This event will focus on the numerical simulation of engineering problems in complex, possibly heterogeneous media, using computational approaches arising from homogenization theory, multiscale science, reduced order modeling, or a combination thereof. All aspects of mathematical modeling and numerical resolution and all aspects related to the practical implementation issues arising from the scientific questions addressed and numerical techniques employed, will be presented. The synergy between these aspects will be highlighted. A special emphasis will also be placed on the importance of this combination of ingredients for the successful solution of engineering problems in an industrial R&D context.
The overall purpose is to introduce, describe and study approaches that allow satisfactory approximation of solutions to these complex problems in a window of time acceptable in an industrial or, more generally, practical context. The focus is on methods that do not necessarily require elaborate and intrusive modifications of existing software that often capitalizes on a significant amount of year.person work (legacy problem). Modern approaches will be presented, that are not dedicated to endlessly obtaining more accurate results at any computational cost, but that achieve a suitable compromise between accuracy, computational cost, and implementation workload. The industrial context and the practical feasibility will be the driving motivation for the problems considered and the solutions proposed.
Winter School
The first part of this event (January 29-31, 2025) is a Winter School on these topics, aimed primarily at early career researchers and graduate and advanced undergraduate students. Students might come from various backgrounds (applied mathematics, computational mechanics, aerospace engineering, etc). The school will include three block courses:
Sonia Fliss (CNRS-Inria-ENSTA, France) will speak about problems of diffraction by thin layers of periodic and random materials;
Ulrich Hetmaniuk (founder of Shift-Invert Ltd, USA) will speak about concrete software issues for multiscale computational science;
Anthony T. Patera (MIT, USA) will speak about perspectives on engineering estimation via the heat transfer dunking problem.
Applications for the Winter School closed on November 11, 2024. Higher priority will be given to applicants who are able to attend in-person. Decisions will be issued in early December.
Research Workshop
The second part of the event (February 3-7, 2025) is a Research Workshop devoted to this area. Participation in the winter school is not a prerequisite for attending the workshop (and vice versa), although the two parts of the event have been designed as a whole.
Review of funding requests for the workshop will begin the week of November 18, 2024.
Lightning talks and Poster Session
The research workshop will include lightning talks and a poster session for early career researchers (including graduate students). If accepted, you will be asked to do both.
In order to propose a lightning session talk and a poster, you must first register for the workshop, and then submit a proposal using the form that will become available on this page after you register. The registration form should not be used to propose a lightning session talk or poster.
The deadline for proposing is January 6, 2025. If your proposal is accepted, you should plan to attend the event in-person.
Winter School Lecture Descriptions
Sonia Fliss (CNRS-Inria-ENSTA, France): Problems of diffraction by thin layers of periodic and random materials
The lectures are focused on problems of acoustic or electromagnetic diffraction by an obstacle covered by a thin layer of microstructured material with periodic or random physical properties. When the thickness of the layer and the typical size of the microstructure are of the same order and small compared to the wavelength, a numerical calculation in force can become prohibitive, especially when electromagnetic waves are involved. The course will explain how, with the help of multi-scale asymptotic developments, it is possible to build an effective model in which the thin layer is replaced by an equivalent boundary condition. The treatment of thin films is often referred to as surface homogenization, as opposed to the volume homogenization discussed in Course 2 below. In particular, the differences and links between the two types of problem and the associated numerical methods will be highlighted. It will be shown how the approximation error can also be quantified under certain assumptions about the microstructure (in particular, assumptions of stationarity and ergodicity are necessary in the case where the layer characteristics follow a probability law). Finally, the numerical resolution of this effective model is obviously simpler and much less costly, but requires additional calculations. Indeed, in these equivalent boundary conditions, a number of constants appear that are characteristic of the layer. These constants depend on “correctors”, solutions to static problems posed in unbounded domains. This raises some interesting numerical questions, which will also be addressed during the course.
Ulrich Hetmaniuk (founder of Shift-Invert Ltd, USA): Concrete software issues for multiscale computational science
The lectures and the computer tutorials will illustrate state-of-the-art techniques to numerically solve problems where a small scale is present, such as computational homogenization with Representative Volume Element (RVE), the Multiscale Finite Element Method (MsFEM), the Heterogeneous Multiscale Method (HMM), and the Local Orthogonal Decomposition (LOD) method. These methods will be applied to the diffusion equation and to the heat equation. Extension to the Helmholtz equation will be discussed in coordination with Course 1. Practical aspects for transferring these techniques to industrial codes will be considered: non-intrusive implementation, implementation for heterogeneous architectures, and three-dimensional simulations.
