Focus on:
All days
Apr 3, 2024
Apr 4, 2024
Apr 5, 2024
All sessions
Contributed Talk
Invited Talk
Plenary Talk
Hide Contributions
Indico style
Indico style  inline minutes
Indico style  numbered
Indico style  numbered + minutes
Indico Weeks View
Back to Conference View
Choose Timezone
Use the event/category timezone
Specify a timezone
Africa/Abidjan
Africa/Accra
Africa/Addis_Ababa
Africa/Algiers
Africa/Asmara
Africa/Bamako
Africa/Bangui
Africa/Banjul
Africa/Bissau
Africa/Blantyre
Africa/Brazzaville
Africa/Bujumbura
Africa/Cairo
Africa/Casablanca
Africa/Ceuta
Africa/Conakry
Africa/Dakar
Africa/Dar_es_Salaam
Africa/Djibouti
Africa/Douala
Africa/El_Aaiun
Africa/Freetown
Africa/Gaborone
Africa/Harare
Africa/Johannesburg
Africa/Juba
Africa/Kampala
Africa/Khartoum
Africa/Kigali
Africa/Kinshasa
Africa/Lagos
Africa/Libreville
Africa/Lome
Africa/Luanda
Africa/Lubumbashi
Africa/Lusaka
Africa/Malabo
Africa/Maputo
Africa/Maseru
Africa/Mbabane
Africa/Mogadishu
Africa/Monrovia
Africa/Nairobi
Africa/Ndjamena
Africa/Niamey
Africa/Nouakchott
Africa/Ouagadougou
Africa/PortoNovo
Africa/Sao_Tome
Africa/Tripoli
Africa/Tunis
Africa/Windhoek
America/Adak
America/Anchorage
America/Anguilla
America/Antigua
America/Araguaina
America/Argentina/Buenos_Aires
America/Argentina/Catamarca
America/Argentina/Cordoba
America/Argentina/Jujuy
America/Argentina/La_Rioja
America/Argentina/Mendoza
America/Argentina/Rio_Gallegos
America/Argentina/Salta
America/Argentina/San_Juan
America/Argentina/San_Luis
America/Argentina/Tucuman
America/Argentina/Ushuaia
America/Aruba
America/Asuncion
America/Atikokan
America/Bahia
America/Bahia_Banderas
America/Barbados
America/Belem
America/Belize
America/BlancSablon
America/Boa_Vista
America/Bogota
America/Boise
America/Cambridge_Bay
America/Campo_Grande
America/Cancun
America/Caracas
America/Cayenne
America/Cayman
America/Chicago
America/Chihuahua
America/Ciudad_Juarez
America/Costa_Rica
America/Creston
America/Cuiaba
America/Curacao
America/Danmarkshavn
America/Dawson
America/Dawson_Creek
America/Denver
America/Detroit
America/Dominica
America/Edmonton
America/Eirunepe
America/El_Salvador
America/Fort_Nelson
America/Fortaleza
America/Glace_Bay
America/Goose_Bay
America/Grand_Turk
America/Grenada
America/Guadeloupe
America/Guatemala
America/Guayaquil
America/Guyana
America/Halifax
America/Havana
America/Hermosillo
America/Indiana/Indianapolis
America/Indiana/Knox
America/Indiana/Marengo
America/Indiana/Petersburg
America/Indiana/Tell_City
America/Indiana/Vevay
America/Indiana/Vincennes
America/Indiana/Winamac
America/Inuvik
America/Iqaluit
America/Jamaica
America/Juneau
America/Kentucky/Louisville
America/Kentucky/Monticello
America/Kralendijk
America/La_Paz
America/Lima
America/Los_Angeles
America/Lower_Princes
America/Maceio
America/Managua
America/Manaus
America/Marigot
America/Martinique
America/Matamoros
America/Mazatlan
America/Menominee
America/Merida
America/Metlakatla
America/Mexico_City
America/Miquelon
America/Moncton
America/Monterrey
America/Montevideo
America/Montserrat
America/Nassau
America/New_York
America/Nome
America/Noronha
America/North_Dakota/Beulah
America/North_Dakota/Center
America/North_Dakota/New_Salem
America/Nuuk
America/Ojinaga
America/Panama
America/Paramaribo
America/Phoenix
America/PortauPrince
America/Port_of_Spain
America/Porto_Velho
America/Puerto_Rico
America/Punta_Arenas
America/Rankin_Inlet
America/Recife
America/Regina
America/Resolute
America/Rio_Branco
America/Santarem
America/Santiago
America/Santo_Domingo
America/Sao_Paulo
America/Scoresbysund
America/Sitka
America/St_Barthelemy
America/St_Johns
America/St_Kitts
America/St_Lucia
America/St_Thomas
America/St_Vincent
America/Swift_Current
America/Tegucigalpa
America/Thule
America/Tijuana
America/Toronto
America/Tortola
America/Vancouver
America/Whitehorse
America/Winnipeg
America/Yakutat
Antarctica/Casey
Antarctica/Davis
Antarctica/DumontDUrville
Antarctica/Macquarie
Antarctica/Mawson
Antarctica/McMurdo
Antarctica/Palmer
Antarctica/Rothera
Antarctica/Syowa
Antarctica/Troll
Antarctica/Vostok
Arctic/Longyearbyen
Asia/Aden
Asia/Almaty
