Error estimate of the Non Intrusive Reduced Basis method with finite volume schemes

The context of this paper is the simulation of parameter-dependent partial differential equations (PDEs). When the aim is to solve such PDEs for a large number of parameter values, Reduced Basis Methods (RBM) are often used to reduce computational costs of a classical high fidelity code based on Finite Element Method (FEM), Finite Volume (FVM) or Spectral methods. The efficient implementation of most of these RBM requires to modify this high fidelity code, which cannot be done, for example in an industrial context if the high fidelity code is only accessible as a "black-box" solver. The Non Intrusive Reduced Basis method (NIRB) has been introduced in the context of finite elements as a good alternative to reduce the implementation costs of these parameter-dependent problems. The method is efficient in other contexts than the FEM one, like with finite volume schemes, which are more often used in an industrial environment. In this case, some adaptations need to be done as the degrees of freedom in FV methods have different meenings. At this time, error estimates have only been studied with FEM solvers. In this paper, we present a generalisation of the NIRB method to Finite Volume schemes and we show that estimates established for FEM solvers also hold in the FVM setting. We first prove our results for the hybrid-Mimetic Finite Difference method (hMFD), which is part the Hybrid Mixed Mimetic methods (HMM) family. Then, we explain how these results apply more generally to other FV schemes. Some of them are specified, such as the Two Point Flux Approximation (TPFA)

The non-intrusive reduced basis two-grid method applied to sensitivity analysis

This paper deals with the derivation of Non-Intrusive Reduced Basis (NIRB) techniques for sensitivity analysis, more specifically the direct and adjoint state methods. For highly complex parametric problems, these two approaches may become too costly. To reduce computational times, Proper Orthogonal Decomposition (POD) and Reduced Basis Methods (RBMs) have already been investigated. The majority of these algorithms are however intrusive in the sense that the High-Fidelity (HF) code must be modified. To address this issue, non-intrusive strategies are employed. The NIRB two-grid method uses the HF code solely as a ``black-box'', requiring no code modification. Like other RBMs, it is based on an offline-online decomposition. The offline stage is time-consuming, but it is only executed once, whereas the online stage is significantly less expensive than an HF evaluation. In this paper, we propose new NIRB two-grid algorithms for both the direct and adjoint state methods. On the direct method, we prove on a classical model problem, the heat equation, that HF evaluations of sensitivities reach an optimal convergence rate in $L^{\infty}(0,T;H^1(\Omega))$, and then establish that these rates are recovered by the proposed NIRB approximation. These results are supported by numerical simulations. We then numerically demonstrate that a Gaussian process regression can be used to approximate the projection coefficients of the NIRB two-grid method. This further reduces the computational costs of the online step while only computing a coarse solution of the initial problem. All numerical results are run with the model problem as well as a more complex problem, namely the Brusselator system.Comment: 37 pages, 3 figures. arXiv admin note: text overlap with arXiv:2211.0889

A mathematical model for meniscus cartilage regeneration

We propose a continuous model for meniscus cartilage regeneration triggered by two populations of cells migrating and (de)differentiating within an artificial scaffold with a known structure. The described biological processes are influenced by a fluid flow and therewith induced deformations of the scaffold. Numerical simulations are done for the corresponding dynamics within a bioreactor which was designed for performing the biological experiments.Comment: GAMM2023, May 2023, Dresden (GERMANY), German

Merging the Structural Motifs of Functionalized Amino Acids and α-Aminoamides: Compounds with Significant Anticonvulsant Activities

Functional amino acids (FAAs) and α-aminoamides (AAAs) are two classes of antiepileptic drugs (AEDs) that exhibit pronounced anticonvulsant activities. We combined key structural pharmacophores present in FAAs and AAAs to generate a new series of compounds and document that select compounds exhibit activity superior to either the prototypical FAA (lacosamide) or the prototypical AAA (safinamide) in the maximal electroshock (MES) seizure model in rats. A representative compound, (R)-N-4′-((3″-fluoro)benzyloxy)benzyl 2-acetamido-3-methoxypropionamide ((R)-10), was tested in the MES (mice, ip), MES (rat, po), psychomotor 6 Hz (32 mA) (mice, ip), and hippocampal kindled (rat, ip) seizure tests providing excellent protection with ED50 values of 13, 14, ~10 mg/kg, and 12 mg/kg, respectively. In the rat sciatic nerve ligation model (ip), (R)-10 (12 mg/kg) provided an 11.2-fold attenuation of mechanical allodynia. In the mouse biphasic formalin pain model (ip), (R)-10 (15 mg/kg) reduced pain responses in the acute and the chronic inflammatory phases

