Date/Time : 11 Feb 2026 15:00
Venue : S17 #06-11 (Seminar Room 6)
Speaker : Ansgar Jungel, TU Vienna, Austria

Many thermodynamic mixture and biological multicomponent models can be described by cross-diffusion systems. Although the diffusion matrices are generally neither symmetric and nor positive definite, the systems often possess an entropy (or free energy) structure. We aim to “translate” this entropy structure to two-point approximation finite-volume discretizations. The main difficulty is to adapt the nonlinear chain rule to the discrete level.

In this talk, we present two strategies to define a discrete chain rule, assuming either that the entropy is
the sum of individual entropies or that the entropy describes volume-filling models. Both strategies use suitable
mean formulas, based on the mean-value theorem and the convexity of the entropy functional. This leads to convergent and structure-preserving finite-volume schemes that preserve the local flux, total mass, positivity, and dissipate the entropy. Examples include models for segregating populations and Maxwell-Stefan systems for gas mixtures.

Add to calendar :

Date/Time : 05 Jan 2026 15:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Qinglin Tang, Sichuan University

The discrete normalized gradient flow with semi-implicit time discretization (GFSI), introduced in [W. Bao and Q. Du, SIAM J. Sci. Comput., 25 (2004), pp. 1674-1697], has become one of the most widely used numerical method for computing the ground states (GS) of Bose-Einstein condensates (BECs). In spite of strong numerical evidence of its energy dissipation and global convergence, a rigorous proof of these two properties for the GFSI remains an open challenge. In this talk, by reformulating the GFSI into an equivalent form in terms of the normalized numerical solution, we will rigorously establish the energy dissipation property of the algorithm. Based on this, we then establish its global convergence to a stationary state. Furthermore, revisiting the equivalent form in its continuous version, we will propose a more efficient semi-implicit scheme to compute the GS. Mass-preserving and energy-diminishing properties of the scheme will also be proved.

Add to calendar :

Date/Time : 11 Dec 2025 15:00
Venue : S17 #05-12 (Seminar Room 4)
Speaker : Guillaume Bal, University of Chicago

A seemingly widely accepted conjecture in the physical literature states that classical wave-fields propagating in random media over large distances eventually follow a complex circular Gaussian distribution. In this limit, the wave intensity becomes exponentially distributed, which corroborates the speckle patterns of, e.g., laser light observed in experiments. This talk reports on a recent result settling the conjecture in the weak-coupling, paraxial regime of wave propagation, which is accurate and routinely used in the application of laser light propagation in turbulent atmospheres. The limiting macroscopic Gaussian wave-field is fully characterized by a correlation function that satisfies an anomalous diffusion equation. Numerical simulations illustrate the theoretical findings.

This is joint work with Anjali Nair.

Add to calendar :

Date/Time : 24 Jan 2025 15:00
Venue : S17 #05-11 (SR5)
Speaker : Douglas Zhou, Shanghai Jiao Tong University

Grasping causal connectivity within a network is essential for uncovering its functional dynamics. However, the inferred causal connections are fundamentally influenced by the choice of causality measure, which may not always correspond to the network’s actual structural connectivity. The relationship between causal and structural connectivity, particularly how different causality measures influence the inferred causal links, warrants further exploration. In this talk, we first theoretically establish a mapping between structural and causal connectivity based on voltage signals from simulated Hodgkin-Huxley and integrate-and-fire neuronal networks. To extend our results to practical cases, we then examine nonlinear networks with pulse signal outputs, such as spiking neural networks, employing four prevalent causality measures: time-delayed correlation coefficient, time-delayed mutual information, Granger causality, and transfer entropy in the context of pairwise treatment. We conduct a theoretical analysis to elucidate the interconnections among these measures when applied to pulse signals. Through case studies involving both a simulated Hodgkin-Huxley network and an empirical mouse brain network, we validate the quantitative relationships between these causality measures. Our findings demonstrate a strong correspondence between the causal connectivity derived from these measures and the actual structural connectivity, thus establishing a direct link between them. We underscore that structural connectivity in networks with pulse outputs can be reconstructed pairwise, circumventing the need for global information from all network nodes and effectively avoiding the curse of dimensionality. Our approach provides a robust and practical methodology for reconstructing networks based on pulse outputs, offering significant implications for understanding and mapping neural circuitry.

