2021-03-17: Luigi Brugnano, Università di Firenze
Spectral numerical solution of evolutionary problems
Abstract:
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 [1], has then been extended to the numerical solution of Hamiltonan PDEs [2], general ODE-IVPs [3], and fractional ODEs [4]. A theoretical analysis of the methods has been given in [5].
[1] 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.
[2] L.Brugnano, F.Iavernaro, J.I.Montijano, L.Randez. Spectrally accurate space-time solution of Hamiltonian PDEs. Numerical Algorithms 81 (2019) 1183-1202.
[3] P.Amodio, L.Brugnano, F.Iavernaro, C.Magherini. Spectral solution of ODE-IVPs using SHBVMs. AIP Conference Proceedings 2293 (2020) 100002.
[4] P.Amodio, L.Brugnano, F.Iavernaro. Spectrally accurate solutions of nonlinear fractional initial value problems. AIP Conference Proceedings 2116 (2019) 140005.
[5] 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.
2021-03-10: Patrick Henning, Ruhr-Universität Bochum
The approximation and conservation of energy in nonlinear Schrödinger equations
Abstract:
In this talk we consider the numerical treatment of nonlinear Schrödinger equations as they appear for example in quantum physics and fluid dynamics. We give numerical examples that demonstrate the influence of the discrete energy on the accuracy of numerical approximations and that a spurious energy can create artificial phenomena such as drifting particles. In order to conserve the exact energy of the equation as accurately as possible, we propose a Crank-Nicolson-type time discretization that is combined with a suitable generalized finite element discretization in space. The space discretization is based on the technique of Localized Orthogonal Decompositions (LOD) and allows to capture general time invariants with a 6th order accuracy with respect to the chosen mesh size H. This accuracy is preserved due to the conservation properties of the time stepping method. The computational efficiency of the method is demonstrated for a numerical benchmark problem with known exact solution, which is however not solvable with traditional methods on long time scales.
2021-03-03: Raphael Kruse, Martin-Luther-University Halle-Wittenberg
Discretization of Elliptic PDEs with the Finite Element Method and Randomized Quadrature Formulas
Abstract:
The implementation of the finite element method for linear elliptic partial differential equations (PDE) requires to assemble the stiffness matrix and the load vector. In general, the entries of this matrix-vector system are not known explicitly but need to be approximated by quadrature rules. However, if the coefficient functions of the differential operator or the forcing term are irregular, then standard quadrature formulas, such as the barycentric quadrature rule, may not be reliable. In this talk we discuss the application of two randomized quadrature formulas to the finite element method for such elliptic PDE with irregular coefficient functions. We derive detailed error estimates for these methods, discuss their implementation in in numerical experiments.
This talk is based on joint work with Nick Polydorides (U Edinburgh) and Yue Wu (U Oxford).
2021-02-24: Balázs Kovács, Universität Regensburg
$L^2$ error estimates for wave equations with dynamic boundary conditions
Abstract:
In this talk we will discuss $L^2$ norm error estimates of
semi- and full discretisations, using bulk--surface finite elements
and Runge--Kutta methods, of wave equations with dynamic boundary
conditions. The presented analysis resides on an abstract formulation
and error estimates, via energy techniques, within this abstract
setting. Four prototypical linear wave equations with dynamic boundary
conditions are analysed which fit into the abstract framework. For
problems with velocity terms, or with acoustic boundary conditions we
prove surprising results: for such problems the spatial convergence
order is shown to be less than two.
These can also be observed in the numerical experiments which we will present.
The talk is based on joint work with D. Hipp (previously KIT, Germany).
2021-02-17: Benoit Dherin, Google Cloud Dublin
Implicit Gradient Regularization
Abstract:
Large deep neural networks used in modern supervised learning have a large submanifold of interpolating solutions, most of which are not good. However, it has been observed experimentally that gradient descent tends to converge in the vicinity of flat interpolating solutions producing trained models that generalize well to new data points, and the more so
as the learning rate increases. Using backward error analysis, we will show that gradient descent actually follows the exact gradient flow of a modified loss surface, which can be described by a regularized loss preferring optimization paths with shallow slopes, and in which the learning rate plays the role of a regularization rate.
