A Hierarchy of Network Models Giving Bistability Under Triadic Closure

Abstract: Triadic closure describes the tendency for new friendships to form between individuals who already have friends in common. It has been argued heuristically that the triadic closure effect can lead to bistability in the formation of large-scale social interaction networks. Here, depending on the initial state and the transient dynamics, the system may evolve towards either of two long-time states. In this work, we study a hierarchy of network evolution models that incorporate triadic closure, building on the work of Grindrod, Higham and Parsons [Internet Mathematics, 8, 2012, 402--423]. In a macroscale regime, we show rigorously that a bimodal steady state distribution is admitted. Computational simulations will be used to support the analysis. This is joint work work with Stefano Di Giovacchino (L'Aquila) and Kostas Zygalakis (Edinburgh).

Slides: https://drive.google.com/file/d/1WuS4QZUvxn5TubbI1sUFUCqChN3U3tXy/view?usp=sharing

Video: https://drive.google.com/file/d/1X-YK4Ct-FPWSr8rRt0Usv-cSYA-fnz4r/view?usp=sharing

Modern semi-Lagrangian solvers for high-dimensional kinetic equations

Abstract: Kinetic equations, such as the Vlasov equation, are important in a range of applications from plasma physics to radiative heat transfer. The main challenge comes from the fact that such equations are posed in an up to six-dimensional phase space. Moreover, when numerical integrating such equations one has to deal with small scale structures, a stringent CFL condition, and the desire not to introduce too much numerical diffusion.

In this talk, we will consider some recent advances in constructing appropriate numerical methods (both time and space discretization) for such problems. Our focus will be on structure-preserving algorithms, which are often essential to capture the relevant dynamics accurately, and how such algorithms can be efficiently implemented on modern computer systems. We will also touch on the question of numerical stability for exponential integrators, which turns out to be dramatically different than in the parabolic case.

Slides: https://drive.google.com/file/d/1_NKXT2c1XsaVVLRbmvx6U9l-oLrXznC-/view?usp=sharing

Video: https://drive.google.com/file/d/1p1M3uZltDBfzjGGW_eCAQRovePUvJQLJ/view?usp=sharing

Time integration of dispersive problems at low regularity

Abstract: Standard numerical integrators such as splitting methods or exponential integrators suffer from order reduction when applied to semi-linear dispersive problems with non-smooth initial data. In this talk, we focus on the cubic nonlinear Schrödinger equation with periodic boundary conditions. For such problems, we present and analyze a class of (filtered) Fourier integrators that exhibit (superior) convergence rates at low regularity.

This is joint work with K. Schratz (Sorbonne U, Paris), F. Rousset (U Paris-Sud) and M. Knöller (KIT, Karlsruhe).

Video: https://drive.google.com/file/d/1b9re_Q2o7b37XUaO61imir3VITBFTQ71/view?usp=sharing

Perlin noise for automatic generation of complex spatial patterns: an application to cardiac fibrosis

Abstract: Fibrosis, the pathological excess of fibroblast activity, is a significant health issue that hinders the function of many organs in the body, in some cases fatally. However, the severity of fibrosis-derived conditions depends on both the positioning of fibrotic affliction, and the microscopic patterning of fibroblast-deposited matrix proteins within afflicted regions. Variability in an individual’s manifestation of a type of fibrosis is an important factor in explaining differences in symptoms, optimum treatment and prognosis, but a need for ex vivo procedures and a lack of experimental control over conflating factors has meant this variability remains poorly understood. Here we present a computational methodology for the generation of patterns of fibrosis microstructure, demonstrating the technique using histological images of four types of cardiac fibrosis. Our generator and automated tuning method prove flexible enough to capture these very distinct patterns, allowing for rapid generation of new realisations for high-throughput computational studies. We also demonstrate via simulation, using the generated fibrotic patterns, the importance of micro-scale variability by showing significant differences in electrophysiological impact even within a single class of fibrosis. Finally, we discuss new techniques for homogenisation to tackle heterogenous, multi-scale problems arising in modelling fibrotic tissue, including a new approach based on graph-based homogenisation.