Asia/Amman
Asia/Anadyr
Asia/Aqtau
Asia/Aqtobe
Asia/Ashgabat
Asia/Atyrau
Asia/Baghdad
Asia/Bahrain
Asia/Baku
Asia/Bangkok
Asia/Barnaul
Asia/Beirut
Asia/Bishkek
Asia/Brunei
Asia/Chita
Asia/Choibalsan
Asia/Colombo
Asia/Damascus
Asia/Dhaka
Asia/Dili
Asia/Dubai
Asia/Dushanbe
Asia/Famagusta
Asia/Gaza
Asia/Hebron
Asia/Ho_Chi_Minh
Asia/Hong_Kong
Asia/Hovd
Asia/Irkutsk
Asia/Jakarta
Asia/Jayapura
Asia/Jerusalem
Asia/Kabul
Asia/Kamchatka
Asia/Karachi
Asia/Kathmandu
Asia/Khandyga
Asia/Kolkata
Asia/Krasnoyarsk
Asia/Kuala_Lumpur
Asia/Kuching
Asia/Kuwait
Asia/Macau
Asia/Magadan
Asia/Makassar
Asia/Manila
Asia/Muscat
Asia/Nicosia
Asia/Novokuznetsk
Asia/Novosibirsk
Asia/Omsk
Asia/Oral
Asia/Phnom_Penh
Asia/Pontianak
Asia/Pyongyang
Asia/Qatar
Asia/Qostanay
Asia/Qyzylorda
Asia/Riyadh
Asia/Sakhalin
Asia/Samarkand
Asia/Seoul
Asia/Shanghai
Asia/Singapore
Asia/Srednekolymsk
Asia/Taipei
Asia/Tashkent
Asia/Tbilisi
Asia/Tehran
Asia/Thimphu
Asia/Tokyo
Asia/Tomsk
Asia/Ulaanbaatar
Asia/Urumqi
Asia/UstNera
Asia/Vientiane
Asia/Vladivostok
Asia/Yakutsk
Asia/Yangon
Asia/Yekaterinburg
Asia/Yerevan
Atlantic/Azores
Atlantic/Bermuda
Atlantic/Canary
Atlantic/Cape_Verde
Atlantic/Faroe
Atlantic/Madeira
Atlantic/Reykjavik
Atlantic/South_Georgia
Atlantic/St_Helena
Atlantic/Stanley
Australia/Adelaide
Australia/Brisbane
Australia/Broken_Hill
Australia/Darwin
Australia/Eucla
Australia/Hobart
Australia/Lindeman
Australia/Lord_Howe
Australia/Melbourne
Australia/Perth
Australia/Sydney
Canada/Atlantic
Canada/Central
Canada/Eastern
Canada/Mountain
Canada/Newfoundland
Canada/Pacific
Europe/Amsterdam
Europe/Andorra
Europe/Astrakhan
Europe/Athens
Europe/Belgrade
Europe/Berlin
Europe/Bratislava
Europe/Brussels
Europe/Bucharest
Europe/Budapest
Europe/Busingen
Europe/Chisinau
Europe/Copenhagen
Europe/Dublin
Europe/Gibraltar
Europe/Guernsey
Europe/Helsinki
Europe/Isle_of_Man
Europe/Istanbul
Europe/Jersey
Europe/Kaliningrad
Europe/Kirov
Europe/Kyiv
Europe/Lisbon
Europe/Ljubljana
Europe/London
Europe/Luxembourg
Europe/Madrid
Europe/Malta
Europe/Mariehamn
Europe/Minsk
Europe/Monaco
Europe/Moscow
Europe/Oslo
Europe/Paris
Europe/Podgorica
Europe/Prague
Europe/Riga
Europe/Rome
Europe/Samara
Europe/San_Marino
Europe/Sarajevo
Europe/Saratov
Europe/Simferopol
Europe/Skopje
Europe/Sofia
Europe/Stockholm
Europe/Tallinn
Europe/Tirane
Europe/Ulyanovsk
Europe/Vaduz
Europe/Vatican
Europe/Vienna
Europe/Vilnius
Europe/Volgograd
Europe/Warsaw
Europe/Zagreb
Europe/Zurich
GMT
Indian/Antananarivo
Indian/Chagos
Indian/Christmas
Indian/Cocos
Indian/Comoro
Indian/Kerguelen
Indian/Mahe
Indian/Maldives
Indian/Mauritius
Indian/Mayotte
Indian/Reunion
Pacific/Apia
Pacific/Auckland
Pacific/Bougainville
Pacific/Chatham
Pacific/Chuuk
Pacific/Easter
Pacific/Efate
Pacific/Fakaofo
Pacific/Fiji
Pacific/Funafuti
Pacific/Galapagos
Pacific/Gambier
Pacific/Guadalcanal
Pacific/Guam
Pacific/Honolulu
Pacific/Kanton
Pacific/Kiritimati
Pacific/Kosrae
Pacific/Kwajalein
Pacific/Majuro
Pacific/Marquesas
Pacific/Midway
Pacific/Nauru
Pacific/Niue
Pacific/Norfolk
Pacific/Noumea
Pacific/Pago_Pago
Pacific/Palau
Pacific/Pitcairn
Pacific/Pohnpei
Pacific/Port_Moresby
Pacific/Rarotonga
Pacific/Saipan
Pacific/Tahiti
Pacific/Tarawa
Pacific/Tongatapu
Pacific/Wake
Pacific/Wallis
US/Alaska
US/Arizona
US/Central
US/Eastern
US/Hawaii
US/Mountain
US/Pacific
UTC
Save
Europe/Rome
English (United States)
Deutsch (Deutschland)
English (United Kingdom)
English (United States)
Español (España)
Français (France)
Italiano (Italia)
Polski (Polska)
Português (Brasil)
Türkçe (Türkiye)
Čeština (Česko)
Монгол (Монгол)
Українська (Україна)
中文 (中国)
Login
Exploiting Algebraic and Geometric Structure in TimeIntegration Methods
from
Wednesday, April 3, 2024 (8:30 AM)
to
Friday, April 5, 2024 (3:45 PM)
Monday, April 1, 2024
Tuesday, April 2, 2024
Wednesday, April 3, 2024
8:30 AM
Registration
Registration
8:30 AM  9:20 AM
Room: Sala degli Stemmi, Scuola Normale Superiore
9:20 AM
Opening Remarks
Opening Remarks
9:20 AM  9:30 AM
Room: Sala degli Stemmi, Scuola Normale Superiore
9:30 AM
Why Parallel in Time (PinT) methods are different for Parabolic and Hyperbolic Problems