(This is joint work with David Barrett from DeepMind).
2021-02-10: Charles-Edouard Bréhier, Université Lyon 1
Asymptotic preserving schemes for a class of SDEs
Abstract:
I will present a class of numerical methods for SDEs with
multiple time scales. When the time-scale separation parameter goes to
0, the slow component converges to an averaged or homogenized equation.
We design asymptotic preserving schemes: passing to the limit in the
scheme provides a consistent approximation of the limiting equation.
This is a joint work with Shmuel Rakotonirina-Ricquebourg
(https://arxiv.org/abs/2011.02341).
2021-02-03: Larisa Beilina, Chalmers & GU
An adaptive finite element method for solution of an ill-posed problem with applications in microwave thermometry
Abstract:
We will present an adaptive finite element method for solution of a linear Fredholm integral equation of the first kind. We derive a posteriori error estimates in the functional to be minimized and in the regularized solution to
this functional, and formulate corresponding adaptive algorithms. Balancing principle for optimal choice of the regularization parameter will be presented. Finally, numerical experiments will show the efficiency of a posteriori estimates applied to data measured in microwave thermometry.
2021-01-27: Anthony Nouy, Centrale Nantes
Approximation with tensor networks
Abstract:
Tensor networks (TNs) are prominent model classes for the approximation of high-dimensional functions in computational and data science. Tree-based TNs based, also known as tree-based tensor formats, can be seen as particular feed-forward neural networks. After an introduction to approximation tools based on tree tensor networks, we introduce their approximation classes and present some recent results on their properties. In particular, we show that classical smoothness (Besov) spaces are continuously embedded in TNs approximation classes. For such spaces, TNs achieve (near to) optimal rate that is usually achieved by classical approximation tools, but without requiring to adapt the tool to the regularity of the function. The use of deep networks with free depth is shown to be essential for obtaining this property. Also, it is shown that exploiting sparsity of tensors allows to obtain optimal rates achieved by classical nonlinear approximation tools, or to better exploit structured smoothness (anisotropic or mixed) for multivariate approximation. We also show that approximation classes of tensor networks are not contained in any Besov space, unless one restricts the depth of the tensor network. That reveals again the importance of depth and the potential of tensor networks to achieve approximation or learning tasks for functions beyond standard regularity classes.
References:
[1] M. Ali and A. Nouy. Approximation with Tensor Networks. Part I: Approximation Spaces. arXiv:2007.00118
[2] M. Ali and A. Nouy. Approximation with Tensor Networks. Part II: Approximation Rates for Smoothness Classes. arXiv:2007.00128
[3] M. Ali and A. Nouy. Approximation with Tensor Networks. Part III: Multivariate approximation.
[4] B. Michel and A. Nouy. Learning with tree tensor networks: complexity estimates and model selection. arXiv:2007.01165.
2021-01-20: Roland Maier, Chalmers & GU
Multiscale scattering in nonlinear Kerr-type media
Abstract:
Wave propagation in heterogeneous and nonlinear media has arisen growing interest in the last years since corresponding materials can lead to unusual and interesting effects and therefore come with a wide range of applications. An important example for such materials are Kerr-type media, where the intensity of a wave directly influences the refractive index. In the time-harmonic regime, this effect can be modelled with a nonlinear Helmholtz equation. If underlying material coefficients are highly oscillatory on a microscopic scale, the numerical approximation of corresponding solutions can be a delicate task.
In this talk, a multiscale technique is presented that allows one to deal with microscopic coefficients in a nonlinear Helmholtz equation without the need for global fine-scale computations. The method is based on an iterative and adaptive construction of appropriate multiscale spaces based on the multiscale method known as Localized Orthogonal Decomposition, which works under minimal structural assumptions.
This talk is based on joint work with Barbara Verfürth (KIT, Karlsruhe)
2021-01-13: Yvon Maday, Sorbonne Université
A few more things I learned about modelling the Covid epidemic 19
Abstract:
After the presentation that I had entitled "Two or three things that came out of the Maths-4-Covid-19 working group" at the seminar of the Laboratoire Jacques-Louis Lions last June
(see https://www.youtube.com/watch?v=QphZv1kytnQ&list=PL2W1YCsKIaN5g7x9QtirnR14jPxFAWYTV&index=3),
my knowledge on the subject has benefited from collaboration with several colleagues on various projects, some recent results of which I would like to present.