An in-silico approach to meniscus tissue regeneration: Modeling, numerical simulation, and experimental analysis

We develop a model the dynamics of human mesenchymal stem cells (hMSCs) and chondrocytes evolving in a nonwoven polyethylene terephtalate (PET) scaffold impregnated with hyaluron and supplied with a differentiation medium. The scaffold and the cells are assumed to be contained in a bioreactor with fluid perfusion. The differentiation of hMSCs into chondrocytes favors the production of extracellular matrix (ECM) and is influenced by fluid stress. The model takes deformations of ECM and PET scaffold into account. The scaffold structure is explicitly included by statistical assessment of the fibre distribution from CT images. The effective macroscopic equations are obtained by appropriate upscaling from dynamics on lower (microscopic and mesoscopic) scales and feature in the motility terms an explicit cell diffusion tensor encoding the assessed anisotropic scaffold structure. Numerical simulations show its influence on the overall cell and tissue dynamics

Synthesis and Anticonvulsant Activities of ( R )- N -(4′-Substituted)benzyl 2-Acetamido-3-methoxypropionamides

The structure-activity relationship (SAR) for the N-benzyl group in the clinical antiepileptic agent (R)-lacosamide ((R)-N-benzyl 2-acetamido-3-methoxypropionamide, (R)-3) has been explored. Forty-three compounds were prepared and then evaluated at the National Institute of Neurological Disorders and Stroke Anticonvulsant Screening Program for seizure protection in the maximal electroshock (MES) and subcutaneous Metrazol models. Comparing activities for two series of substituted aryl regioisomers (2′, 3′, 4′) showed that 4′-modified derivatives had the highest activity. Significantly, structural latitude existed at the 4′-site. The SAR indicated that non-bulky 4′-substituted (R)-3 derivatives exhibited superb activity, independent of their electronic properties. Activities in the MES test of several compounds were either comparable with or exceeded that of (R)-3, and surpassed the activities observed for the traditional antiepileptic agents phenytoin, phenobarbital, and valproate

La substitution successorale sous le prisme du droit fiscal

Cet article illustre, au travers des exemples chiffrés, les incidences en droit fiscal de la substitution successorale et s'interroge sur l'intérêt éventuel de transposer ce mécanisme en droit fiscal

L'option héréditaire revisitée par la loi Pot Pourri V

L'option héréditaire revisitée par la loi Pot Pourri

Variations et autres développements sur la méthode de base réduite non intrusive à deux grilles

The purpose of this thesis is the analysis and the development of low-cost numerical tools. The first chapter is a review on Non-Intrusive Reduced Basis methods (NIRB) studied during this thesis. We contributed to the elaboration of an open library on NIRB methods, and we present several numerical results. In particular, we introduce the two-grid method which is analyzed in the second chapter. Its aims to recover an accurate approximation of the solution of a parameterized problem as if we had used a high-fidelity code for instance with the Finite Element Method (FEM) while significantly reducing the degrees of freedom. Thus, it reduces the complexity. In the second chapter, we proceed with several further analyses such as its adaptation to parabolic problems. We then apply the method to domain with singularities. Subsequently, we analyse the method in the context of finite volume schemes. The third chapter is concerned with the development of two new NIRB methods. One tool allows us to consider truncated domains and to further reduce the runtimes. The last part of this thesis is about an application of the two-grid method on offshore wind farm simulations. This part is a collaboration with EDF and the purpose is to test the two-grid method on highly complex problems.Le but de cette thèse est l'analyse et le développement d'outils numériques à faible coûts. La première partie présente les méthodes de Bases Réduites (MBR) étudiée pendant cette thèse. Nous avons contribué à l'élaboration d'une bibliothèque open-source sur les MBR et nous présentons plusieurs résultats numériques. Nous introduisons la méthode deux grilles. Son but est de trouver une bonne approximation d'une solution d'un problème paramétrique, aussi proche de la solution exacte que si nous avions utilisé un code haute-fidélité avec une méthode classique, tout en réduisant considérablement les degrés de liberté. Cela induit donc une réduction des temps de calculs. Après son analyse pour les équations elliptiques dans le contexte MEF, nous présentons plusieurs analyses supplémentaires comme son adaptation aux équations paraboliques. Nous appliquons ensuite la méthode à des domaines contenant des singularités. Par la suite, nous réalisons son analyse dans le contexte des volumes finis. La troisième partie de cette thèse porte sur le développement de deux nouveaux outils non intrusifs. Un des outils permet de considérer des domaines tronqués et ainsi de réduire drastiquement les temps de calculs des simulations. La dernière partie de cette thèse concerne l'application de ces méthodes à la simulation d'un champ d'éoliennes offshores. Cette partie est une collaboration avec EDF