We consider in this paper multiphase flow models involving two, three or four fields, in total mechanical and thermodynamical disequilibrium. Thus several pressure fields arise, and we precisely focus here on the pressure relaxation process, while restricting to four distinct multiphase flow models. The first two models only involve immiscible compressible components, while the last two hybrid models involve both miscible and immiscible components. It is shown that some -weak- restrictions may occur on pressure gaps, which are unlikely to appear in practice. Evenmore, three-phase flow models may involve a non-monotone behaviour in the return to pressure equilibrium.
{"title":"Pressure relaxation in some multiphase flow models","authors":"J. Hérard, Guillaume Jomée","doi":"10.1051/proc/202372019","DOIUrl":"https://doi.org/10.1051/proc/202372019","url":null,"abstract":"We consider in this paper multiphase flow models involving two, three or four fields, in total mechanical and thermodynamical disequilibrium. Thus several pressure fields arise, and we precisely focus here on the pressure relaxation process, while restricting to four distinct multiphase flow models. The first two models only involve immiscible compressible components, while the last two hybrid models involve both miscible and immiscible components. It is shown that some -weak- restrictions may occur on pressure gaps, which are unlikely to appear in practice. Evenmore, three-phase flow models may involve a non-monotone behaviour in the return to pressure equilibrium.","PeriodicalId":53260,"journal":{"name":"ESAIM Proceedings and Surveys","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81704129","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
B. Delacroix, M. Bouyges, Ghislain Blanchard, C. Laurent, P. Villedieu
This paper presents a new shallow-water type model suitable for the simulation of partially wetting liquid films without the need for very fine resolution of the contact line phenomena, which is particularly suitable for industrial applications. This model is based on the introduction of a color function, propagated at the averaged velocity of the bulk flow, and equal to one where there is a liquid film and zero in the dry zone, which implies a non zero gradient only at the interface. This approach has the advantage of easily locating the interface, allowing to model macroscopically the forces acting at the contact line, which is essential for the simulation of partial wetting phenomena. The formal derivation of this model is based on the principle of least action known as Hamilton's principle. Here this principle is applied in a full Eulerian form to derive the complete system of equations with the color function. This method proves to be particularly suitable for this type of development and as an illustration it is also applied to recover another model proposed by Lallement et al. [39, 73]. Finally, both models are compared from a theoretical point of view and the advantages of the new color function based model are discussed.
{"title":"Derivation via Hamilton's principle of a new shallow-water model using a color function for the macroscopic description of partial wetting phenomena","authors":"B. Delacroix, M. Bouyges, Ghislain Blanchard, C. Laurent, P. Villedieu","doi":"10.1051/proc/202372064","DOIUrl":"https://doi.org/10.1051/proc/202372064","url":null,"abstract":"This paper presents a new shallow-water type model suitable for the simulation of partially wetting liquid films without the need for very fine resolution of the contact line phenomena, which is particularly suitable for industrial applications. This model is based on the introduction of a color function, propagated at the averaged velocity of the bulk flow, and equal to one where there is a liquid film and zero in the dry zone, which implies a non zero gradient only at the interface. This approach has the advantage of easily locating the interface, allowing to model macroscopically the forces acting at the contact line, which is essential for the simulation of partial wetting phenomena. The formal derivation of this model is based on the principle of least action known as Hamilton's principle. Here this principle is applied in a full Eulerian form to derive the complete system of equations with the color function. This method proves to be particularly suitable for this type of development and as an illustration it is also applied to recover another model proposed by Lallement et al. [39, 73]. Finally, both models are compared from a theoretical point of view and the advantages of the new color function based model are discussed.","PeriodicalId":53260,"journal":{"name":"ESAIM Proceedings and Surveys","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73206753","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a parameterized-background data-weak (PBDW) approach [Y Maday, AT Patera, JD Penn, M Yano, Int J Numer Meth Eng, 102(5), 933–965] to the steady-state variational data assimilation (DA) problem for systems modeled by partial differential equations (PDEs) and characterized by multiple interconnected components, with emphasis on vascular flows. We focus on the problem of reconstructing the state of the system in one specific component, based on local measurements. The PBDW approach does not require the solution of any PDE model at prediction stage (projection-by-data) and, as such, enables local state estimates on single components, as long as good background and update spaces for the estimation can be constructed. We discuss the application of PBDW to a two-dimensional steady Navier-Stokes problem for a family of parameterized geometries, and investigate instead the effects of enforcing no-slip boundary conditions and incompressibility constraints on the background and update spaces to enhance the state estimation. Furthermore, we show an actionable strategy to train local reduced-order bases (ROBs) for the background space that can later be used for DA tasks.