The main contribution is about deterministic compression of information derived from the knowledge of epidemiologists and infectious disease dynamics exerts. This reduction of complexity involves reduced basis methods, in particular in the frame of positive functions where the approximation garanties the preservation of the positivity.
2020-12-09: Antoine Gloria, Sorbonne Université
The structure of fluctuations as a computational tool in stochastic homogenization
Abstract:
Consider an elliptic equation in divergence form with random coefficients. The solution of the equation is itself a random field. When the correlation length of the coefficient field is small with respect to the length scale of the (deterministic) forcing term, homogenization occurs in form of an ergodic theorem for the solution : the latter looks deterministic at large scales (that is, compared to the correlation length). The gradient of the solution, however, both oscillates and fluctuates. In this talk I will describe the so-called path wise structure of fluctuations, and explain how it leads to a reduction of complexity that can be taken advantage of for numerical purposes.
This is based on joint works with Mitia Duerinckx (CNRS, Orsay) and Felix Otto (MPI Leipzig).
2020-12-02: Mike Pereira, Chalmers and GU
The STONE Project: A multidisciplinary approach to road traffic modeling
Abstract:
Understanding efficiency and behavior aspects in partially automated (vehicular) technology in large-scale (traffic) context is an unsolved problem nowadays. In this context, the STONE project aims at developing learning methods for uncertain traffic networks while relying on interdisciplinary approaches between mathematical sciences (stochastic Partial Differential Equations), traffic flow theory (hyperbolic conservation vehicular laws, network efficiency) and probabilistic machine leaning concepts.
In this talk, we will present the motivations of the project and the modeling choices we are making, as well as the technical challenges we wish to overcome.
2020-11-25: Salvador Ortiz-Latorre, University of Oslo
High order discretizations for the solution of the nonlinear filtering problem
Abstract:
The solution of the continuous time stochastic filtering problem can be represented as a ratio of two expectations of certain functionals of the signal process that are parametrized by the observation path. In this talk I will introduce a class of discretization schemes of these functionals of arbitrary order. For a given time interval partition, we construct discretization schemes with convergence rates that are proportional with the mth power of the mesh of the partition for arbitrary natural number m. The result generalizes the classical work of Picard, who introduced first order discretizations to the filtering functionals. Moreover, the result paves the way for constructing high order numerical approximation for the solution of the filtering problem.
This talk is based in a joint work with Dan Crisan (Imperial College) recently published in Stochastics and Partial Differential equations: Analysis and Computations Vol. 8, Issue 4, December 2020.
2020-11-18: Kristian Debrabant, University of Southern Denmark
Order conditions for generalized exponential stochastic partitioned Runge—Kutta methods
Abstract:
In Molecular Physics, Volume 118, 2020 - Issue 8, Grønbech-Jensen presented a complete set of stochastic Verlet-type methods for asymptotically statistically correct Langevin simulations. In this talk, we will discuss how this class of methods can be interpreted as stochastic partitioned Runge—Kutta methods with non-linear coefficients, generalizing exponential methods. Based on B-series, we derive then order conditions both for strong and weak convergence and analyse the order of the Grønbech-Jensen methods.
The content of this talk is based on joint work with Anne Kværnø.
2020-11-11: Lehel Banjai, Heriot-Watt
A tensor finite element method for a space fractional wave equation
Abstract:
We study solution techniques for an evolution equation involving second order derivative in time and the spectral fractional powers of symmetric, coercive, linear, elliptic, second-order operators in bounded spatial domains. We realize fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a semi-infinite cylinder. We thus rewrite our evolution problem as a quasi-stationary elliptic problem with a dynamic boundary condition and derive space, time, and space-time regularity estimates for its solution. The latter problem exhibits an exponential decay in the extended dimension and thus suggests a truncation that is suitable for numerical approximation. We propose and analyze two fully discrete schemes. The discretization in time is based on finite difference discretization techniques: trapezoidal and leapfrog schemes. The discretization in space relies on the tensorization of a first-degree FEM in with a suitable hp-FEM in the extended variable. For both schemes we derive stability and error estimates and present numerical results.
