Speaker: Massimiliano Fasi, Durham University, UK.
Date: Thursday, December 15, at 13.15, T213, Teknikhuset
Title: Computational Graphs for Matrix Functions
Abstract: Many numerical methods for evaluating matrix functions can be naturally viewed as computational graphs. Rephrasing these methods as directed acyclic graphs (DAGs) is a particularly effective approach to study existing techniques, improve them, and eventually derive new ones. The accuracy of these matrix techniques can be characterized by the accuracy of their scalar counterparts, thus designing algorithms for matrix functions can be regarded as a scalar-valued optimization problem. The derivatives needed during the optimization can be calculated automatically by exploiting the structure of the DAG, in a fashion analogous to backpropagation. GraphMatFun.jl is a Julia package that offers the means to generate and manipulate computational graphs, optimize their coefficients, and generate Julia, MATLAB, and C code to evaluate them efficiently at a matrix argument. The software also provides tools to estimate the accuracy of a graph-based algorithm and thus obtain numerically reliable methods. For the exponential, for example, using a particular form (degree-optimal) of polynomials produces implementations that in many cases are cheaper, in terms of computational cost, than the Padé-based techniques typically used in mathematical software. This is joint work with Elias Jarlebring (KTH) and Emil Ringh (Ericsson Research).
Speaker: Francesco Tudisco, GSSI Gran Sasso Science Institute, L’Aquila, Italy
Date: Friday, December 9, at 13.15, Zoom meeting
Title: Fast and efficient neural networks’ training via low-rank gradient flows
Abstract: Neural networks have achieved tremendous success in a large variety of applications. However, their memory footprint and computational demand can render them impractical in application settings with limited hardware or energy resources. At the same time, overparametrization seems to be necessary in order to overcome the highly nonconvex nature of the optimization problem. An optimal trade-off is then to be found in order to reduce networks’ dimensions while maintaining high performance.
Popular approaches in the literature are based on pruning techniques that look for “winning tickets”, smaller subnetworks achieving approximately the initial performance. However, these techniques are not able to reduce the memory footprint of the training phase and can be unstable with respect to the input weights. In this talk we will present a training algorithm that looks for “low-rank lottery tickets” by interpreting the training phase as a continuous ODE and by integrating it within the manifold of low-rank matrices. The low-rank subnetworks and their ranks are determined and adapted during the training phase, allowing the overall time and memory resources required by both training and inference phases to be reduced significantly. We will illustrate the efficiency of this approach on a variety of fully connected and convolutional networks.
The talk is based on:
S Schotthöfer, E Zangrando, J Kusch, G Ceruti, F Tudisco
Low-rank lottery tickets: finding efficient low-rank neural networks via matrix differential equations
https://arxiv.org/pdf/2205.13571.pdf
(to appear on NeurIPS 2022)
Speakers: Andrey Kiselev, AASS, Örebro University and Stina Liivamäe, AI Sweden
Date: Friday, November 4, at 13.15, in T213
Title: AI++: Autonomous Systems in Industrial Applications
Abstract: The term AI can have many different meaning in different contexts and this presentation is going to look at the application of various AI methods in industrial applications. We will look into several examples of previous and current projects of various size and see how those projects are implemented from ideas to active collaboration and possibly to market-ready products. We will also discuss how various research activities are supported on the university level and through national wide networks to increase the impact.
Speaker: François Rousse, Örebro University
Date: Friday, October 14, at 13.15, in T213
Title: Phase-space representation for fermionic Quantum Dynamics
Abstract: Phase-space representations for bosonic Quantum Dynamics were introduced in Quantum Optics (i.e. for photons) in the 60’s. Some more practical computational maturity was achieved first in the 80-90’s through the development of the positive-P representation and the Truncated Wigner Approximation (TWA), and through new numerical implementation of the corresponding stochastic equations. Then experimental progress in the field of Bose-Einstein condensates motivated applications of phase-space representations for (massive) bosonic particles, such as cold dilute atoms around 2000.
The more general understanding of matter also requires investigations of fermionic particles and of electronic structures, i.e. there is a need to develop simulation methods for fermions. Phase-space methods maps the Hamiltonians to multidimensional Partial Differential Equations (PDE) for a quasi-probability density representing the Quantum Dynamics in an overcomplete basis. The PDEs are then mapped to stochastic differential equations (SDE) for stochastic sampling of physical observables. This last step is a standard procedure but allows for different types of so-called gauge degrees of freedoms, that can be explored to increase the practical numerical performance of the methods.
We have explored a new phase-space representation for fermions, the fermionic Truncated Wigner Approximation (fTWA), first presented in 2017. fTWA is an approximative method which we have shown have major advantages over the standard mean-field method, in capturing higher order correlations of the quantum states. The first fTWA method have been explored on a large hexagonal 2D lattice, resembling the electronic structure of graphene. While the latter exact numerical-matrix-equation-based simulation have only been carried out on minimalistic 1D systems so far.