本文提出了一种参数化背景数据弱化(PBDW)方法[Y Maday, AT Patera, JD Penn, M Yano, Int J .数值模拟学报，32(5)，933-965]，用于解决偏微分方程(PDEs)系统稳态变分数据同化(DA)问题。我们关注的是基于局部测量，在一个特定组件中重构系统状态的问题。PBDW方法不需要在预测阶段求解任何PDE模型(按数据投影)，因此，只要可以为估计构建良好的背景和更新空间，就可以对单个组件进行局部状态估计。我们讨论了PBDW在一类参数化几何的二维稳态Navier-Stokes问题中的应用，并研究了在背景空间和更新空间上施加无滑移边界条件和不可压缩约束以增强状态估计的效果。此外，我们还展示了一种可操作的策略来为背景空间训练局部降阶基(ROBs)，这些背景空间可以稍后用于数据处理任务。
{"title":"A component-based data assimilation strategy with applications to vascular flows","authors":"D. Bui, P. Mollo, F. Nobile, T. Taddei","doi":"10.1051/proc/202373089","DOIUrl":"https://doi.org/10.1051/proc/202373089","url":null,"abstract":"We present a parameterized-background data-weak (PBDW) approach [Y Maday, AT Patera, JD Penn, M Yano, Int J Numer Meth Eng, 102(5), 933–965] to the steady-state variational data assimilation (DA) problem for systems modeled by partial differential equations (PDEs) and characterized by multiple interconnected components, with emphasis on vascular flows. We focus on the problem of reconstructing the state of the system in one specific component, based on local measurements. The PBDW approach does not require the solution of any PDE model at prediction stage (projection-by-data) and, as such, enables local state estimates on single components, as long as good background and update spaces for the estimation can be constructed. We discuss the application of PBDW to a two-dimensional steady Navier-Stokes problem for a family of parameterized geometries, and investigate instead the effects of enforcing no-slip boundary conditions and incompressibility constraints on the background and update spaces to enhance the state estimation. Furthermore, we show an actionable strategy to train local reduced-order bases (ROBs) for the background space that can later be used for DA tasks.","PeriodicalId":53260,"journal":{"name":"ESAIM Proceedings and Surveys","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90292542","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This note is concerned with the rigorous justification of the so-called hard congestion limit from a compressible system with singular pressure towards a mixed compressible-incompressible system modeling partially congested dynamics, for small data in the framework of BV solutions. We present a first convergence result for perturbations of a reference state represented by a single propagating large interface front, while the study of a more general framework where the reference state is constituted by multiple interface fronts is announced in the conclusion and will be the subject of a forthcoming paper. A key element of the proof is the use of a suitably weighted Glimm functional that allows to obtain precise estimates on the BV norm of the front-tracking approximation.
{"title":"Hard-congestion limit of the p-system in the BV setting","authors":"F. Ancona, Roberta Bianchini, C. Perrin","doi":"10.1051/proc/202372041","DOIUrl":"https://doi.org/10.1051/proc/202372041","url":null,"abstract":"This note is concerned with the rigorous justification of the so-called hard congestion limit from a compressible system with singular pressure towards a mixed compressible-incompressible system modeling partially congested dynamics, for small data in the framework of BV solutions. We present a first convergence result for perturbations of a reference state represented by a single propagating large interface front, while the study of a more general framework where the reference state is constituted by multiple interface fronts is announced in the conclusion and will be the subject of a forthcoming paper. A key element of the proof is the use of a suitably weighted Glimm functional that allows to obtain precise estimates on the BV norm of the front-tracking approximation.","PeriodicalId":53260,"journal":{"name":"ESAIM Proceedings and Surveys","volume":"75 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86398608","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Recent force-fatigue mathematical models in biomechanics [7] allow to predict the muscular force response to functional electrical stimulation (FES) and leads to the optimal control problem of maximizing the force. The stimulations are Dirac pulses and the control parameters are the pulses amplitudes and times of application, the number of pulses is physically limited and the model leads to a sampled data control problem. The aim of this article is to present and compare two methods. The first method is a direct optimization scheme where a further refined numerical discretization is applied on the dynamics. The second method is an indirect scheme: first-order Pontryagin type necessary conditions are derived and used to compute the optimal sampling times.
