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1. Finite element-based invariant-domain preserving approximation of hyperbolic systems: Beyond second-order accuracy in space

   https://doi.org/10.1016/j.cma.2023.116470  

   Guermond, Jean-Luc; Nazarov, Murtazo; Popov, Bojan (January 2024, Computer Methods in Applied Mechanics and Engineering)

   This paper proposes an invariant-domain preserving approximation technique for nonlinear conservation systems that is high-order accurate in space and time. The algorithm mixes a high order finite element method with an invariant-domain preserving low-order method that uses the closest neighbor stencil. The construction of the flux of the low-order method is based on an idea from Abgrall et al. (2017). The mass flux of the low-order and the high-order methods are identical on each finite element cell. This allows for mass preserving and invariant-domain preserving limiting. 

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2. Invariant-domain preserving and locally mass conservative approximation of the Lagrangian hydrodynamics equations

   https://doi.org/10.1016/j.cma.2025.117927  

   Guermond, Jean-Luc; Popov, Bojan; Saavedra, Laura; Sheridan, Madison (June 2025, Computer methods in applied mechanics and engineering)

   In this paper we construct an explicit approximation for the Lagrangian hydrodynamics equations equipped with an arbitrary equation of state. The approximation of the state variable is done with piecewise constant finite elements and the approximation of the mesh motion is done with higher-order continuous finite elements. The method is invariant-domain preserving and locally mass conservative. The purpose of this method is to be used in combination with higher-order methods to make them invariant domain preserving as well. 

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3. First-Order Greedy Invariant-Domain Preserving Approximation for Hyperbolic Problems: Scalar Conservation Laws, and p-System

   https://doi.org/10.1007/s10915-024-02592-4  

   Guermond, Jean-Luc; Maier, Matthias; Popov, Bojan; Saavedra, Laura; Tomas, Ignacio (August 2024, Journal of Scientific Computing)

   The paper focuses on first-order invariant-domain preserving approximations of hyperbolic systems. We propose a new way to estimate the artificial viscosity that has to be added to make explicit, conservative, consistent numerical methods invariant-domain preserving and entropy inequality compliant. Instead of computing an upper bound on the maximum wave speed in Riemann problems, we estimate a minimum wave speed in the said Riemann problems such that the approximation satisfies predefined invariant-domain properties and predefined entropy inequalities. This technique eliminates non-essential fast waves from the construction of the artificial viscosity, while preserving pre-assigned invariant-domain properties and entropy inequalities. 

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4. Label-Noise Robust Domain Adaptation

   Xiyu Yu, Tongliang Liu (January 2020, International Conference on Machine Learning)

   null (Ed.)

   Domain adaptation aims to correct the classifiers when faced with distribution shift between source (training) and target (test) domains. State-of-the-art domain adaptation methods make use of deep networks to extract domain-invariant representations. However, existing methods assume that all the instances in the source domain are correctly labeled; while in reality, it is unsurprising that we may obtain a source domain with noisy labels. In this paper, we are the first to comprehensively investigate how label noise could adversely affect existing domain adaptation methods in various scenarios. Further, we theoretically prove that there exists a method that can essentially reduce the side-effect of noisy source labels in domain adaptation. Specifically, focusing on the generalized target shift scenario, where both label distribution 𝑃𝑌 and the class-conditional distribution 𝑃𝑋|𝑌 can change, we discover that the denoising Conditional Invariant Component (DCIC) framework can provably ensures (1) extracting invariant representations given examples with noisy labels in the source domain and unlabeled examples in the target domain and (2) estimating the label distribution in the target domain with no bias. Experimental results on both synthetic and real-world data verify the effectiveness of the proposed method. 

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5. Asymptotic and Invariant-Domain Preserving Schemes for Scalar Conservation Equations with Stiff Source Terms and Multiple Equilibrium Points

   https://doi.org/10.1007/s10915-024-02628-9  

   Ern, Alexandre; Guermond, Jean-Luc; Wang, Zuodong (September 2024, Journal of Scientific Computing)

   We propose an operator-splitting scheme to approximate scalar conservation equations with stiff source terms having multiple (at least two) stable equilibrium points. The scheme com- bines a (reaction-free) transport substep followed by a (transport-free) reaction substep. The transport substep is approximated using the forward Euler method with continuous finite elements and graph viscosity. The reaction substep is approximated using an exponential integrator. The crucial idea of the paper is to use a mesh-dependent cutoff of the reaction time-scale in the reaction substep. We establish a bound on the entropy residual motivating the design of the scheme. We show that the proposed scheme is invariant-domain preserv- ing under the same CFL restriction on the time step as in the nonreactive case. Numerical experiments in one and two space dimensions using linear, convex, and nonconvex fluxes with smooth and nonsmooth initial data in various regimes show that the proposed scheme is asymptotic preserving. 

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