Slides: https://drive.google.com/file/d/1cV9spwfkDjOmIVyWuac1D9GuhUG7X80F/view?usp=sharing

Video: https://drive.google.com/file/d/1GmOFVT3kmOzQyd3KTN7SvOP9p05g26mO/view

Runge-Kutta TASE schemes for the numerical solution of stiff systems

Abstract: TASE operators have been introduced by Bassenne, Fu and Mani (Journal of Computational Physics, 2021) to stabilize a stiff IVP so that the modified differential equation can be solved by an explicit Runge-Kutta method in a stable way. The operators of order k are defined in terms of the inverse of k matrices. Here, we propose a new family of TASE operators of order k that depends on k free parameters in contrast with Bessenne's family which depends only on one parameter to be chosen for stability and accuracy requirements. A complete study of A-stability properties is carried out for explicit RK schemes supplemented with TASE operators (RK TASE methods) with order k ≤ 4. Then, we present a new family of singly Tase operators that are defined in terms of the inverse of just one matrix, which provides more efficient methods. The absolute stability properties of the resulting RKTASE schemes is carried out and some particular methods are proposed.

Slides: https://drive.google.com/file/d/1ogHDCUCW7OPjzNXHm-3t4fb7O04VipWm/view?usp=sharing

Video: https://drive.google.com/file/d/1oVS24uHTvjmWqMN4aqPSIoBo0_Oujwfy/view?usp=sharing

Time-renormalization of gravitational N-body problems

Abstract: This work considers the gravitational (Newtonian) N-body problem and introduces global time-renormalization functions that allow the efficient numerical integration with fixed time-steps, regardless of the occurrence of close encounters. We first consider the Taylor expansion of the solution of the problem with non-collisional initial values, and apply the method of majorants to estimate its coefficients and bound from below its radius of convergence. Next, we rewrite the N-body problem with a fictitious time related to the physical time in terms of a so-called time-renormalization function. Such a renormalization function is chosen so that the radius of convergence of the Taylor expansion of the solution in the fictitious time is uniformly bounded from below (independently of the actual initial values). This allows the efficient numerical integration of the time-renormalized N-body problem with constant time-steps. As a by-product, a global power series representation of the solutions of the N-body problem is obtained, generalising Sundman’s global solution of the three-body problem.

Slides: https://drive.google.com/file/d/1p_k_WnITpK5WBGV3mwFjZC1r8-1IxkC4/view?usp=sharing

Video: https://drive.google.com/file/d/1IpdQVasfhWmAkhnTlitbMyv1LVVp4gik/view?usp=sharing

Splitting scheme for Schrödinger equations with white noise dispersion

Abstract: We analyze a splitting integrator for the time discretization of the Schrödinger equation with nonlocal interaction cubic nonlinearity and white noise dispersion. We prove that the splitting scheme preserves the L² -norm and has a strong order of convergence one. This is a joint work with Charles-Edouard Bréhier (Université Claude Bernard Lyon 1).

Slides: https://drive.google.com/file/d/1uYYHtgdJ6OlcE9-hmfE7-FWx8gCGDS61/view?usp=sharing

Video: https://drive.google.com/file/d/1JVNqlfs12obLKdZhH4Ry8rf33gEM06T_/view?usp=sharing

Solving Structured Matrix Equations Encountered in the Analysis of Stochastic Processes

Abstract: Here

Slides: https://drive.google.com/file/d/1vlhIqxTwPYhsS6uqMgNXMj_sJFlqb3_M/view?usp=sharing

Video: https://drive.google.com/file/d/1L8_CFOWIQuC_-CrL1_ywvD6yZc_WZQ19/view?usp=sharing

Unified error analysis for certain full discretizations of wave-type problems

Abstract: In this talk we consider the full discretization of a class of linear wave-type problems in first order formulation. Important examples of this class are Maxwell's equations, the acoustic wave equation or the advection equation.

We follow a method of lines approach, where we first discretize in space via the discontinuous Galerkin (dG) method. Subsequently, we use one of four schemes for time integration

- the Crank-Nicolson scheme,

- the leapfrog scheme,

- a locally implicit scheme,

- the Peaceman-Rachford scheme.

and then investigate the resulting fully discrete schemes. We show their stability (subject to an appropriate CFL condition where necessary) and error bounds that are optimal in space and time and robust under mesh refinement. These bounds are derived within a unified error analysis based on the fact that all schemes can be interpreted as perturbations of the Crank--Nicolson scheme.

This is joint work with Jonas Köhler, Karlsruhe.

Slides: https://drive.google.com/file/d/1y50pI6f3kKQk74NtlID9WFFx0KqdHqHA/view?usp=sharing

Video: https://drive.google.com/file/d/1LY34vbsUyimq8Pqvp1d45L4NuuaZE3ur/view?usp=sharing

Variationally evolving Gaussians revisited

Abstract: The semiclassically scaled multi-particle time-dependent Schrödinger equation describes, e.g., quantum dynamics of nuclei in a molecule. It poses the combined computational challenges of high oscillations and high dimension. This talk reviews Gaussian wave packets that evolve according to the Dirac-Frenkel time-dependent variational principle for the semiclassical Schrödinger equation. This provides a simple yet fundamental approximation that is basic for many more elaborate approximations.