Anthony T. Patera (MIT, USA): Perspectives on Engineering Estimation via the Heat Transfer Dunking Problem
We consider the dunking problem: a solid body, initially at thermal equilibrium in a first environment at temperature $T_i$, is abruptly placed — “dunked” — at time $t = 0$ in a second environment, characterized by far field fluid/enclosure temperature $T_\infty$. The Quantities of Interest (QoI) describe the spatial distribution of the temperature as a function of time. The high-fidelity mathematical model for the dunking problem — a very detailed conjugate heat transfer formulation — is not practical for typical engineering studies. However, the QoI can be predicted, inexpensively and reasonably accurately, by well-established engineering estimation procedures: simplification of the high-fidelity mathematical model; subsequent approximate solution of the simplified mathematical model, often informed by archived experimental data. An important aspect of model simplification is the treatment of multiscale phenomena both in time — rapid variations in the fluid relative to slow variations in the solid — and in space — due to heterogeneous material composition. We shall first present the classical estimation procedures and then proceed to newer work on rigorous error estimation. The ultimate goal is Autonomous Heat Transfer Estimation (AHTE). We briefly describe and (in the hands-on part of the program) illustrate the AHTE framework with particular reference to underlying software components.Work in collaboration with Theron Guo (MIT), Kento Kaneko (MIT), and Claude Le Bris (ENPC). Research supported by the Office of Naval Research.
Moritz Hauck
Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University
P
H
Patrick Henning
Ruhr University Bochum
D
K
David Knezevic
Akselos Company
G
L
Guanglian Li
University of Hong Kong
R
M
Roland Maier
Karlsruhe Institute of Technology
D
P
Diego Paredes
University of Concepcion
D
P
Daniel Peterseim
Augsburg University
L
R
Lukas Renelt
Institut für Analysis und Numerik, Universität Münster
M
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Marcus Sarkis
Worcester Polytechnic Institute
K
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Kathrin Smetana
Stevens Institute of Technology
T
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Timo Sprekeler
Texas A&M University
Schedule
Monday, February 3, 2025
8:30-9:00 CST
Check-In and Breakfast
9:00-9:45 CST
Multicontinuum homogenization and applications
Speaker: Yalchin Efendiev (Texas A&M University)
In this talk, I will discuss a general approach for homogenization that uses multiple macroscopic continua. I will discuss its relation to existing methods, derivation to multiscale methods, and show some applications.
9:45-10:00 CST
Q&A
10:00-10:30 CST
Coffee Break
10:30-11:15 CST
Parametric Model-Order-Reduction for Turbulent-Flow Applications
Speaker: Paul Fischer (Argonne National Laboratory)
We present recent developments in parametric model-order-reduction (pMOR) for buoyancy-driven flows in the open source Navier-Stokes code, Nek5000/RS. The main idea behind pMOR is to leverage high-fidelity simulations (aka full-order models, or FOMs) of turbulent thermal/fluid problems that run on DOE's leading-edge supercomputers to build reduced-order models (ROMs) that can run on a laptop. There are two essential elements to the approach. One is the reproduction problem, in which we build a model that is capable of tracking the time evolution of quantities of interest (QOIs) at a given point in parameter space (e.g., thermal loading conditions) using a low-rank ordinary differential equation that governs representative solution modes. The modes typically come from a proper orthogonal decomposition of FOM solution snapshots but it is also possible to consider augmenting this basis with higher wavenumber modes derived from nonlinear interactions of the POD modes [1]. The second element of pMOR is to run the ROM at different test points in the parametric domain in order to track QOI dependencies without rerunning the FOM.
This talk will primarily explore successes and limitations in ROM reproduction of turbulent flow examples. We illustrate cases where pMOR is viable and also examples where it is unable to extend beyond the training space. To aid in understanding the pMOR/ROM process and the fundamental fluid mechanics itself we seek to identify and characterize differences between flows where pMOR succeeds and those where it fails.
[1] Kento Kaneko and Paul Fischer. Augmented reduced order models for turbulence. Front. Phys. 10:905392. doi: 10.3389/fphy.2022.905392 (2022).