{"title":"Direct and indirect methods to optimize the muscular force response to a pulse train of electrical stimulation","authors":"T. Bakir, B. Bonnard, L. Bourdin, J. Rouot","doi":"10.1051/proc/202171101","DOIUrl":"https://doi.org/10.1051/proc/202171101","url":null,"abstract":"Recent force-fatigue mathematical models in biomechanics [7] allow to predict the muscular force response to functional electrical stimulation (FES) and leads to the optimal control problem of maximizing the force. The stimulations are Dirac pulses and the control parameters are the pulses amplitudes and times of application, the number of pulses is physically limited and the model leads to a sampled data control problem. The aim of this article is to present and compare two methods. The first method is a direct optimization scheme where a further refined numerical discretization is applied on the dynamics. The second method is an indirect scheme: first-order Pontryagin type necessary conditions are derived and used to compute the optimal sampling times.","PeriodicalId":53260,"journal":{"name":"ESAIM Proceedings and Surveys","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73654958","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the sensitivity, essentially the differentiability, of the so-called “intermediate point” c in the classical mean value theorem $ frac{f(a)-f(b)}{b-a}={f}^{prime}(c)$we provide the expression of its gradient ∇c(d,d), thus giving the asymptotic behavior of c(a, b) when both a and b tend to the same point d. Under appropriate mild conditions on f, this result is “universal” in the sense that it does not depend on the point d or the function f. The key tool to get at this result turns out to be the Legendre-Fenchel transformation for convex functions.
{"title":"Sensitivity of the “intermediate point” in the mean value theorem: an approach via the Legendre-Fenchel transformation","authors":"J. Hiriart-Urruty","doi":"10.1051/proc/202171114","DOIUrl":"https://doi.org/10.1051/proc/202171114","url":null,"abstract":"We study the sensitivity, essentially the differentiability, of the so-called “intermediate point” c in the classical mean value theorem $ frac{f(a)-f(b)}{b-a}={f}^{prime}(c)$we provide the expression of its gradient ∇c(d,d), thus giving the asymptotic behavior of c(a, b) when both a and b tend to the same point d. Under appropriate mild conditions on f, this result is “universal” in the sense that it does not depend on the point d or the function f. The key tool to get at this result turns out to be the Legendre-Fenchel transformation for convex functions.","PeriodicalId":53260,"journal":{"name":"ESAIM Proceedings and Surveys","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74084939","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We provide an overview of a class of iterative convex approximation methods for nonlinear optimization problems with convex-over-nonlinear substructure. These problems are characterized by outer convexities on the one hand, and nonlinear, generally nonconvex, but differentiable functions on the other hand. All methods from this class use only first order derivatives of the nonlinear functions and sequentially solve convex optimization problems. All of them are different generalizations of the classical Gauss-Newton (GN) method. We focus on the smooth constrained case and on three methods to address it: Sequential Convex Programming (SCP), Sequential Convex Quadratic Programming (SCQP), and Sequential Quadratically Constrained Quadratic Programming (SQCQP). While the first two methods were previously known, the last is newly proposed and investigated in this paper. We show under mild assumptions that SCP, SCQP and SQCQP have exactly the same local linear convergence – or divergence – rate. We then discuss the special case in which the solution is fully determined by the active constraints, and show that for this case the KKT conditions are sufficient for local optimality and that SCP, SCQP and SQCQP even converge quadratically. In the context of parameter estimation with symmetric convex loss functions, the possible divergence of the methods can in fact be an advantage that helps them to avoid some undesirable local minima: generalizing existing results, we show that the presented methods converge to a local minimum if and only if this local minimum is stable against a mirroring operation applied to the measurement data of the estimation problem. All results are illustrated by numerical experiments on a tutorial example.
{"title":"Survey of sequential convex programming and generalized Gauss-Newton methods","authors":"F. Messerer, Katrin Baumgärtner, M. Diehl","doi":"10.1051/proc/202171107","DOIUrl":"https://doi.org/10.1051/proc/202171107","url":null,"abstract":"We provide an overview of a class of iterative convex approximation methods for nonlinear optimization problems with convex-over-nonlinear substructure. These problems are characterized by outer convexities on the one hand, and nonlinear, generally nonconvex, but differentiable functions on the other hand. All methods from this class use only first order derivatives of the nonlinear functions and sequentially solve convex optimization problems. All of them are different generalizations of the classical Gauss-Newton (GN) method. We focus on the smooth constrained case and on three methods to address it: Sequential Convex Programming (SCP), Sequential Convex Quadratic Programming (SCQP), and Sequential Quadratically Constrained Quadratic Programming (SQCQP). While the first two methods were previously known, the last is newly proposed and investigated in this paper. We show under mild assumptions that SCP, SCQP and SQCQP have exactly the same local linear convergence – or divergence – rate. We then discuss the special case in which the solution is fully determined by the active constraints, and show that for this case the KKT conditions are sufficient for local optimality and that SCP, SCQP and SQCQP even converge quadratically. In the context of parameter estimation with symmetric convex loss functions, the possible divergence of the methods can in fact be an advantage that helps them to avoid some undesirable local minima: generalizing existing results, we show that the presented methods converge to a local minimum if and only if this local minimum is stable against a mirroring operation applied to the measurement data of the estimation problem. All results are illustrated by numerical experiments on a tutorial example.","PeriodicalId":53260,"journal":{"name":"ESAIM Proceedings and Surveys","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91079575","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we present the optimization procedure for computing the discrete boxconstrained minimax classifier introduced in [1, 2]. Our approach processes discrete or beforehand discretized features. A box-constrained region defines some bounds for each class proportion independently. The box-constrained minimax classifier is obtained from the computation of the least favorable prior which maximizes the minimum empirical risk of error over the box-constrained region. After studying the discrete empirical Bayes risk over the probabilistic simplex, we consider a projected subgradient algorithm which computes the prior maximizing this concave multivariate piecewise affine function over a polyhedral domain. The convergence of our algorithm is established.