Old and new results on the Gaussian approximation to the wave function are given, in particular an L^2 error bound that goes back to Hagedorn (1980) in a non-variational setting, and a new error bound for averages of observables, which shows the double approximation order in the semiclassical scaling parameter in comparison with the norm estimate.

The variational equations of motion in Hagedorn's parametrization of the Gaussian are presented. They show a perfect quantum-classical correspondence and allow us to read off directly that the Ehrenfest time is determined by the Lyapunov exponent of the classical equations of motion.

A variational splitting integrator is formulated and its remarkable conservation and approximation properties are discussed. A new result shows that the integrator approximates averages of observables with the full order in the time stepsize, with an error constant that is uniform in the semiclassical parameter.

The material presented here for variational Gaussians is part of an Acta Numerica 2020 review article with Caroline Lasser on computational methods for quantum dynamics in the semiclassical regime.

Slides: https://drive.google.com/file/d/1WVlo1JGsieVXa1cnY0Tz1vGZ5IZn2AnA/view?usp=sharing

Video: https://drive.google.com/file/d/1pTuhwes9tt0_htOtaVlz6nTxIz1JApNa/view?usp=sharing

Backward error analysis for symmetric solutions of PDEs

Abstract: In backward error analysis (BEA), an approximate solution to an equation is interpreted as the exact solution of a nearby "modified" equation up to any power of the discretisation parameter. If the differential equation has a geometric property, then the modified equation may share it. In this way, known properties of differential equations apply to the approximation.

While techniques for the ODE case are well established, I will point out some problems that arise in attempts for PDEs and show an approach to BEA for symmetric solutions of discretised PDEs. In particular, I will show how to compute modified Hamiltonian systems or modified 1st order Lagrangians that formally govern symmetric solutions of variationally discretised PDEs. This will be illustrated on the example of rotating travelling waves in the nonlinear wave equation.

Slides: https://drive.google.com/file/d/11lLZkf1wUgKIX0Gl_cbBUJnEcos65lp5/view?usp=sharing

Video: https://drive.google.com/file/d/1Ox36_Ff-OtLXZI6kcuNMhlIPoDyIY8zn/view?usp=sharing

Splitting integrators to reduce the number of rejections in Hamiltonian Monte Carlo sampling

Abstract: The leapfrog integrator is routinely used within the Hamiltonian Monte Carlo method. We give strong numerical evidence that alternative splitting integrators, as easily implementable as leapforg, yield fewer rejections with a given computational effort. When the dimensionality of the target distribution is high, the number of accepted proposals may be multiplied by a factor of three or more without impairing in any way the quality of the samples. We also provide theoretical results validating the derivation of the proposed integrators.

Slides: https://drive.google.com/file/d/12l07UPrTUNuaztgLhqccrC3EKBYILw3y/view?usp=sharing

Video: https://drive.google.com/file/d/1MbAPPLq-fsabVjldsAAc_Vu4_G0Y0Sj4/view?usp=sharing

Efficient explicit schemes for stochastic chemical reaction networks with time-scale disparity

Abstract: Chemical reaction systems typically involve wide ranges of time scales. Mathematically, a chemical reaction system with fast/slow time scales and stochasticity taken into account can be modeled by stiff chemical Langevin equations (CLEs). In this talk I will introduce a stable and efficient explicit time-scale splitting algorithm for stiff CLEs. The algorithm is based on the concept of computational singular perturbation and allows time integration step sizes much larger than those required by typical explicit numerical methods for stiff stochastic differential equations. Numerical results will also be presented to illustrate the accuracy and stability of the proposed method.

Slides: https://drive.google.com/file/d/16Af0v9f2ZrNLewE0EGzsWOYq0QKS_ZZO/view?usp=sharing

Video: https://drive.google.com/file/d/1UgZ0bgKjyyyOHpQ2dGW75Kn7KUVBJrHs/view?usp=sharing

Extensions of the Deep Galerkin Method: Fokker-Planck, Hamilton-Jacobi-Bellman and Path-Dependent