Paul Fischer (UIUC) Viral Shah (UIUC) Nicholas Christensen (UIUC) Kento Kaneko (M.I.T.) Ping-Hsuan Tsai (Virginia Tech)
11:15-11:30 CST
Q&A
11:30-11:35 CST
Tech Break
11:35-12:20 CST
Finite element approximation of Fokker–Planck–Kolmogorov equations with application to numerical homogenization
Speaker: Timo Sprekeler (Texas A&M University)
We begin by discussing the finite element approximation of stationary Fokker--Planck--Kolmogorov (FPK) equations subject to periodic boundary conditions in two settings: one with weakly differentiable coefficients, and one with merely essentially bounded measurable coefficients under a Cordes-type condition. These problems arise as governing equations for the invariant measure in the homogenization of nondivergence-form equations with large drifts. We then suggest and rigorously analyze an approximation scheme for the effective diffusion matrix in both settings, based on the finite element scheme for stationary FPK problems developed in the first part. This is joint work with Endre Süli (University of Oxford) and Zhiwen Zhang (The University of Hong Kong).
12:20-12:35 CST
Q&A
12:35-13:45 CST
Lunch Break
13:45-14:30 CST
Achieving higher-order convergence rates in numerical homogenization
Speaker: Roland Maier (Karlsruhe Institute of Technology)
This talk is about the construction of higher-order multiscale methods in the framework of the Localized Orthogonal Decomposition approach. We show how to achieve higher-order convergence rates in the elliptic setting without restrictive regularity assumptions on the domain, the coefficient, or the exact solution. Further, we discuss extensions to time-dependent problems where appropriate adaptations are required. Numerical examples are presented to illustrate the theoretical findings.
14:30-14:45 CST
Q&A
14:45-14:50 CST
Tech Break
14:50-15:20 CST
Lightning Talks part 1
15:20-16:30 CST
Poster Session part 1 and Social Hour
Tuesday, February 4, 2025
8:30-9:00 CST
Check-In and Breakfast
9:00-9:45 CST
Component-based Reduced Order Modeling for Digital Twins of Large Scale Industrial Assets
Speaker: David Knezevic (Akselos Company)
Akselos provides Digital Twins of industrial equipment in a range of industries, including energy (oil & gas, wind, hydro), marine, mining, chemicals, and aerospace. The Akselos platform is based on RB-FEA, which is a unique combination of the Reduced Basis method for fast and accurate reduced order modeling of parametrized PDEs and a domain decomposition framework that enables large-scale component-based analysis. In this presentation we will discuss the foundations of RB-FEA --- including its similarities and differences to other related Scientific Machine Learning methods --- and demonstrate applications to industrial Digital Twins.
9:45-10:00 CST
Q&A
10:00-10:30 CST
Coffee Break
10:30-11:15 CST
Randomized Multiscale Methods for Heterogeneous Nonlinear Partial Differential Equations
Speaker: Kathrin Smetana (Stevens Institute of Technology)
Heterogeneous problems that take place at multiple scales are ubiquitous in science and engineering. Examples are wind turbines made from composites or groundwater flow relevant e.g., for the design of flood prevention measures. However, finite element or finite volume methods require an often prohibitively large amount of computational time for such tasks. Multiscale methods that are based on ansatz functions which incorporate the local behavior of the (numerical) solution of the partial differential equations (PDEs) have been developed to tackle these heterogeneous problems. Localizable multiscale methods that allow controlling the error due to localization and the (global) approximation error at a (quasi-optimal) rate and do not rely on structural assumptions such as scale separation or periodicity have only been developed within the last decade. Here, localizable multiscale methods allow the efficient construction of the basis functions by solving the PDE (in parallel) on several small subdomains at low cost. While there has been a significant progress in recent years for these types of multiscale methods for linear PDEs, very few results have been obtained so far for nonlinear PDEs. In this talk, we will show how randomized methods and their probabilistic numerical analysis can be exploited for the construction and numerical analysis of such types of multiscale methods for nonlinear PDEs.