Martin J. Gander
(
University of Geneva
)
Why Parallel in Time (PinT) methods are different for Parabolic and Hyperbolic Problems
Martin J. Gander
(
University of Geneva
)
9:30 AM  10:30 AM
Room: Sala degli Stemmi, Scuola Normale Superiore
Spacetime parallel methods, also known more recently under the name PinT methods, have received a lot of attention over the past two decades, driven by the parallel hardware architectures that have now millions of cores, and acceleration often saturates when one parallelizes in space only. Research focus has therefore shifted to trying to parallelize also the time direction. However, for parallelization, time is very different from space, because evolution problems satisfy a causality principle: the future is dependent on the past, and not the other way round. It is therefore not clear a priori if useful numerical work can be done simultaneously in the near and far future. I will first show in my presentation why intuitively there is a fundamental difference when one parallelizes hyperbolic or parabolic problems in spacetime, and which key properties need to be taken into account to be successful. I will then give examples of PinT methods which use the physical properties of the evolution problem to their advantage to parallelize in spacetime. For hyperbolic problems, effective PinT methods are Domain Decomposition methods of Waveform Relaxation type, culminating in Unmapped Tent Pitching methods. Very powerfull techniques are also the ParaDiag methods, and direct time parallel methods like ParaExp. Several of these methods can also be very effectively used for parabolic problems, but for such problems there are also very successful multilevel methods, like Parareal, and the currently best ones are spacetime multigrid methods. These multilevel methods struggle however when applied to hyperbolic problems.
10:30 AM
ParallelInTime solver for the allatonce RungeKutta discretization