Abstract: In this talk, we will present several extensions of the Deep Galerkin Method (DGM), originally introduced in Sirignano and Spiliopoulos (2018), where they propose a deep learning algorithm to solve Partial Differential Equations. In the Fokker-Planck setting, our approach involves reparameterizing the solution as the exponential of a neural network appropriately normalized to ensure that it is both positive and integrates one. We also extend the DGM algorithm to solve for the value function and the optimal control simultaneously in the Hamilton-Jacobi-Bellman PDEs by characterizing both as deep neural networks. Training the networks is performed by taking alternating stochastic gradient descent steps for the two functions, a technique similar in spirit to policy improvement algorithms. Finally, we present a novel numerical method for Path-Dependent Partial Differential Equations (PPDEs). These equations firstly appeared in the seminal work of Dupire [2009], where the functional Itô calculus was developed to deal with path-dependence in Mathematical Finance. More specifically, we generalize the DGM to deal with these equations. The method, which we call Path-Dependent DGM (PDGM), consists of using a combination of feed-forward and Long Short-Term Memory architectures to model the solution of the PPDE. We then analyze several numerical examples that show the capabilities of the method under very different situations.

Slides: https://drive.google.com/file/d/16MViynFkUytn70lRp0GNdFZp9br7y3Ko/view?usp=sharing

Video: https://drive.google.com/file/d/16CxA9ps_Gv_whqtcIVjz4ymrC5Tn8XsX/view?usp=sharing

Power boundedness in the maximum norm of stability matrices for ADI-type methods

Abstract: Here

Slides: https://drive.google.com/file/d/1-eGembhhWiBizZdQt_Vey4YL6CbC64I0/view?usp=sharing

Video: https://drive.google.com/file/d/16PofK5cKbUHzD_7twFWtOYGGxA1tuzSX/view?usp=sharing

A stochastic model of infection: Francisella tularensis

Abstract: With a mouse infection model, agent-based computation and mathematical analysis, we study the pathogenesis of Francisella tularensis infection. A small initial number of bacteria enter host cells and proliferate inside them, eventually destroying the host cell and releasing numerous copies that infect other cells. Our analysis of disease progression is based on a stochastic model of a population of infectious agents inside one host cell, extending the birth-and-death process by the occurrence of catastrophes: cell rupture events that affect all bacteria in a cell simultaneously. Closed expressions are obtained for the survival function of an infected macrophage, the number of bacteria released as a function of time after infection, and total bacterial load. We compare our analysis with the results of agent-based computation and, via Approximate Bayesian Computation, with experimental measurements carried out after of murine aerosol infection with the virulent SCHU S4 strain of the bacterium. The posterior distribution is consistent with the estimate that the time between rounds of bacterial division is less than 6 hours in vivo.

Video: https://drive.google.com/file/d/18HXmnWAjU6EYV4d7Hn2kCsfmhxx-t7Dc/view?usp=sharing

Computational models in immunology

Abstract: Different questions in immunology require different types of computational model, by which I mean an idealised dynamical system, usually stochastic, that allows us to explore the consequences of hypotheses. Because there are approximately 400000000000 naive CD4 T cells in a human body, it may be tempting to assume that fluctuations can be ignored. However, the number of cells sharing one T-cell receptor is a small integer that increases or decreases by one cell at a time, when cells divide or die. T cells are often classified, based on conjectured links between measured cell-surface molecules and immunological function. We may simulate this using agent-based (here, cell-based) models: cells have both mutable and non-mutable attributes; no two cells are identical. Heterogeneous cell populations can be maintained by stochastic rules defined at the cell level.

Video: https://drive.google.com/file/d/17OZkLzVz-mjtUeFG7rTq3altniMY75vm/view?usp=sharing

Resonances as a computational tool

Abstract: In recent years, a large toolbox of numerical schemes for dispersive equations has been established, based on different discretisation techniques such as discretising the variation of constants formula (e.g., exponential integrators) or splitting the full equation into a series of simpler subproblems (e.g., splitting methods). In many situations these classical schemes allow a precise and efficient approximation. This, however, drastically changes whenever non-smooth phenomena enter the scene such as for problems at low-regularity and/or with high oscillations. Classical schemes fail indeed to capture the oscillatory nature of the solution, a fact that leads to severe instabilities and loss of convergence. In this talk I present a new class of resonance based schemes. The key idea in the construction of these new schemes is to tackle and deeply embed the underlying structure of resonances into the numerical discretisation. As in the continuous case, these resonances are central to structure preservation and provide the new schemes with strong geometric properties at low regularity.