{"title":"Box-constrained optimization for minimax supervised learning","authors":"Cyprien Gilet, Susana Barbosa, L. Fillatre","doi":"10.1051/proc/202171109","DOIUrl":"https://doi.org/10.1051/proc/202171109","url":null,"abstract":"In this paper, we present the optimization procedure for computing the discrete boxconstrained minimax classifier introduced in [1, 2]. Our approach processes discrete or beforehand discretized features. A box-constrained region defines some bounds for each class proportion independently. The box-constrained minimax classifier is obtained from the computation of the least favorable prior which maximizes the minimum empirical risk of error over the box-constrained region. After studying the discrete empirical Bayes risk over the probabilistic simplex, we consider a projected subgradient algorithm which computes the prior maximizing this concave multivariate piecewise affine function over a polyhedral domain. The convergence of our algorithm is established.","PeriodicalId":53260,"journal":{"name":"ESAIM Proceedings and Surveys","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90014458","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study two-player one-dimensional discrete Hotelling pure location games assuming that demand f(d) as a function of distance d is constant or strictly decreasing. We show that this game admits a best-response potential. This result holds in particular for f(d) = wd with 0 < w ≤ 1. For this case special attention will be given to the structure of the equilibrium set and a conjecture about the increasingness of best-response correspondences will be made.
{"title":"Discrete hotelling pure location games: potentials and equilibria","authors":"Takuya Iimura, P. Mouche","doi":"10.1051/proc/202171163","DOIUrl":"https://doi.org/10.1051/proc/202171163","url":null,"abstract":"We study two-player one-dimensional discrete Hotelling pure location games assuming that demand f(d) as a function of distance d is constant or strictly decreasing. We show that this game admits a best-response potential. This result holds in particular for f(d) = wd with 0 < w ≤ 1. For this case special attention will be given to the structure of the equilibrium set and a conjecture about the increasingness of best-response correspondences will be made.","PeriodicalId":53260,"journal":{"name":"ESAIM Proceedings and Surveys","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82414449","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work is part of the development of a two-phase multi-objective differentiable optimization method. The first phase is classical: it corresponds to the optimization of a set of primary cost functions, subject to nonlinear equality constraints, and it yields at least one known Pareto-optimal solution xA*. This study focuses on the second phase, which is introduced to permit to reduce another set of cost functions, considered as secondary, by the determination of a continuum of Nash equilibria, {x̅ε} (ε≥ 0), in a way such that: firstly, x̅0=xA* (compatibility), and secondly, for ε sufficiently small, the Pareto-optimality condition of the primary cost functions remains O(ε2), whereas the secondary cost functions are linearly decreasing functions of ε. The theoretical results are recalled and the method is applied numerically to a Super-Sonic Business Jet (SSBJ) sizing problem to optimize the flight performance.
{"title":"Prioritized optimization by Nash games : towards an adaptive multi-objective strategy","authors":"J. Désidéri, R. Duvigneau","doi":"10.1051/proc/202171106","DOIUrl":"https://doi.org/10.1051/proc/202171106","url":null,"abstract":"This work is part of the development of a two-phase multi-objective differentiable optimization method. The first phase is classical: it corresponds to the optimization of a set of primary cost functions, subject to nonlinear equality constraints, and it yields at least one known Pareto-optimal solution xA*. This study focuses on the second phase, which is introduced to permit to reduce another set of cost functions, considered as secondary, by the determination of a continuum of Nash equilibria, {x̅ε} (ε≥ 0), in a way such that: firstly, x̅0=xA* (compatibility), and secondly, for ε sufficiently small, the Pareto-optimality condition of the primary cost functions remains O(ε2), whereas the secondary cost functions are linearly decreasing functions of ε. The theoretical results are recalled and the method is applied numerically to a Super-Sonic Business Jet (SSBJ) sizing problem to optimize the flight performance.","PeriodicalId":53260,"journal":{"name":"ESAIM Proceedings and Surveys","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89845932","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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