Luca Bergamaschi
(
University of Padua
)
ParallelInTime solver for the allatonce RungeKutta discretization
Luca Bergamaschi
(
University of Padua
)
10:30 AM  11:00 AM
Room: Sala degli Stemmi, Scuola Normale Superiore
Timedependent PDEs arise quite often in many scientific areas, such as mechanics, biology, economics, or chemistry. Of late, researchers have devoted their effort in devising parallelintime methods for the numerical solution of such PDEs, adding a new dimension of parallelism and allowing to speedup the solution process on modern supercomputers. In this talk, we present a fully parallelizable preconditioner for the allatonce linear system arising when employing a RungeKutta (RK) method in time. The resulting system is solved iteratively for the numerical solution and for the stages. The proposed preconditioner results in a blockdiagonal solve for the stages, accelerated by a novel blockpreconditioner based on the SVD of the RK coefficient matrix, and a Schur complement obtained by solving again systems for the stages. Parallel results show the robustness of the preconditioner with respect to the discretization parameters and to the number of stages, as well as very promising scalability and parallel efficiency indices.
11:00 AM
Coffee break
Coffee break
11:00 AM  11:30 AM
Room: Chiostro Scuola Normale
11:30 AM
A new ParaDiag timeparallel time integration method

Davide Palitta
(
University of Bologna
)
A new ParaDiag timeparallel time integration method
Davide Palitta
(
University of Bologna
)
11:30 AM  12:00 PM
Room: Sala degli Stemmi, Scuola Normale Superiore
Many parallelintime (PinT) paradigms have been developed in the last decades to efficiently solve timedependent partial differential equations (PDEs). In this talk we focus on the ParaDiag scheme whose most peculiar feature consists in the explicit diagonalization of the matrix stemming from the adopted time integrator. However, certain classes of time integrators lead to a discrete operator which is not diagonalizable. For instance, this is the case of Backward Differentiation Formulas (BDFs) like the backward Euler scheme. With the aim of overcoming such a drawback, different approaches have been developed in the literature. In this talk, we illustrate a novel technique. In particular, we show how to exploit the possible circulantpluslowrank structure of the discrete time integrator to design a new, successful implementation of the ParaDiag paradigm. Notice that this peculiar structure arises in many different families of time integrators. The efficiency of our original scheme is displayed by several numerical examples.
12:00 PM
A preconditioned MINRES method for block lower triangular Toeplitz systems with application to solving allatonce systems of linear timedependent PDEs

Sean Hon
(
Hong Kong Baptist University
)
A preconditioned MINRES method for block lower triangular Toeplitz systems with application to solving allatonce systems of linear timedependent PDEs
Sean Hon
(
Hong Kong Baptist University
)
12:00 PM  12:30 PM
Room: Sala degli Stemmi, Scuola Normale Superiore
In this study, a novel preconditioner based on the absolutevalue block αcirculant matrix approximation is developed, specifically designed for nonsymmetric dense block lower triangular Toeplitz (BLTT) systems that emerge from the numerical discretization of evolutionary equations. Our preconditioner is constructed by taking the absolute value of a block αcirculant matrix approximation to the BLTT matrix. To apply our preconditioner, the original BLTT linear system is converted into a symmetric form by applying a timereversing permutation transformation. Then, with our preconditioner, the preconditioned minimal residual method (MINRES) solver is employed to solve the symmetrized linear system. With a properly chosen α, the eigenvalues of the preconditioned matrix are proven to be clustered around ±1 without any significant outliers. With the clustered spectrum, we demonstrate that the preconditioned MINRES solver for the preconditioned system has a convergence rate independent of system size. Our preconditioner can be implemented in a parallelintime manner. The efficacy of the proposed preconditioner is corroborated by our numerical experiments, which reveal that it attains optimal convergence.
12:30 PM
Lunch break
Lunch break
12:30 PM  2:30 PM
2:30 PM
Global spacetime lowrank methods for the timedependent Schrödinger equations

Virginie Ehrlacher
(
Ecole des Ponts ParisTech
)
Global spacetime lowrank methods for the timedependent Schrödinger equations
Virginie Ehrlacher
(
Ecole des Ponts ParisTech
)
2:30 PM  3:30 PM
Room: Sala degli Stemmi, Scuola Normale Superiore
The aim of this talk is to present novel global spacetime methods for the approximation of the timedependent Schrödinger equation, using Kato theory. The latter can be used in conjunction with lowrank tensor formats (such as Tensor Trains for instance) to derive new variational principles to compute dynamical lowrank approximations of the solution, which are different from the DiracFrenkel principle. One significant advantage of this new variational formulation is that the existence of a dynamical lowrank approximation for any finitetime horizon can be proved, whereas dynamical lowrank approximations constructed with the DiracFrenkel principle can usually be porved to exist only locally in time. Illustrative numerical results will be presented to highlight the differences between the dynamical lowrank approximations obtained with these different approaches. This is joint work with Clément Guillot and MiSong Dupuy.
3:30 PM
Parallel timedependent variational principle algorithm for matrix product states

Sergey Dolgov
(
University of Bath
)
Parallel timedependent variational principle algorithm for matrix product states
Sergey Dolgov
(
University of Bath
)
3:30 PM  4:00 PM
Room: Sala degli Stemmi, Scuola Normale Superiore
Combining the timedependent variational principle (TDVP) algorithm with the parallelization scheme introduced by Stoudenmire and White for the density matrix renormalization group (DMRG), we present the first parallel matrix product state (MPS) algorithm capable of time evolving onedimensional (1D) quantum lattice systems with longrange interactions. We benchmark the accuracy and performance of the algorithm by simulating quenches in the longrange Ising and XY models. We show that our code scales well up to 32 processes, with parallel efficiencies as high as 86%.
4:00 PM
Coffee break
Coffee break
4:00 PM  4:30 PM
Room: Chiostro Scuola Normale
4:30 PM
Parallel implementation of block circulant type preconditioner for allatonce systems of linear timedependent PDEs