11:15-11:30 CST
Q&A
11:30-11:35 CST
Tech Break
11:35-12:20 CST
Multiscale Spectral Generalised Finite Elements: Preconditioning and Model Reduction Beyond SPD
Speaker: Christian Alber (University of Heidelberg)
Multiscale PDEs with heterogeneous, highly oscillatory coefficients pose severe challenges for standard numerical methods. Two prominent approaches to tackle such problems are numerical multiscale methods with problem-adapted coarse spaces and structured (approximate) inversion techniques that exploit a low-rank property of the associated Green’s function. They can also be used to precondition the resulting large-scale and typically very ill-conditioned linear equation systems. This work presents an abstract framework for the design and analysis of the Multiscale-Spectral Generalized FEM (MS-GFEM), a partition of unity method based on optimal local approximation spaces constructed from local eigenproblems, which is closely related to the GenEO coarse space more familiar in the DD community. We establish a general local approximation theory demonstrating, under certain assumptions, an exponential convergence w.r.t the local degrees of freedom and an explicit dependence on key parameters. Our framework applies to a broad class of multiscale PDEs with L∞-coefficients, including convection-dominated diffusion or high-frequency Helmholtz/Maxwell. Notably, we prove a nearly exponential, local convergence rate on all those problems. As a corollary, the MS-GFEM space can be used within a robust two-level DD preconditioner to achieve condition numbers arbitrarily close to one. Numerical experiments support the theoretical results and demonstrate huge efficiency gains.
12:20-12:35 CST
Q&A
12:35-13:45 CST
Lunch Break
13:45-14:30 CST
Energy-adaptive Riemannian gradient methods for computing rotating Bose-Einstein condensates
Speaker: Patrick Henning (Ruhr-Universität Bochum)
In this talk we investigate the numerical approximation of ground states of rotating Bose-Einstein condensates. This problem requires the minimization of the Gross-Pitaevskii energy functional on a Hilbert manifold. To find a In corresponding minimizer, we use a generalized Riemannian gradient method that is based on the concept of Sobolev gradients in combination with an adaptively changing metric on the manifold. By a suitable choice of the metric, global energy dissipation for the arising gradient method can be proved. The energy dissipation property in turn implies global convergence to the density of a ground or excited state of the system. Furthermore, we present a precise characterization of the local convergence rates in a neighborhood of each ground state and how these rates depend on spectral gaps. Our findings are validated in numerical experiments.
14:30-14:45 CST
Q&A
14:45-15:15 CST
Coffee Break
15:15-16:00 CST
Iterative methods for heterogeneous Timoshenko beam network models
Speaker: Moritz Hauck (Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University)
This talk deals with the numerical solution of Timoshenko beam network models, i.e., Timoshenko beam equations at each edge of the network, coupled at the nodes of the network by rigid joint conditions. A prominent application of such models is the simulation of fiber-based materials such as paper or cardboard. Through hybridization, we can equivalently reformulate the problem as a symmetric positive definite system of linear equations posed at the network nodes. This is possible because the nodes to which the beam equations are coupled are zero-dimensional objects. To discretize the beam network model, we apply a hybridizable discontinuous Galerkin method that can achieve arbitrary orders of convergence under mesh refinement without increasing the size of the global system matrix. As a preconditioner for the typically very poorly conditioned global system matrix, we employ a two-level overlapping additive Schwarz method. We prove uniform convergence of the corresponding preconditioned conjugate gradient method under appropriate connectivity assumptions on the network. Numerical experiments demonstrate the practical performance of the method.
16:00-16:15 CST
Q&A
Wednesday, February 5, 2025
8:30-9:00 CST
Check-In and Breakfast
9:00-9:45 CST
Data driven reduced order modeling for first order hyperbolic systems with application to waveform inversion
Speaker: Liliana Borcea (University of Michigan)
Waveform inversion seeks to estimate an inaccessible heterogeneous medium by using sensors to probe the medium with signals and measure the generated waves. It is an inverse problem for a hyperbolic system of equations, with the sensor excitation modeled as a forcing term and the heterogeneous medium described by unknown, variable coefficients. The traditional formulation of the inverse problem, called full waveform inversion (FWI), estimates the unknown coefficients via nonlinear least squares data fitting. For typical band limited and high frequency data, the data fitting objective function has spurious local minima near and far from the true coefficients. This is why FWI implemented with gradient based optimization can fail, even for good initial guesses. We propose a different approach to waveform inversion: First, use the data to ``learn” a good algebraic model, called a reduced order model (ROM), of how the waves propagate in the unknown medium. Second, use the ROM to obtain a good approximation of the wave field inside the medium. Third, use this approximation to solve the inverse problem. I will give a derivation of such a ROM for a general first order hyperbolic system satisfied by all linear waves in lossless media (sound, electromagnetic or elastic). I will describe the properties of the ROM and will use it to solve the inverse problem for sound waves.