Add to calendar :

Date/Time : 20 Dec 2024 15:00
Venue : S17 #04-05 (SR2)
Speaker : Shouhong Wang, Indiana University

The first part of the talk is to introduce the general framework of phase transition dynamics for deterministic dissipative systems, with various applications to statistical physics and fluid flows. This is based on joint work with Dr. Tian Ma. In the second part, we study transitions in stochastic non-equilibrium systems. As we know, a central challenge in physics is to describe non-equilibrium systems driven by randomness. For deterministic systems, the center manifold theory has shown a prodigious efficiency to often completely characterize how the onset of linear instability translates into the emergence of nonlinear patterns, associated with genuine physical regimes. In presence of random fluctuations, the underlying reduction principle to the center manifold is seriously challenged due to large excursions caused by the noise. We present an alternative framework to cope with these difficulties exploiting the approximation theory of stochastic invariant manifolds, on one hand, and energy estimates measuring the defect of parameterization of the high-modes, on the other. As a result, the approach enables us to predict, from reduced equations of the stochastic fluid problem, the occurrence in large probability of a stochastic analogue to the pitchfork bifurcation. This is joint work with Mickael Chekroun, Honghu Liu, and James McWilliams [JDE, 346(2023), 145-204].

Add to calendar :

Date/Time : 12 Dec 2024 14:00
Venue : S17 #04-05 (SR2)
Speaker : Yihong Du, University of New England, Australia

A new phenomenon in nonlocal diffusion models is that accelerated propagation may happen, that is, the propagation speed could be infinite, which never occurs in the corresponding local diffusion model with compactly supported initial data. In this talk, we will first briefly review the history of the KPP model used to describe the propagation of biological/chemical species, and then look at some very recent results on the KPP equation with nonlocal diffusion and free boundaries. For several natural classes of kernel functions appearing in the nonlocal diffusion term, we will show how the exact rate of acceleration can be determined. The talk is based on joint works with several collaborators.

Add to calendar :

Date/Time : 10 Jun 2024 15:00
Venue : S17 #04-05 (SR2)
Speaker : Ionut Danaila, University of Rouen Normandy

We present several numerical tools using classical finite elements with mesh adaptivity to solve the Bogoliubov-de Gennes (BdG) eigenvalue problem, representing a linearization of the Gross-Pitaevskii (GP) equation. The BdG model is commonly used to assess the stability of stationary solutions of the GP equation. Revealing the existence of such solutions (wave configurations, exotic vortices or vortex structures) in Bose-Einstein condensates is an active and timely research topic in experimental and theoretical physics.

For the computation of the GP stationary solutions we use a Newton algorithm coupled with a continuation method exploring the parameter space (the chemical potential or the interaction constant). Bogoliubov-de Gennes equations are then solved using dedicatedlibraries (ARPACK or SLEPc) for the associated eigenvalue problem. For 3D configurations, we use domain decomposition techniques enabling parallel computing.

The programs are written as toolboxes for FreeFem++ (www.freefem.org), a free finite-element software, allowing to easily implement various numerical algorithms. Validations and illustrations are presented for computing difficult configurations with vortices observed in physical experiments: dark solitons, single-line vortex, dark/anti-dark solitons in one or two-component Bose-Einstein condensates.

References:

I. Danaila, M. A. Khamehchi, V. Gokhroo, P. Engels and P. G. Kevrekidis, Vector dark-antidark solitary waves in multicomponent
Bose-Einstein condensates, Phys. Rev. A 94, 053617, 2016.

G. Sadaka, V. Kalt, I. Danaila and F. Hecht, A finite element toolbox for the
Bogoliubov-de Gennes stability analysis of Bose-Einstein condensates, Computer Physics Communications, 294, 108948(1-17), 2024.

G. Sadaka, P. Jolivet, E. G. Charalampidis, I. Danaila, Parallel finite-element codes for the Bogoliubov-de Gennesstability analysis of Bose-Einstein condensates, submitted, 2024.