Ryo Yoda
(
The University of Tokyo
)
Parallel implementation of block circulant type preconditioner for allatonce systems of linear timedependent PDEs
Ryo Yoda
(
The University of Tokyo
)
4:30 PM  5:00 PM
Room: Sala degli Stemmi, Scuola Normale Superiore
Parallelintime approaches solve allatonce systems obtained by solving all timedependent PDEs at once in order to extract temporal parallelism. Assuming linear and constantintime integrators for all time steps, the resulting systems have a block Toeplitz structure. Consequently block circulant preconditioners have attracted much attention for these systems. In particular, block epsiloncirculant preconditioners, introducing a weighting coefficient, have achieved convergence independent from the spatial size, thus they are promising parallelintime approaches. This work focuses on parallel implementations of block circulant type preconditioners. The primary operations of these preconditioners are FFTs on onedimensional timestepsized vectors and solving spatialsized linear systems with complexvalued coefficient matrices. We use FFTW and Trilinos packages with their MPI implementation and investigate their parallel performance. Additionally, we propose an alternative parallelization strategy to execute the onedimensional FFTs. Numerical experiments demonstrate good scaling behavior for linear diffusion and advectiondiffusion problems.
5:00 PM
Efficiently solving nonlinear timedependent problems from the geosciences

Gabriel Wittum
(
King Abdullah University of Science and Technology
)
Efficiently solving nonlinear timedependent problems from the geosciences
Gabriel Wittum
(
King Abdullah University of Science and Technology
)
5:00 PM  5:30 PM
Room: Sala degli Stemmi, Scuola Normale Superiore
The presentation investigates the efficient use of a linearly implicit stiff integrator for the numerical solution of density driven flow problems. Upon choosing a onestep method of extrapolation type (code LIMEX), the use of full Jacobians and reduced approximations are discussed. Numerical experiments include nonlinear density flow problems such as diffusion from a salt dome (2D), a (modified) Elder problem (3D), the saltpool benchmark (3D) and a real life salt dome problem (2D). The arising linear equations are solved using either a multigrid preconditioner from the software package UG4 or the sparse matrix solver SuperLU. Based on these component, this work devises guidelines for the design of an efficient solver.
Thursday, April 4, 2024
9:00 AM
Regularized dynamical parametric approximation

Christian Lubich
(
University of Tuebingen
)
Regularized dynamical parametric approximation
Christian Lubich
(
University of Tuebingen
)
9:00 AM  10:00 AM
Room: Sala degli Stemmi, Scuola Normale Superiore
This paper studies the numerical approximation of solutions to initial value problems of highdimensional ordinary differential equations or evolutionary partial differential equations such as the Schr\"odinger equation by nonlinear parametrizations $u(t)=\Phi(q(t))$ with timedependent parameters $q(t)$, which are to be determined in the computation. The motivation comes from approximations in quantum dynamics by multiple Gaussians and approximations of various dynamical problems by tensor networks and by neural networks. In all these cases, the parametrization is typically irregular: the derivative $\Phi'(q)$ can have arbitrarily small singular values and may have varying rank. We derive approximation results for a regularized approach in the timecontinuous case as well as in timediscretized cases. With a suitable choice of the regularization parameter and the time stepsize, this approach can still be successfully applied in such irregular situations, even if it runs counter to the basic principle in numerical analysis to avoid solving illposed subproblems when aiming for a stable algorithm. Numerical experiments with sums of Gaussians for approximating laserinduced quantum dynamics and with neural networks for approximating the flow map of a system of ordinary differential equations illustrate and complement the theoretical results. The talk is based on joint work with Caroline Lasser, Jörg Nick and Michael Feischl.
10:00 AM
Randomized lowrank approximation for time and parameterdependent problems

Daniel Kressner
(
EPFL
)
Randomized lowrank approximation for time and parameterdependent problems
Daniel Kressner
(
EPFL
)
10:00 AM  10:30 AM
Room: Sala degli Stemmi, Scuola Normale Superiore
A variety of applications gives rise to matrices $A(t)$ that admit good lowrank approximation for each time (or parameter value) $t$, including dynamical systems, spectral density estimation, and Gaussian process regression, In this talk, we discuss the benefits of randomized methods for approximating $A(t)$ simultaneously for many values of $t$. We describe and analyze parameterdependent extensions of two popular randomized algorithms, the randomized singular value decomposition and the generalized Nyström method. Both, the theoretical results and numerical experiments, show that these methods reliably return quasibest lowrank approximations. This talk is based on joint work with Hysan Lam.
10:30 AM
Coffee break
Coffee break
10:30 AM  11:00 AM
Room: Chiostro Scuola Normale
11:00 AM
Adaptive rational Krylov methods for exponential RungeKutta integrators

