of interest to study how the shape changes the sensitivity of the numerical method with respect to the Reynolds number. You will be using existing C++ code which you will have to modify for carrying out the simulations. |
| Gustaf Lorentzon | Projektförslag: Effektiv Beräkning av Matrisfunktioner | I detta projekt utforskar vi hur klassiska metoder för att beräkna matrixexponentialen kan förbättras och anspassas genom användning av andra tekniker och tillämpning på andra funktioner, såsom sin(A) och cos(A). Som utgångspunkt använder vi artikeln "Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later". Projektets mål är att implementera och anpassa alternativa tekniker för polynomevaluering, och undersöka deras precision och effektivitet vid beräkning av olika matrisfunktioner, inklusive förbättring av evaluering av sin(A) och cos(A) med hjälp av Eulers formel |
| Sebastian Myrbäck/Sara Zahedi | Iterative Solvers for CutFEM | The cut finite element method (CutFEM) is a modern finite element method that allows solving partial differential equations (PDEs) without fitting the mesh to the geometry. To keep the method stable, one adds special “ghost penalty” stabilization terms. These terms are necessary, but they can also affect how well iterative solvers (like Conjugate Gradient or GMRES) perform. In this project, you will study how iterative solvers behave for CutFEM problems and compare them to direct solvers. You will investigate: 1) How solver convergence changes when varying ghost-penalty stabilization constants; 2) How iterative solvers affect certain quantities related to the numerical solution, such as mass conservation, as compared to when using direct solvers; and 3) If time allows, investigate ways to use preconditioning to improve the convergence of the iterative solvers. |
| Jiayu Bian | Interpretable low-rank matrix decomposition for data analysis | Matrix decompositions are central to modern data analysis, with the singular value decomposition (SVD) and principal components analysis (PCA) being standard techniques for dimensionality reduction. However, the singular vectors produced are often hard to interpret, as they are abstract linear combinations of all data points. This project will study the paper CUR matrix decompositions for improved data analysis by Mahoney and Drineas (2009), which approximates a data matrix using a small set of actual columns and rows. The work will include a review of the mathematical foundations, implementation of CUR algorithms (with statistical leverage–based sampling), and experiments on synthetic and real-world datasets. The objective is to compare CUR with classical SVD-based methods in terms of accuracy, efficiency, and interpretability for tasks such as clustering and exploratory data analysis. |
