2021-04-28: Eric Cancès, Ecole des Ponts ParisTech and INRIA Paris
Iterative methods in quantum chemistry and first-principle materials science
Electronic structure calculation is one of the major application fields of scientific computing. It is used daily in any chemistry or materials science department, and it accounts for a high percentage of machine occupancy in supercomputing centers. Current challenges include the study of complex molecular systems and processes (e.g. photosynthesis, high-temperature superconductivity...), and the building of large, reliable databases for the design of materials and drugs.
The most commonly used models are the Kohn-Sham Density Functional Theory (DFT), and the (post) Hartree-Fock models. The Hartree-Fock and Kohn-Sham models have similar mathematical structures. They consist in minimizing an energy functional on the Sobolev space $(H^1(R^3))^N$ under $L^2$-orthonormality constraints. The associated Euler-Lagrange equations are systems on nonlinear elliptic PDEs. After discretization in a Galerkin basis, one obtains smooth optimization problems on matrix manifolds, or on convex hulls of matrix manifolds.
Solving these problems is easy for small simple molecular systems, but very challenging for large or complex systems. Two classes of numerical methods compete in the field: constrained direct minimization of the energy functional, and self-consistent field (SCF) iterations to solve the Euler-Lagrange equations. In this talk, I will present a comparative study of these two approaches, as well as new efficient algorithms for systems with spin symmetries.
2021-04-21: Adrian Muntean, Karlstad University
Does vesicle micro-dynamics enhance the signalling among plants macro-transport? A modeling with measures approach
We study a transport problem for signalling among plants in the context of measure-valued equations. We report on preliminary results concerning the modelling and mathematical analysis of a reaction-diffusion scenario involving the macroscopic diffusion of signalling molecules enhanced by the presence of a finite number of microscopic vesicles - pockets with own dynamics able to capture and release signals as a relay system. The coupling between the macroscopic and microscopic spatial scales relies on the use of a two-scale transmission condition and benefits of the posing of the problem in terms of measures. Mild solutions to our problem will turn to exist and will also be positive weak solutions. A couple of open questions at the modeling, mathematical analysis, and simulation levels will be pointed out. This is a joint work with Sander Hille (Leiden, NL) and is supported financially by the KVA’s G. S. Magnussonsfond.
2021-04-14: Aretha Teckentrup, The University of Edinburgh
Convergence and Robustness of Gaussian Process Regression
We are interested in the task of estimating an unknown function from data, given as a set of point evaluations. In this context, Gaussian process regression is often used as a Bayesian inference procedure, and we are interested in the convergence as the number of data points goes to infinity. Hyper-parameters appearing in the mean and covariance structure of the Gaussian process prior, such as smoothness of the function and typical length scales, are often unknown and learnt from the data, along with the posterior mean and covariance. We work in the framework of empirical Bayes', where a point estimate of the hyper-parameters is computed, using the data, and then used within the standard Gaussian process prior to posterior update. Using results from scattered data approximation, we provide a convergence analysis of the method applied to a fixed, unknown function of interest.
 A.L. Teckentrup. Convergence of Gaussian process regression with estimated hyper-parameters and applications in Bayesian inverse problems. SIAM/ASA Journal on Uncertainty Quantification, 8(4), p. 1310-1337, 2020.
2021-04-07: Monika Eisenmann, Lund University
Sub-linear convergence of stochastic
optimization methods in Hilbert space
In order to solve a minimization problem, a possible approach is to find the steady state of the corresponding gradient flow initial value problem through a long time integration. The well-known stochastic gradient descent (SGD) method then corresponds to the forward Euler scheme with a stochastic approximation of the gradient. Our goal is to find more suitable schemes that work well in the stochastic setting.
In the talk, we first present a stochastic version of the proximal point algorithm. This method corresponds to the backward Euler method with a stochastic approximation of the gradient. While it is an implicit method, it has better stability properties than the SGD method and advantages can be seen if the implicit equation can be solved within an acceptable time frame. Secondly, we present a stochastic version of the tamed Euler scheme in this context. This method is fully explicit but it is more stable for larger step sizes. We provide convergence results with a sub-linear rate also in an infinite-dimensional setting. We will illustrate the theoretical results on numerical examples.
A typical application for such optimization problems is supervised learning.
The talk is based on a joint work with Tony Stillfjord and Måns Williamson (both Lund University).
2021-03-24: Maria Lopez Fernandez, University of Malaga
Directional H2-matrices for lossy Helmholtz problems
The sparse approximation of high-frequency Helmholtz-type integral operators has many important physical applications such as problems in wave propagation and wave scattering. The discrete system matrices are huge and densely populated and hence their sparse approximation is of outstanding importance. We generalize the directional H2-matrix techniques from the "pure" Helmholtz operator, with imaginary frequency, to general complex frequencies with a positive real part. In this case, the fundamental solution decreases exponentially for large arguments. We develop a new admissible condition which contains the real part of the frequency in an explicit way and introduce the approximation of the integral kernel function on admissible blocks in terms of frequency-dependent directional expansion functions. We develop an error analysis which is explicit with respect to the expansion order and with respect to the real and the imaginary parts of the frequency. This allows us to choose the variable expansion order in a quasi-optimal way. The complexity analysis shows how higher values of the real part of the frequency reduce the complexity. In certain cases, it even turns out that the discrete matrix can be replaced by its near field part. Numerical experiments illustrate the sharpness of the derived estimates and the efficiency of our sparse approximation.
In this talk, I will briefly recall recent results on the use of spectral methods for the efficient numerical solution of evolutionary problems. Their main feature is the ability of obtaining full machine accuracy despite the use of (relatively) large time-steps. This approach, at first devised for solving highly-oscillatory problems , has then been extended to the numerical solution of Hamiltonan PDEs , general ODE-IVPs , and fractional ODEs . A theoretical analysis of the methods has been given in .
 L.Brugnano, J.I.Montijano, L.Rández. On the effectiveness of spectral methods for the numerical solution of multi-frequency highly-oscillatory Hamiltonian problems. Numerical Algorithms 81 (2019) 345-376.
 L.Brugnano, F.Iavernaro, J.I.Montijano, L.Randez. Spectrally accurate space-time solution of Hamiltonian PDEs. Numerical Algorithms 81 (2019) 1183-1202.
 P.Amodio, L.Brugnano, F.Iavernaro, C.Magherini. Spectral solution of ODE-IVPs using SHBVMs. AIP Conference Proceedings 2293 (2020) 100002.
 P.Amodio, L.Brugnano, F.Iavernaro. Spectrally accurate solutions of nonlinear fractional initial value problems. AIP Conference Proceedings 2116 (2019) 140005.
 P.Amodio, L.Brugnano, F.Iavernaro. Analysis of Spectral Hamiltonian Boundary Value Methods (SHBVMs) for the numerical solution of ODE problems. Numerical Algorithms 83 (2020) 1489-1508.