Kai Bergermann
(
TU Chemnitz
)
Adaptive rational Krylov methods for exponential RungeKutta integrators
Kai Bergermann
(
TU Chemnitz
)
11:00 AM  11:30 AM
Room: Sala degli Stemmi, Scuola Normale Superiore
We consider the solution of large stiff systems of ordinary differential equations with explicit exponential RungeKutta integrators. These problems arise from semidiscretized semilinear parabolic partial differential equations on continuous domains or on inherently discrete graph domains. A series of results reduces the requirement of computing linear combinations of $\varphi$functions in exponential integrators to the approximation of the action of a smaller number of matrix exponentials on certain vectors. Stateoftheart computational methods use polynomial Krylov subspaces of adaptive size for this task. They have the drawback that the required number of Krylov subspace iterations to obtain a desired tolerance increase drastically with the spectral radius of the discrete linear differential operator, e.g., the problem size. We present an approach that leverages rational Krylov subspace methods promising superior approximation qualities. We prove a novel aposteriori error estimate of rational Krylov approximations to the action of the matrix exponential on vectors for single time points, which allows for an adaptive approach similar to existing polynomial Krylov techniques. We discuss pole selection and the efficient solution of the arising sequences of shifted linear systems by direct and preconditioned iterative solvers. Numerical experiments show that our method outperforms the state of the art for sufficiently large spectral radii of the discrete linear differential operators. The key to this are approximately constant numbers of rational Krylov iterations, which enable a nearlinear scaling of the runtime with respect to the problem size. We will also discuss the special algebraic structure of the supraLaplacian operator appearing in multilayer network analysis.
11:30 AM
Energypreserving splitting integration for Hamiltonian Monte Carlo method with adaptive tuning

Fasma Diele
(
IAC
)
Energypreserving splitting integration for Hamiltonian Monte Carlo method with adaptive tuning
Fasma Diele
(
IAC
)
11:30 AM  12:00 PM
Room: Sala degli Stemmi, Scuola Normale Superiore
Splitting schemes provide a promising alternative to the classical StormerVerlet method in Hamiltonian Monte Carlo (HMC) methodology. Within the family of oneparameter secondorder splitting procedures, we demonstrate that using a designated function of the free parameter to select the step size ensures stability and Hamiltonian preservation when sampling from Gaussian distributions. This guarantees no sample rejections in the HMC process, a key factor for superior performance compared to recent similar methods. The effectiveness of the proposed approach for sampling from general nonGaussian distributions is assessed, incorporating a simple adaptive selection technique for the free parameter to improve HMC performance. Benchmark examples from literature and experiments, including the LogGaussian Cox process and Bayesian Logistic Regression, highlight the effectiveness of the approach.
12:00 PM
Detecting the numerical of ill posedness in delay differential equations

Miryam Gnazzo
(
GSSI
)
Detecting the numerical of ill posedness in delay differential equations
Miryam Gnazzo
(
GSSI
)
12:00 PM  12:30 PM
Room: Sala degli Stemmi, Scuola Normale Superiore
Given a set of matrices $A_i \in \mathbb{C}^{n \times n}$ and a set of analytic functions $f_i : \mathbb{C} \mapsto \mathbb{C}$, we consider a regular matrixvalued function $\mathcal{F}(\lambda)=\sum_{i=0}^d f_i(\lambda) A_i$, that is $\det \left( \mathcal{F}(\lambda) \right)$ is not identically zero for $\lambda \in \mathbb{C}$. An interesting problem consists in the computation of the nearest singular function $\widetilde{\mathcal{F}}(\lambda)=\sum_{i=0}^d f_i(\lambda) \left( A_i + \Delta A_i \right)$, with respect to the Frobenius norm. For example, this problem has particular importance in the context of delay differential algebraic equations, where a function in the form $\mathcal{D}(\lambda)=\lambda E  A  B e^{\tau \lambda}$ is studied. Indeed in this setting and in presence of small delays $\tau$, the ill posedness of the problem may be connected with the numerical singularity of the function $\mathcal{D}(\lambda)$, even if the pencil $\lambda E A$ is regular. We will provide a general overview of the problem, describing the possible issues connected with the lack of robustness of the differential equation, associated with destabilizing perturbations of $\mathcal{D}(\lambda)$. Moreover we propose a method for the numerical approximation of the function $\mathcal{F}(\lambda)$, which rephrases the matrix nearness problem for the matrixvalued function into an equivalent optimization problem. Nevertheless this problem turns out to be highly nonconvex. To solve it, we propose a two level procedure, which introduces a constrained gradient system of differential equations in the inner iteration and a Newtonlike method for the optimization of the perturbation size in the outer one. This is a joint work with Nicola Guglielmi (GSSI).
12:30 PM
Lunch
Lunch
12:30 PM  2:30 PM
2:30 PM
Highorder conservative and accurately dissipative numerical integrators via finite elements in time

Patrick Farrell
(
University of Oxford
)
Highorder conservative and accurately dissipative numerical integrators via finite elements in time
Patrick Farrell
(
University of Oxford
)
2:30 PM  3:30 PM
Room: Sala degli Stemmi, Scuola Normale Superiore
Numerical methods for the simulation of transient systems with structurepreserving properties are known to exhibit greater accuracy and physical reliability, in particular over long durations. These schemes are often built on powerful geometric ideas for broad classes of problems, such as Hamiltonian or reversible systems. However, there remain difficulties in devising higherorderintime structurepreserving discretizations for nonlinear problems, and in conserving higherorder invariants. In this work we propose a general framework for the construction of structurepreserving timesteppers via finite elements in time and the systematic introduction of auxiliary variables. The framework reduces to Gauss methods where those are structurepreserving, but extends to generate arbitraryorder structurepreserving schemes for nonlinear problems, and allows for the construction of schemes that conserve multiple higherorder invariants. We demonstrate the ideas by devising novel schemes that exactly conserve all known invariants of the Kepler and Kovalevskaya problems, highorder energyconserving schemes for the compressible NavierStokes equations, and highorder energy and helicitypreserving schemes for the incompressible magnetohydrodynamics equations.
3:30 PM
Convolution Quadrature for the quasilinear subdiffusion equation

