Low-Rank and Tensor Network Methods for Differential Equations (VT 2026, 4 ECTS)

Overview

Modern scientific computing faces a fundamental challenge: the number of degrees of freedom required to resolve solutions of partial differential equations (PDEs) in two or more spatial dimensions grows exponentially with dimension, a phenomena called curse of dimensionality. Data compression techniques, originally developed in signal processing, linear algebra, and data science, offer a powerful antidote. When the solution of a differential equation possesses low-rank or low-complexity structure, it can be represented and evolved in a compressed format at a fraction of the cost of traditional full-rank approaches.

This course gives PhD students a rigorous and practical introduction to the use of data compression, low-rank matrix methods, and tensor network techniques for the numerical solution of differential equations. Students will learn both the mathematical foundations, singular value decomposition, CUR/cross approximations, Tucker and tensor-train formats; and how these tools are integrated into modern numerical schemes for time-dependent and high-dimensional PDEs. The course is structured around three interconnected themes:

1. Low-rank methods for differential equations; 
2. CUR decompositions, cross approximation, and the discrete empirical interpolation method (DEIM) for nonlinear problems; and 
3. Tensor network formats (Tucker, Tensor-Train) and their application to PDEs in three and higher dimensions. 

Throughout, emphasis is placed on sub-linear scaling in storage and arithmetic, practical implementation, and the analysis of convergence and rank growth.

Objectives

Upon successful completion of this course, students will be able to:

• Implement and analyze CUR matrix factorizations, cross approximation, and DEIM-based interpolation, including the Cross-DEIM algorithm for Tucker tensors.
• Derive and implement time-stepping methods, for time-dependent matrix and tensor
  differential equations.
• Solve nonlinear matrix and tensor equations using, e.g. low-rank Anderson acceleration (lrAA) and Tucker-Anderson acceleration (Tucker-AA), including the role of warm-starting and adaptive rank truncation.
• Work with tensor network formats, Tucker and Tensor-Train --- for the representation and computation of functions and solutions in three or more spatial dimensions.
• Critically read and present current research literature at the intersection of numerical
  analysis, data science, and scientific computing.

Dates

The course runs for four weeks, starting 18th May 2026.

Personnel

• Daniel Appelö - Professor, Department of Mathematics, Virginia Tech.
• Yingda Cheng - Professor, Department of Mathematics, Virginia Tech.

Prerequisites

• Linear algebra at the level of an undergraduate or introductory graduate course (vector spaces, matrix factorizations, eigenvalue problems, norms).
• Numerical analysis or numerical methods for PDEs (finite differences or finite elements, stability, consistency, convergence).
• Basic ordinary or partial differential equations.
• Programming experience in Python, MATLAB, or Julia (course assignments may be completed in any of these languages).
• An introductory graduate course in numerical linear algebra is recommended but not required.

Further Information

Registration

Use the registration form to register for the course.   Registration is set to close on 6th May 2026.

Registration form for the course (on sunet)

Credit

Successful candidates will receive a certificate of completion.  LADOK credit has to be arranged by your supervisor in your home department.