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Le 2 mai 2024
à 14:00
Séminaire de Calcul Scientifique et Modélisation
Salle 2
Victor Michel-Dansac Inria Strasbourg
Recent advances on fully well-balanced methods: high-order accuracy, hydrodynamic reconstruction and hybridization with machine learning
This talk is dedicated to the presentation of several recent papers, all dealing with the development of fully well-balanced (FWB) methods, i.e. numerical methods which exactly (or approximately) preserve the steady solutions of a system of hyperbolic balance laws. In addition, the schemes we describe share another property: they do not require the costly inversion of nonlinear systems.
Namely, we will present results from https://hal.science/hal-03271103/document, https://hal.science/hal-04083181/document and https://hal.science/hal-04246991/document:
1/ A high-order FWB scheme obtained by introducing a straightforward correction method, applicable to schemes of order 2 or higher, such as MUSCL-type schemes. This correction ensures exact preservation of steady solutions without the need to invert the underlying nonlinear system. This technique ends up being a way of making any first-order scheme exactly well-balanced, but it relies on a first-order FWB scheme to fall back to.
2/ To that end, we also present an extension of the well-known hydrostatic reconstruction to preserve steady solutions of the shallow water system with nonzero velocity, without the need for specific numerical fluxes, and without having to solve a nonlinear system.
3/ Finally, relaxing the constraint on "exact" preservation of the steady solution, we design new discontinuous Galerkin (DG) basis functions able to either exactly or approximately preserve steady solutions. The DG basis is enriched with a prior computed by a Physics-Informed Neural Network (PINN), maintaining the same convergence order but improving the error constant.
Le 13 mai 2024
à 15:30
Séminaire de Calcul Scientifique et Modélisation
Salle 1
Eitan Tadmor Fondation Sciences Mathematiques de Paris\, LJLL\, Sorbonne University and University of Maryland\, College Park
Swarm-Based Gradient Descent Method for Non-Convex Optimization
Le 13 juin 2024
à 14:00
Séminaire de Calcul Scientifique et Modélisation
Salle 2
Firas Dhaouadi University of Trento
.
Le 20 juin 2024
à 14:00
Séminaire de Calcul Scientifique et Modélisation
Salle 2
Dmitri Kuzmin Université de Dortmund
.
Le 27 juin 2024
à 14:00
Séminaire de Calcul Scientifique et Modélisation
Salle 2
Davide Torlo SISSA Trieste
Structure preserving methods via Global Flux quadrature: divergence-free preservation with continuous Finite Element
In many problems, the emergence of physical structures and equilibrium solutions, such as divergence-free solutions in contexts like shallow water and magneto-hydrodynamics, poses a significant challenge. A simple linear approximation of such systems that already show these behavior is the linear acoustic system of equations. We focus on Cartesian grid discretizations of such system in 2 dimensions and in the preservation of stationary solutions that arise due to a truly multidimensional balance of terms, which corresponds to the divergence-free solutions for acoustic systems.
Conventional methods, like the continuous Finite Element SUPG, face limitations in maintaining these structures due to the stabilization techniques employed, which do not effectively vanish when the discrete divergence is zero.
What we propose is to use the Global Flux procedure, which has proven to be successful in preserving 1-dimensional equilibria [1,2], to define some auxiliary variables guiding a suitable discretization of both the divergence and stabilization operators [3]. This approach enables the natural preservation of divergence-free solutions and more intricate equilibria involving various sources. Moreover, this strategy facilitates the identification of discrete equilibria of the scheme that verify boundary or initial conditions. We use the Deferred Correction time discretization, obtaining explicit arbitrarily high order methods.
Numerous numerical tests validate the accuracy of our proposed scheme compared to classical approaches. Our method not only excels in preserving (discretely) divergence-free solutions and their perturbations but also maintains the original order of accuracy on smooth solutions.
[1] Y. Cheng, A. Chertock, M. Herty, A. Kurganov and T. Wu. A new approach for designing moving-water equilibria preserving schemes for the shallow water equations. J. Sci. Comput. 80(1): 538–554, 2019.