9:45-10:00 CST
Q&A
10:00-10:30 CST
Coffee Break
10:30-11:15 CST
Multiscale approximations of minimizers in the Ginzburg-Landau model
Speaker: Christian Doeding (University of Bonn)
Superconductors are described by minimizers of the Ginzburg-Landau energy, which can exhibit fascinating macroscopic phenomena such as the formation of Abrikosov vortex lattices under an external magnetic field. The numerical approximation of these states is challenging due to stringent requirements on the computational mesh resolution, which can be expressed mathematically by error bounds of the discrete minimizers involving the mesh size and a material parameter in the Ginzburg-Landau energy functional. In this work, we present a framework based on Localized Orthogonal Decomposition, a multiscale technique for constructing problem-adapted approximation spaces. Tailored to the Ginzburg-Landau setting, this approach relaxes the mesh size and material parameter resolution constraints compared to standard finite element methods, allowing for more efficient and accurate approximations of vortex lattice structures in superconductors.
11:15-11:30 CST
Q&A
11:30-11:35 CST
Tech Break
11:35-12:20 CST
Multiscale Hybrid Methods: Theoretical Foundations and Computational Analysis
Speaker: Diego Paredes (Universidad de Concepción)
In this talk, we revisit the fundamentals of the Multiscale Hybrid-Mixed (MHM) method, initially proposed for a multiscale Laplace equation in [1] and further analyzed in [2,3]. The MHM method couples a global problem, defined on a skeleton mesh, with a collection of uncoupled local problems, each associated with the elements of a polytopal partition. These problems are derived from a static condensation process combined with a space decomposition applied to the hybrid formulation of the original problem. This decomposition ensures the well-posedness of the global and local problems by defining them as constrained problems. The second part of this talk introduces a novel variational methodology that bypasses the need for such a space decomposition, directly formulating the methods through elliptic local and global problems. This new approach forms the basis of the Multiscale Hybrid (MH) method, introduced in [4]. We present a comprehensive theoretical framework, numerical experiments, and a computational performance analysis, comparing both methods. [1] Harder, C., Paredes, D. and Valentin, F. "A Family of Multiscale Hybrid Mixed Finite Element Methods for the Darcy Equation with Rough Coefficients", Journal of Computational Physics, Vol. 245, pp. 107-130, 2013 [2] Araya, R., Harder, C., Paredes, D. and Valentin, F. "Multiscale Hybrid-Mixed Methods", SIAM Journal on Numerical Analysis, Vol. 51(6), pp. 3505-3531, 2013 [3] Barrenechea, G., Jaillet, F., Paredes, D., and Valentin, F. "The Multiscale Hybrid Mixed Method in General Polygonal Meshes", Numerische Mathematik, Vol. 145, pp. 197-237, 2020 [4] Barrenechea, G. R., Gomes, A. T. A., Paredes, D. "A Multiscale Hybrid Method". SIAM Journal on Scientific Computing, Vol. 46, pp. A1628-A1657, 2024
12:20-12:35 CST
Q&A
12:35-13:45 CST
Lunch Break
13:45-14:30 CST
On some non-intrusive implementations of ROM techniques for the effective simulation of engineering problems
Speaker: Ludovic Chamoin (ENS Paris-Saclay)
The talk will present two engineering applications in which non-intrusive implementations of ROM techniques in commercial codes have been performed. The first application deals with the prediction of remaining useful life (RUL) of IGBT power electronic modules, which are essential components to numerous electrical systems. During their operation, losses generate heat within the module, leading to thermal stress, and eventually resulting in component failure. In order to obtain a physics-based computational model which is compatible with real-time RUL prediction, and which accounts for the numerous uncertainty sources, a parametrized multi-physics (electro-thermo-mechanical) reduced model is developed for IGBT power modules. It is based on the Proper Generalized Decomposition (PGD) method, and it is implemented in a non-intrusive version in Ansys. RUL estimation and uncertainty quantification are then performed from the assimilation of experimental measures, by means of Bayesian inference and transport maps sampling. The second application tackles the challenge of effective modeling and simulation for large mechanical structures exhibiting numerous local complex behaviors, here spot welds in automative crash numerical analysis. We propose a non-intrusive local/global model coupling strategy, in which the local model is a neural network-based reduced model, specifically a Physics-Guided Neural Network (PGANN). The proposed strategy, implemented in OpenRadioss, does not modify the global solver. It enables accurate simulations on complex 3D industrial structures with multiple spot welds, while maintaining computational efficiency. During the talk, methodological and technical aspects of these two non-intrusive ROM applications will be discussed.