Add to calendar :

Date/Time : 28 May 2024 15:00
Venue : S17 #04-05 (SR2)
Speaker : Schwab Christoph, ETH

Centered Gaus­sian ran­dom fields (GRFs) in­dexed by com­pacta such as smooth, bounded do­mains in Eu­c­lidean space or by smooth, com­pact and ori­ent­able man­i­folds are de­term­ined by their co­v­ari­ance op­er­at­ors. We con­sider centered GRFs given sample-​wise as vari­ational solu­tions to col­or­ing op­er­ator equa­tions driven by spa­tial white noise, with pseudodif­fer­en­tial col­or­ing op­er­ator be­ing el­liptic, self-​adjoint and pos­it­ive from the Hörmander class. This in­cludes the Mat\’ern class of GRFs as a spe­cial case. Us­ing mi­cro­local tools and biortho­gonal mul­tiresol­u­tion ana­lyses on the man­i­fold, we prove that the pre­ci­sion and co­v­ari­ance op­er­at­ors, re­spect­ively, may be iden­ti­fied with bi-​infinite matrices. Fi­nite sec­tions of the covariances may be di­ag­on­ally pre­con­di­tioned ren­der­ing the con­di­tion num­ber in­de­pend­ent of the di­men­sion p of the sec­tion. We prove that a taper­ing strategy by threshold­ing as e.g. in [Bickel, P.J. and Lev­ina, E. Co­v­ari­ance reg­u­lar­iz­a­tion by threshold­ing, Ann. Stat­ist., 36 (2008), 2577-​2604] ap­plied on fi­nite sec­tions of the bi-​infinite pre­ci­sion and co­v­ari­ance matrices res­ults in op­tim­ally nu­mer­ic­ally sparse ap­prox­im­a­tions.

Nu­mer­ical sparsity here sig­ni­fies that only asymp­tot­ic­ally lin­early many nonzero mat­rix entries are suf­fi­cient to ap­prox­im­ate the ori­ginal sec­tion of the bi-​infinite co­v­ari­ance or pre­ci­sion mat­rix us­ing this taper­ing strategy to ar­bit­rary pre­ci­sion. The taper­ing strategy is non-​adaptive and the loc­a­tions of these nonzero mat­rix entries are known a pri­ori. The tapered co­v­ari­ance or pre­ci­sion matrices may also be op­tim­ally di­ag­onally pre­con­di­tioned.

Ana­lysis of the re­l­at­ive size of the entries of the tapered co­v­ari­ance matrices mo­tiv­ates novel, mul­ti­level Monte Carlo (MLMC) or­acles for co­v­ari­ance es­tim­a­tion, in sample com­plex­ity that scales log-​linearly with re­spect to the num­ber p of para­met­ers.  This ex­tends [Bickel, P.J. and Lev­ina, E. Reg­u­lar­ized Es­tim­a­tion of Large Co­v­ari­ance Matrices, Ann. Stat., 36 (2008), pp. 199-​227] to es­tim­a­tion of (fi­nite sec­tions of) pseudodif­fer­en­tial co­v­ari­ances for GRFs by this fast MLMC method.

As­sum­ing at hand sec­tions of the bi-​infinite co­v­ari­ance mat­rix in wave­let co­ordin­ates, we pro­pose and ana­lyze a novel com­press­ive al­gorithm for sim­u­lat­ing and kri­ging of GRFs. The com­plex­ity (work and memory vs. ac­cur­acy) of these three al­gorithms scales near-​optimally in terms of the num­ber of para­met­ers p of the sample-​wise ap­prox­im­a­tion of the GRF in So­bolev scales. 

Add to calendar :

Date/Time : 22 Feb 2024 14:00
Venue : S17 #05-11 (SR5)
Speaker : Sebastian Pokutta, Zuse Institute Berlin (ZIB) and TU Berlin

Conditional Gradient methods are an important class of methods to minimize (non-)smooth convex functions over (combinatorial) polytopes. Recently these methods received a lot of attention as they allow for structured optimization and hence learning, incorporating the underlying polyhedral structure into solutions. In this talk I will give a broad overview of these methods, their applications, as well as present some recent results both in traditional optimization and learning as well as in deep learning.

Add to calendar :