This is joint work with E. Otarola.
2020-10-28: Fernando Casas, Jaume I University
Compositions of pseudo-symmetric integrators with complex coefficients in the numerical integration of differential equations
Abstract:
In this talk I will consider composition methods for the time integration of differential equations obtained as double jump compositions with complex coefficients and projection on the real axis. It is shown in particular that the new integrators are time-symmetric and symplectic up to high orders if one uses a time-symmetric and symplectic basic method. This technique requires fewer stages than standard compositions of the same orders and is thus expected to lead to faster methods.
2020-10-21: Máté Gerencsér, IST Austria
Approximation of SDEs - a stochastic sewing approach
Abstract:
We give a new take on the error analysis of approximations of stochastic differential equations (SDEs), utilizing and developing the stochastic sewing lemma of Le (2020). As an alternative to earlier PDE-based works, this approach allows one to go beyond Markovian settings. We discuss the first results on convergence rates of the Euler-Maruyama scheme for SDEs driven by additive fractional noise and irregular drift, as well as the derivation of optimal convergence rates for SDEs driven by multiplicative standard Brownian noise and arbitrary Holder-continuous drift.
Joint work with Oleg Butkovsky and Konstantinos Dareiotis.
2020-10-14: Chus Sanz-Serna, Universidad Carlos III de Madrid: Vibrational resonance: A study with word series
Abstract:
Vibrational resonance occurs when the response of a system to a periodic forcing is enhanced by the presence of an additional fast forcing. In the talk I will study this phenomenon by means of word series: formal series that make it possible to study systematically dynamical systems and also numerical integrators.
2020-09-30: Elena Celledoni, NTNU: Deep learning as optimal control and structure preserving deep learning
Abstract:
Over the past few years, deep learning has risen to the foreground as a topic of massive interest, mainly as a result of successes obtained in solving large-scale image processing tasks. There are multiple challenging mathematical problems involved in applying deep learning.
We consider recent work of Haber and Ruthotto 2017 and Chang et al. 2018, where deep learning neural networks have been interpreted as discretisations of an optimal control problem subject to an ordinary differential equation constraint. We review the first order conditions for optimality, and the conditions ensuring optimality after discretisation. There is a growing effort to mathematically understand the structure in existing deep learning methods and to systematically design new deep learning methods to preserve certain types of structure in deep learning. Examples are invertibility, orthogonality constraints, or group equivariance, and new algorithmic frameworks based on conformal Hamiltonian systems and Riemannian manifolds.
References:
Deep learning as optimal control problems: models and numerical methods
Martin Benning, Elena Celledoni, Matthias J. Ehrhardt, Brynjulf Owren, Carola-Bibiane Schönlieb
Structure preserving deep learning
Elena Celledoni, Matthias J. Ehrhardt, Christian Etmann, Robert I McLachlan, Brynjulf Owren, Carola-Bibiane Schönlieb, Ferdia Sherry
2020-09-23: Anders Szepessy, KTH: Optimal control estimates of residual networks
Abstract:
I will show and explain estimates of the generalization error
to approximate given data by a residual neural network.
2020-09-16. Karl Larsson, Umeå University: Least-Squares Stabilized Nitsche Boundary Conditions for Unfitted Finite Element MethodsAbstract:
Weak enforcement of Dirichlet boundary conditions in finite element methods (FEM) can be done efficiently and with higher order accuracy using so-called Nitsche boundary conditions. This makes Nitsche boundary conditions suitable for unfitted FEM where the domain boundary is allowed to arbitrarily cut through the computational mesh. However, in some cut situations this leads to a method which is non-coercive. The problem is usually dealt with by using a very large penalty parameter in the Nitsche boundary condition or adding some stabilization terms to the method but in this talk we take a different approach. By adding certain consistent least-squares terms to the Nitsche boundary conditions we achieve a method which is proven coercive in every cut situation using only a moderate size penalty parameter.