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.

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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. 

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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.

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Date/Time : 13 Feb 2024 15:00
Venue : S17 #04-05 (SR2)
Speaker : Lei Zhen, Fudan University

More than 20 years ago, Soffer-Weinstein considered a class of nonlinear Klein-Gordon equations with nice potentials. For the first time they proved that spatially localized and time-periodic solutions of the linear problem are destroyed by generic nonlinear Hamiltonian perturbations via slow radiation of energy to infinity, via energy transfer from the discrete to continuum modes, under the condition that the discrete modes are close to the continuous spectral modes. Since then, a long-standing open question is to study the the corresponding problem with small eigenvalues, which was also raised in the paper of Soffer-Weinstein 1999. In this talk, we will report our recent result on this problem which settles the above open question. This is a joint work with my students Jie Liu and Zhaojie Yang.

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Date/Time : 08 Feb 2024 14:00
Venue : S17 #05-11 (SR5)
Speaker : Peter A Markowich, KAUST

We propose a mesoscopic modeling framework for optimal transportation networks with biological applications. The network is described in terms of a joint probability measure on the phase space of tensor-valued conductivity and position in physical space. The energy expenditure of the network is given by a functional consisting of a pumping (kinetic) and metabolic power-law term, constrained by a Poisson equation accounting for local mass conservation. We establish convexity and lower semicontinuity of the functional on appropriate sets. We then derive its gradient flow with respect to the 2-Wasserstein topology on the space of probability measures, which leads to a transport equation, coupled to the Poisson equation. To lessen the mathematical complexity of the problem, we derive a reduced Wasserstein gradient flow, taken with respect to the tensor-valued conductivity variable only. We then construct equilibrium measures of the resulting PDE system. Finally, we derive the gradient flow of the constrained energy functional with respect to the Fisher-Rao (or Hellinger-Kakutani) metric, which gives a reaction-type PDE. We calculate its equilibrium states, represented by measures concentrated on a hypersurface in the phase space.

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