Maria LopezFernandez
(
University of Malaga
)
Convolution Quadrature for the quasilinear subdiffusion equation
Maria LopezFernandez
(
University of Malaga
)
3:30 PM  4:00 PM
Room: Sala degli Stemmi, Scuola Normale Superiore
We construct a Convolution Quadrature (CQ) scheme for the quasilinear subdiffusion equation and supply it with the fast and oblivious implementation. In particular we find a condition for the CQ to be admissible and discretize the spatial part of the equation with the Finite Element Method. We prove the unconditional stability and convergence of the scheme and find a bound on the error. As a passing result, we also obtain a discrete Grönwall inequality for the CQ, which is a crucial ingredient of our convergence proof based on the energy method. The paper is concluded with numerical examples verifying convergence and computation time reduction when using fast and oblivious quadrature.
4:00 PM
Coffee break
Coffee break
4:00 PM  4:30 PM
Room: Chiostro Scuola Normale
4:30 PM
Highorder conservative and accurately dissipative numerical integrators via finite elements in time

Boris Andrews
(
University of Oxford
)
Highorder conservative and accurately dissipative numerical integrators via finite elements in time
Boris Andrews
(
University of Oxford
)
4:30 PM  5:00 PM
Room: Sala degli Stemmi, Scuola Normale Superiore
Numerical methods for the simulation of integrable systems with conservative properties are known to exhibit greater accuracy and physical reliability, in particular over long durations. One important class of schemes is that of Gauss methods, a class of symplectic Runge–Kutta methods. In this talk we propose an alternative, general framework for the construction of conservative schemes via finite elements in time and the systematic introduction of auxiliary variables. For linear problems with quadratic invariants, the scheme is provably equivalent to Gauss methods. However, the alternative framework extends to nonlinear systems with potentially multiple nonquadratic invariants. The framework also allows for the construction of numerical methods that accurately preserve dissipation structures, e.g. energy dissipation in the incompressible Navier–Stokes equations. We demonstrate the ideas by devising novel schemes that exactly conserve all known invariants of Hamiltonian systems with highorder invariants, and mass– and energy–conserving schemes for the compressible Navier–Stokes equations. This talk complements and extends the plenary talk given by Patrick Farrell.
5:00 PM
Geometric integration meets datadriven dynamical systems

Matthew Colbrook
(
University of Cambridge
)
Geometric integration meets datadriven dynamical systems
Matthew Colbrook
(
University of Cambridge
)
5:00 PM  5:30 PM
Room: Sala degli Stemmi, Scuola Normale Superiore
Koopman operators globally linearise nonlinear dynamical systems, and their spectral information can be a powerful tool for analysing and decomposing such systems. There has been a recent flurry of activity in datadriven computations of the spectral properties of Koopman operators. However, Koopman operators are infinitedimensional, making the computation of their spectral information a considerable challenge. In this talk, we will combine ideas from geometric integration with dynamic mode decomposition (DMD), one of the most popular algorithms for analysing Koopman operators. We introduce \textit{measurepreserving extended dynamic mode decomposition} (\texttt{mpEDMD}), the first Galerkin method whose eigendecomposition converges to the spectral quantities of Koopman operators for general measurepreserving dynamical systems. \texttt{mpEDMD} is a datadriven algorithm based on an orthogonal Procrustes problem that enforces measurepreserving truncations of Koopman operators using a general dictionary of observables. It is flexible and easy to use with any preexisting DMDtype method and with different data types. Enforcing this structure is crucial to its convergence and qualitative behaviour. We prove the convergence of \texttt{mpEDMD} for projectionvalued and scalarvalued spectral measures, spectra, and Koopman mode decompositions. For the case of delay embedding (Krylov subspaces), our results include the first convergence rates of the approximation of spectral measures as the size of the dictionary increases. We demonstrate \texttt{mpEDMD} on a range of challenging examples, its increased robustness to noise compared with other DMDtype methods, and its ability to capture the energy conservation and cascade of a turbulent boundary layer flow with a Reynolds number $> 6 \times 10^4$ and a statespace dimension $> 10^5$.
6:00 PM
Poster Session and Apero
Poster Session and Apero
6:00 PM  7:30 PM
Room: Dipartimento di Matematica, main hall
Pietro Deidda Arturo De Marinis Alejandro EscorihuelaTomàs Sebastian Götschel Stefano Sicilia
Friday, April 5, 2024
9:00 AM
A dynamical systems view to deep learning: contractivity and structure preservation

