2020-11-25: Salvador Ortiz-Latorre, University of Oslo
High order discretizations for the solution of the nonlinear filtering problem
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
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
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
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
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
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
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.
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
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 Methods
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.