Video: https://drive.google.com/file/d/18GsHRUpMTswl9vkgEdOvhQU35UksGTY6/view?usp=sharing

Using mathematics and history to predict the future of semi-arid vegetation

Abstract: Vegetation in semi-arid regions has complicated dynamics, with a tendency to self-organise into spatiotemporal patterns. Given the lack of laboratory replicates, and the practical difficulties associated with fieldwork, mathematical modelling plays a key role in understanding these dynamics. In this lecture, Jonathan Sherratt will discuss the ability of simple mathematical models of semi-arid vegetation to provide important and often surprising insights into spatial patterning. One important feature of the phenomenon is that the ecological and environmental parameters do not on their own determine the pattern wavelength, which depends also on the process leading to patterning. For example, the degradation of uniform vegetation and the colonization of bare ground lead to different patterns. This “history-dependence” means that prediction of future vegetation levels requires detailed information on previous vegetation density and pattern. Focussing on the specific case of the Sahel region of Africa, Jonathan Sherratt will show how this can be obtained by combining modelling with data on climate history. Using predictions of future rainfall levels from global climate models, he will go on to discuss the prediction of future vegetation levels in the Sahel, up to the end of the 21st century – an issue with major ecological and socioeconomic importance.

Slides: https://drive.google.com/file/d/1AxaFCLz6ySnvuYAPaNjzLsjKUZMzFz1e/view?usp=sharing

Video: https://drive.google.com/file/d/1AijAGvhPg4_Om1o6Q2Xmw_vmkq4v2-vv/view?usp=sharing

On a randomized backward Euler method for nonlinear evolution equations with time-irregular coefficients

Abstract: In this talk we introduce a randomized version of the backward Euler method that is applicable to stiff ordinary differential equations and nonlinear evolution equations with time-irregular coefficients. In the finite-dimensional case, we consider Carathéodory-type functions satisfying a one-sided Lipschitz condition. After investigating the well-posedness and the stability properties of the randomized scheme, we prove the convergence to the exact solution with a rate of 0.5 in the root-mean-square norm assuming only that the coefficient function is square integrable with respect to the temporal parameter. These results are then extended to the approximation of infinite dimensional evolution equations under monotonicity and Lipschitz conditions. Here, we consider a combination of the randomized backward Euler scheme with a Galerkin finite element method. We obtain error estimates that correspond to the regularity of the exact solution. The practicability of the randomized scheme is also illustrated through several numerical experiments.

This is a joint work with Raphael Kruse (Halle), Moni Eisenmann (Lund) and Stig Larsson (Chalmers).

Slides: https://drive.google.com/file/d/1FonQMLZ6KH0UrHYus5u05CwEAP85v66e/view?usp=sharing

Video: https://drive.google.com/file/d/1EeCy-_WxYYtNN4hTE36WVbcnolGUDEwr/view?usp=sharing

From Runge-Kutta methods to B-series

Abstract: Here

Video: https://drive.google.com/file/d/1H-FBJKGnGzJxd-CFIjR9PAUzhGFR40cb/view?usp=sharing

A review on two-step peer methods

Abstract: In this talk, a review on Peer methods for the numerical integration of first order initial value problems will be considered. This class of numerical schemes only use the first derivative in two consecutive steps. These two-step peer methods were introduced by R. Weiner et al. and combine the advantages of Runge-Kutta and multistep methods to obtain high stage order. We will consider several type of peer methods: functionally fitted, singly implicit and we will focus our attention to explicit methods with re-used stages. In general, two-step s-stages peer methods require s derivative function calls per step, but if the matrices of the peer method have a special structure, it is possible to use less function calls by using previously computed stages that they are re-used in the current step. We will show that this class of peer schemes are competitive in the numerical integration of non-stiff test problems.

Slides: https://drive.google.com/file/d/1M6sgopamaGuuI2JHtFhY6zFltJiONxHL/view?usp=sharing

Video: https://drive.google.com/file/d/1M6sgopamaGuuI2JHtFhY6zFltJiONxHL/view?usp=sharing

Explicit two-step Runge-Kutta methods for computational fluid dynamics solvers

Abstract: Here

Video: https://drive.google.com/file/d/1jhvOrpe-9BxWAflzYNO2GrfLaQxBNYuT/view?usp=sharing

Determining the index of eigenvalues of an elliptic operator

Abstract: Throughout mathematical physics there are many problems dependent on, or consisting of determining eigenvalues of a linear elliptic operator on a given domain with Dirichlet boundary conditions. Some examples are the Schrödinger equation, the wave equation, the linear theory of elasticity, ... Many researchers have invested time and effort in stating and proving theorems about the spectrum of elliptic operators.

In this talk we present a new tool to determine the number of eigenvalues less then a given value. Our theorem will be accompanied with examples and historic context.

Slides: https://drive.google.com/file/d/1Q7N3BykiMDqwENwHw7wknVwmvPbWQV0U/view?usp=sharing

Video: https://drive.google.com/file/d/1_cjIeICQ1RDJGVQy2yaSLgFtw33IcWpm/view?usp=sharing