Elena Celledoni
(
NTNU
)
A dynamical systems view to deep learning: contractivity and structure preservation
Elena Celledoni
(
NTNU
)
9:00 AM  10:00 AM
Room: Sala degli Stemmi, Scuola Normale Superiore
The (discrete) optimal control point of view to neural networks offers an interpretation of deep learning from a dynamical systems and numerical analysis perspective and opens the way to mathematical insight. In this talk we discuss topics of structure preservation and their use in deep learning. Some deep neural networks can be designed to have desirable properties such as invertibility and group equivariance or can be adapted to problems of manifold value data. Contractivity is identified as a desirable property for stability and robustness of neural networks. We discuss classical results of contractivity of numerical ODE integrators, applications to neural networks and recent extensions to Riemannian manifolds.
10:00 AM
Variational (dynamical) discretisations of metriplectic systems

Damiano Lombardi
(
INRIA
)
Variational (dynamical) discretisations of metriplectic systems
Damiano Lombardi
(
INRIA
)
10:00 AM  10:30 AM
Room: Sala degli Stemmi, Scuola Normale Superiore
In this work we present a variational structure preserving discretisation method for metriplectic systems. These are composed of two parts: a Hamiltonian flow and a dissipation. The dissipation is built based on the Casimirs of the Hamiltonian system. A weak formulation of the Hamiltonian flow is defined. Based on this, a variational semidiscretisation in space is performed, leading to a finite dimensional Hamiltonian system. We discuss the possibility of introducing a dynamical variational discretisation by leveraging the geometrical description of the weak Hamiltonian flow, in the spirit of the Dynamical Low Rank method. Work in collaboration with Cecilia Pagliantini.
10:30 AM
Coffee break
Coffee break
10:30 AM  11:00 AM
Room: Chiostro Scuola Normale
11:00 AM
A new fast numerical method for the generalized RosenZener model

Stefano Pozza
(
Charles University
)
A new fast numerical method for the generalized RosenZener model
Stefano Pozza
(
Charles University
)
11:00 AM  11:30 AM
Room: Sala degli Stemmi, Scuola Normale Superiore
In quantum mechanics, the RosenZener model represents a twolevel quantum system. Its generalization to multiple degenerate sets of states leads to larger nonautonomous linear systems of ordinary differential equations (ODEs). We propose a new method for computing the solution operator of this system of ODEs. This new method is based on a recently introduced expression of the solution in terms of an infinite matrix equation, which can be efficiently approximated by combining truncation, fixed point iterations, and lowrank approximation. This expression is possible thanks to the socalled ∗product approach for linear ODEs. In the numerical experiments, the new method’s computing time scales linearly with the model’s size. We provide a first partial explanation of this linear behavior. Joint work with Christian Bonhomme and Niel Van Buggenhout.
11:30 AM
LiePoisson discretization for incompressible magnetohydrodynamics on the sphere

Michael Roop
(
Chalmers University of Technology
)
LiePoisson discretization for incompressible magnetohydrodynamics on the sphere
Michael Roop
(
Chalmers University of Technology
)
11:30 AM  12:00 PM
Room: Sala degli Stemmi, Scuola Normale Superiore
We give a structure preserving spatiotemporal discretization for incompressible magnetohydrodynamics (MHD) on the sphere. Discretization in space is based on the theory of geometric quantization, which yields a spatially discretized analogue of the MHD equations as a finitedimensional LiePoisson system on the dual of the magnetic extension Lie algebra, for which we develop structure preserving time discretizations. The full method preserves the underlying geometry, namely the LiePoisson structure and all the Casimirs. To showcase the method, we apply it to two models for magnetized fluids: incompressible magnetohydrodynamics and Hazeltine's model. This is a joint work with Klas Modin, arXiv:2311.16045.
12:00 PM
Variational discretizations of ideal magnetohydrodynamics in smooth regime using finite element exterior calculus

Valentin Carlier
(
MPI
)
Variational discretizations of ideal magnetohydrodynamics in smooth regime using finite element exterior calculus
Valentin Carlier
(
MPI
)
12:00 PM  12:30 PM
Room: Sala degli Stemmi, Scuola Normale Superiore
We propose a new class of finite element approximations to ideal compressible magnetohydrodynamics equations in smooth regime. Our discretizations are built via a discrete variational principle mimicking the continuous EulerPoincaré principle, with vector fields represented by their action as Lie derivatives on differential forms, to further exploit the geometrical structure of the problem. The resulting semidiscrete approximations are shown to conserve the total mass, entropy and energy of the solutions. In addition the divergencefree nature of the magnetic field is preserved in a pointwise sense, and the scheme is reversible at the fully discrete level. Numerical simulations are conducted to verify the accuracy of our approach and its ability to preserve the semidiscrete invariants for several test problems. An eponym paper will be uploaded to arxiv in the coming days.
12:30 PM
Concluding remarks
Concluding remarks
12:30 PM  12:45 PM
Room: Sala degli Stemmi, Scuola Normale Superiore