In addition we have worked on extending the useful simulation time for another exact phase-space representation for fermions called the Gaussian phase-space representation (GPSR). We have done this by building in the mentioned gauge freedom into numerical matrices, obeying a constrained matrix equation.
Speaker: Andrii Dmytryshyn, Örebro University
Date: Friday, September 16, at 13.15, Zoom meeting
Title: Versal deformations of matrices
Abstract: Jordan canonical form for matrices is well known and studied with various purposes but reduction to this form is an unstable operation: both the corresponding canonical form and the reduction transformation depend discontinuously on the entries of an original matrix. This issue complicates the use of the canonical form for numerical purposes. Therefore V.I. Arnold introduced a normal form to which an arbitrary family of matrices A' close to a given matrix A can be reduced by similarity transformation smoothly depending on the entries of A’. He called such a normal form a versal deformation of A.
In this presentation we will discuss versal deformations and their use in investigation of possible changes in canonical forms (eigenstructures), reduction of unstructured perturbations to structured perturbations, and codimension computations.
Speaker: Fernando De Terán, Universidad Carlos III de Madrid
Date: Friday, May13 , at 13.15, Zoom meeting
Title: On the consistency of the matrix equation X^TAX=B when B is either symmetric or skew
Abstract: In this talk, we analyze the consistency of the matrix equation X^T AX=B, (1) where A in an mxm complex matrix, X in an mxn complex matrix (unknown), and B (complex nxn) is either symmetric or skew-symmetric (and (·)^T means the transpose). In particular, we first provide a necessary condition for (1) to have a solution X. Then, we will prove that this condition is also sufficient for most matrices A and an arbitrary symmetric (or skew) matrix B. We want to emphasize that the question on the consistency of (1), when B is symmetric (respectively, skew), is equivalent to the following problem: given a bilinear form over C^m (represented by the matrix A), find the maximum dimension of a subspace such that the restriction of the bilinear form to this subspace is a symmetric (resp., skew) non-degenerate bilinear form.
Speaker: Johan Hellsvik, KTH
Date: Friday, April 1 , at 13.15, Zoom meeting
Title: The Dardel HPE Cray EX supercomputer at PDC
Abstract: The CPU partition of the HPE Cray EX supercomputer Dardel has now entered regular operation. In this talk I will give an introduction to Dardel, the Cray programming environment, and the experiences learned so far on how to obtain good performance for scientific codes. An overview will be given on some of the scientific application programs that are now running on Dardel and how these how been tuned for AMD EPYC CPUs. In the spring the hardware for the GPU partition will be delivered to PDC. The GPU nodes with AMD Instinct MI250X GPUs will provide the main part of the total performance of Dardel. Examples will be given on some of the software porting activities that are currently ongoing to get codes ready for the GPU partition.
Speaker: Alan Edelman, Massachusetts Institute of Technology
Date: Friday, March 4, at 13.15, Zoom meeting
Title: Generalizing Orthogonal Matrices: On the Structure of the Solutions to the Matrix Equation G*JG=J
Abstract: We study the mathematical structure of the solution set to the matrix equation G*JG=J for a given square matrix J. In the language of pure mathematics, this is a Lie group which is the isometry group for a bilinear (or a sesquilinear) form.
We found that on its own, the related (tangent space) equation X*J+JX= 0 is hard to solve. By throwing into the mix the complementary linear equation X*J−JX= 0, we find that rather than increasing the complexity, we reduce the complexity.
We explicitly demonstrate computation of solutions, visualizations, and closure hierarchies that connect to previous work by Dmytryshyn, Futorny, Kågström, Klimenko, and Sergeichuk.
Joint work with Sung Woo Jeong.
Speaker: Jens Fjelstad, Örebro University
Date: Friday, February 4, at 13.15, Zoom meeting
Title: On quantum representations of mapping class groups from a finite group
Abstract: Topological quantum field theory (TQFT) is a gadget that produces two kinds of data, topological invariants and finite dimensional representations of mapping class groups of surfaces, so called quantum representations. These data are of interest both for Mathematics and Physics, one recent application being in quantum computing. One construction of TQFT is via a certain Hopf algebra constructed from a finite group, producing the quantum representations relevant for this talk. I will briefly introduce the ingredients such as mapping class groups of surfaces and the class of quantum representations associated to a finite group, and then discuss work (partly in collaboration with Jürgen Fuchs) to determine their structure in terms of finite group data. Recent results are focused on the restriction of a quantum representation to the terms in a filtration of a mapping class group known as the Johnson filtration.