[2] M. Ciallella, D. Torlo and M. Ricchiuto. Arbitrary high order WENO finite volume scheme with flux globalization for moving equilibria preservation. Journal of Scientific Computing, 96(2):53, 2023.
[3] W. Barsukow, M. Ricchiuto and D. Torlo. Structure preserving methods via Global Flux quadrature: divergence-free preservation with continuous Finite Element. In preparation, 2024.
Séminaire Calcul Scientifique et Modélisation
Responsables : Christele Etchegaray et Martin Parisot
Namely, we will present results from https://hal.science/hal-03271103/document, https://hal.science/hal-04083181/document and https://hal.science/hal-04246991/document:
1/ A high-order FWB scheme obtained by introducing a straightforward correction method, applicable to schemes of order 2 or higher, such as MUSCL-type schemes. This correction ensures exact preservation of steady solutions without the need to invert the underlying nonlinear system. This technique ends up being a way of making any first-order scheme exactly well-balanced, but it relies on a first-order FWB scheme to fall back to.
2/ To that end, we also present an extension of the well-known hydrostatic reconstruction to preserve steady solutions of the shallow water system with nonzero velocity, without the need for specific numerical fluxes, and without having to solve a nonlinear system.
3/ Finally, relaxing the constraint on "exact" preservation of the steady solution, we design new discontinuous Galerkin (DG) basis functions able to either exactly or approximately preserve steady solutions. The DG basis is enriched with a prior computed by a Physics-Informed Neural Network (PINN), maintaining the same convergence order but improving the error constant.
We discuss a new swarm-based gradient descent (SBGD) method for non-convex optimization. The swarm consists of agents, each is identified with position $x$ and mass $m$. There are three key aspects to the SBGD dynamics: (i) persistent transition of mass from agents at high to lower ground; (ii) a random marching direction, aligned with the steepest gradient descent; and (iii) a time stepping protocol which decreases with $m$.
The interplay between positions and masses leads to dynamic distinction between `heavier leaders’ near local minima, and `lighter explorers’ which explore for improved position with large(r) time steps. Convergence analysis and numerical simulations demonstrate the effectiveness of SBGD method as a global optimizer.
Conventional methods, like the continuous Finite Element SUPG, face limitations in maintaining these structures due to the stabilization techniques employed, which do not effectively vanish when the discrete divergence is zero.
What we propose is to use the Global Flux procedure, which has proven to be successful in preserving 1-dimensional equilibria [1,2], to define some auxiliary variables guiding a suitable discretization of both the divergence and stabilization operators [3]. This approach enables the natural preservation of divergence-free solutions and more intricate equilibria involving various sources. Moreover, this strategy facilitates the identification of discrete equilibria of the scheme that verify boundary or initial conditions. We use the Deferred Correction time discretization, obtaining explicit arbitrarily high order methods.
Numerous numerical tests validate the accuracy of our proposed scheme compared to classical approaches. Our method not only excels in preserving (discretely) divergence-free solutions and their perturbations but also maintains the original order of accuracy on smooth solutions.
[1] Y. Cheng, A. Chertock, M. Herty, A. Kurganov and T. Wu. A new approach for designing moving-water equilibria preserving schemes for the shallow water equations. J. Sci. Comput. 80(1): 538–554, 2019.
[2] M. Ciallella, D. Torlo and M. Ricchiuto. Arbitrary high order WENO finite volume scheme with flux globalization for moving equilibria preservation. Journal of Scientific Computing, 96(2):53, 2023.
[3] W. Barsukow, M. Ricchiuto and D. Torlo. Structure preserving methods via Global Flux quadrature: divergence-free preservation with continuous Finite Element. In preparation, 2024.
Institut de Mathématiques de Bordeaux UMR 5251
Université de Bordeaux
351, cours de la Libération - F 33 405 TALENCE
Tél: +33 (0)5 40 00 60 70
institut
math.u-bordeaux.fr