14:30-14:45 CST
Q&A
14:45-15:15 CST
Coffee Break
15:15-15:45 CST
Lightning Talks part 2
15:45-16:30 CST
Poster Session part 2
Thursday, February 6, 2025
8:30-9:00 CST
Check-In and Breakfast
9:00-9:45 CST
Computational Relaxation Techniques for Enhanced Damage Modeling
Speaker: Daniel Peterseim (Augsburg University)
Damage modeling is critical to understanding and simulating the behavior of materials as they approach failure. Classical continuum damage models often face significant challenges due to the loss of convexity in their incremental variational formulations, leading to unreliable and mesh-dependent solutions. Relaxation techniques provide robust alternatives by modeling homogenized microstructures and providing mesh-independent results.
This talk explores recent advances in the computational implementation of these methods, with a focus on the role of reduced-order modeling techniques in different approaches to semiconvex relaxation. Polyconvexification exploits structural assumptions on the model to achieve substantial dimensionality reduction, while numerical reduction techniques make large-scale simulations feasible in the rank-one case. Together, these advances improve the stability, efficiency, and scalability of damage models, enabling simulations of strain-softening phenomena in soft biological tissues and other complex material behaviors critical to engineering applications.
This is joint work with D. Balzani, M. Köhler (Bochum) and M. Peter, T. Neumeier (Augsburg) and D. Wiedemann (Dortmund).
References:
Balzani, D., Köhler, M., Neumeier, T., & Wiedemann, D. (2024). Multidimensional rank-one convexification of incremental damage models at finite strains. Computational Mechanics, 73(1), 27–47. https://doi.org/10.1007/s00466-023-02354-3
Neumeier, T., Peter, M. A., Peterseim, D., & Wiedemann, D. (2024). Computational polyconvexification of isotropic functions. Multiscale Modeling & Simulation, 22(4), 1402–1420. https://doi.org/10.1137/23M1589773
Köhler, M., Neumeier, T., Peter, M. A., Peterseim, D., & Balzani, D. (2024). Hierarchical rank-one sequence convexification for the relaxation of variational problems with microstructures. Computer Methods in Applied Mechanics and Engineering, 432, Article 117321. https://doi.org/10.1016/j.cma.2024.117321
Köhler, M., Neumeier, T., Melchior, J., Peter, M. A., Peterseim, D., & Balzani, D. (2022). Adaptive convexification of microsphere-based incremental damage for stress and strain softening at finite strains. Acta Mechanica, 233(12), 4347–4364. https://doi.org/10.1007/s00707-022-03320-7
9:45-10:00 CST
Q&A
10:00-10:30 CST
Coffee Break
10:30-11:15 CST
Adaptive LOD-BDDC for elliptic problems with rough coefficients
Speaker: Marcus Sarkis (Worcester Polytechnic Institute)
We consider finite element methods of multiscale type to approximate solutions for two-dimensional symmetric elliptic partial differential equations with very rough coefficients. The methods are of Galerkin type and follow the Variational Multiscale and Localized Orthogonal Decomposition--LOD approaches in the sense that it decouples spaces into multiscale and fine subspaces. In a first method, the multiscale basis functions are obtained by mapping coarse basis functions, based on corners used on primal iterative substructuring methods, to functions of global minimal energy. This approach delivers quasi-optimal a priori error energy approximation with respect to the mesh size, but it is not robust with respect to high-contrast coefficients. In a second method, edge modes based on local generalized eigenvalue problems are added to the corner modes. As a result, optimal a priori error energy estimate is achieved which is mesh and contrast independent. The methods converge at optimal rate even if the solution is only in H1. Numerical experiments will be provided.
11:15-11:30 CST
Q&A
11:30-11:35 CST
Coffee Break
11:35-12:20 CST
Numerical homogenization for time-harmonic Maxwell equations in heterogeneous media with large wavenumber
Speaker: Guanglian Li (University of Hong Kong)
We propose a new numerical homogenization method based upon the edge multiscale method for time-harmonic Maxwell equations in heterogeneous media with large wavenumber. Numerical methods for time-harmonic Maxwell equations in homogeneous media with large wavenumber is very challenging due to the so-called pollution effect: the mesh size should be much smaller than the reciprocal of the wavenumber to obtain a solution with certain accuracy. It is much more challenging for the case with heterogeneous media that occurs often in the practical applications, such as the simulation of metamatrial, since one has to resolve the heterogeineouty for a reasonable numerical solution. We devise a novel approach that do not to resolve the heterogeneouty in the coefficient and has a mesh size linearly depends on the reciprocal of the wavenumber, which has a first order convergence rate. Extensive numerical tests are provided to verify our theoretical findings.
12:20-12:35 CST
Q&A
12:35-13:45 CST
Lunch Break
13:45-14:30 CST
Best Current Parallel in Time Methods for Parabolic and Hyperbolic Problems
Speaker: Martin Gander (Universite de Geneve)
Space-time parallel methods, also known more recently under the name PinT (Parallel in Time) methods, have a long history, but they have received a lot of attention over the past two decades. This is driven by the parallel hardware architectures that have now millions of cores, leading to saturation when parallelizing in space only. Parallelizing also the time direction is tempting, but this is very different from the space direction, because evolution problems satisfy a causality principle: the future is dependent on the past, and not the other way round. I will show in my presentation that successful strategies for PinT methods depend strongly on the nature of the evolution problem. For hyperbolic problems, effective PinT methods are Domain Decomposition methods of Waveform Relaxation type, culminating in Unmapped Tent Pitching methods, ParaDiag methods, and also direct time parallel methods like ParaExp. Most of these methods can also be very effectively used for parabolic problems, but for such problems there are also highly successful multilevel methods, like Parareal and its variants. The currently best ones are however space-time multigrid methods. All these multilevel methods struggle however when applied to hyperbolic problems.
14:30-14:45 CST
Q&A
14:45-15:15 CST
Coffee Break
Friday, February 7, 2025
8:30-9:00 CST
Check-In and Breakfast
9:00-9:45 CST
Model order reduction and localization of Friedrichs’ systems
Speaker: Lukas Renelt (Institut für Analysis und Numerik, Universität Münster)
The Friedrichs' framework, originally introduced by Friedrichs in 1958, generalizes a large class of linear differential operators into an abstract setting. Examples of Friedrichs' operators include advection-reaction or convection-diffusion equations, the time-harmonic Maxwell equations or compressible linear elasticity. In recent years, the unified analysis and discretization of these operators has seen increased interest, in particular due to a well-founded variational theory that as been developed by now. In this talk we will explore the benefits of the framework for the fields of model order reduction (MOR) and for spectral multiscale approaches leveraging localized training.
In a first step, we present a result that allows for the identification of exponentially approximable parametrized Friedrichs' systems (in the sense of Kolmogorov). This also generalizes established approximation-theoretic results to the case of parameter-dependent ansatz or test spaces which naturally occur in the variational formulations. Moreover, a least squares approach which is based on an ultraweak formulation and only requires minimal regularity is presented and subsequently used for the computation of reduced basis functions.
In a second step, the by now well-established construction of local approximation spaces via localized training will be analyzed for Friedrichs' systems. For given subdomains, this procedure solves the underlying equation on an oversampling domain using randomized boundary data, restricts to the inner domain and finally compresses the resulting local solutions. Leveraging our theory, we will give a criterion when this method can be expected to work well.
9:45-10:00 CST
Q&A
10:00-10:30 CST
Coffee Break
10:30-11:15 CST
Robust Multiscale Methods for Helmholtz equations in high contrast heterogeneous media
Speaker: Eric Chung (Chinese University of Hong Kong)
Solving Helmholtz equations with heterogeneous coefficients can be challenging due to the high-contrast structure and pollution effect. In this talk, we present a novel multiscale method in the spirit of the constraint energy minimization generalized multiscale finite element method (CEM-GMsFEM) by tailoring new trial and test spaces. We establish the inf-sup stability to secure the well-posedness of our multiscale problem and prove the error estimate that is independent of the high-contrast coefficient. The theoretical results are validated by numerical tests, which further show that the multiscale technique can effectively capture pertinent physical phenomena. This work is partially supported by the Hong Kong RGC General Research Fund (Projects: 14305423 and 14305222).
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