Search

Deliverable D5.4 in the ROMSOC project

In this work, we address the “freeform optics” inverse problem of designing a reflector surface mapping a prescribed source distribution of light to a prescribed target far-field distribution, for the point light source and the extended light source. When the source is a point source, the light distribution has support only on the optics ray directions. In this setting, the inverse problem is well-posed for arbitrary source and target probability distributions. It can be recast as an optimal transport problem and is a classic example of an optimal transport problem with a non-euclidean displacement cost. We explore the use of entropic Optimal Transport and the associated Sinkhorn algorithm to solve it numerically. As the reflector modeling is based on the Kantorovich potentials, several questions arise. First, on the convergence of the discrete entropic approximation and here we follow the recent work of Berman and in particular the imposed discretization requirements therein. Secondly, the correction of the bias induced by the entropic Optimal Transport using the recent notion of Sinkhorn divergences is shown to be necessary to achieve satisfactory results. For the point source problem, we discuss the necessary mathematical and numerical tools needed to produce and analyze the obtained numerical results. We find that Sinkhorn algorithm may be adapted to the resolution of the point source to far-field reflector problem.We are not aware of any similar mathematical formulation in the extended source case: i.e. the source has an “étendue” with support in the product space: physical domain-ray directions. We propose to leverage the well-posed variational formulation of the point source problem to build a smooth parameterization of the reflector and the map modeling the reflection. Under this parametrization, we can construct a smooth cost function to optimize for the best solution in this class of reflectors. Both steps, the parameterization and the cost function, are related to entropic optimal transport distances. We also take advantage of recent progress in the optimization techniques and the efficient implementations of Sinkhorn algorithm to perform a numerical study.; Dans ce travail, nous abordons un problème inverse en optique anidolique consistant à déterminer une surface capable de réflechir une distribution de lumière source à une distribution cible en champ lointain, toutes deux prescrites. La source lumineuse peut être ponctuelle ou étendue. Lorsque la source est une source ponctuelle , la distribution est supportée uniquement sur les directions des rayons optiques. Dans ce contexte, le problème inverse est bien posé pour des distributions de probabilité source et cible arbitraires. Il peut être reformulé comme un problème de transport optimal et constitue un exemple célébre de transport optimal sous un coût de déplacement non euclidien. Nous explorons l’utilisation du transport optimal entropique et de l’algorithme Sinkhorn associé pour le résoudre numériquement. La modélisation du réflecteur étant basée sur les potentiels de Kantorovich, plusieurs questions se posent. Premièrement, sur la convergence de l’approximation entropique discrète et nous suivons ici les travaux récents de Berman et en particulier les exigences de discrétisation qui y sont imposées. Deuxièmement, nous montrons que la correction du biais induit par le transport entropique Optimal peut être atteinte en utilisant la notion récente de divergences Sinkhorn. Pour le problème de source ponctuelle, nous discutons des outils mathématiques et numériques nécessaires pour produire et analyser les résultats numériques obtenus. Nous trouvons que l’algorithme Sinkhorn peut être adapté à la résolution du problème de la source ponctuelle au réflecteur en champ lointain.Nous ne connaissons pas de formulation mathématique similaire dans le cas de la source étendue : la distribution de lumière source a support sur l’espace produit: domaine physique-directions des rayons. Nous proposons de tirer parti de la formulation variationnelle bien posée du problème de source ponctuelle pour construire une paramétrisation lisse du réflecteur et de l’application modélisant la réflexion. Sous cette paramétrisation, nous pouvons construire une fonction de coût lisse à optimiser pour trouver la meilleure solution dans cette classe de réflecteurs. Les deux étapes, la paramétrisation et la fonction de coût, sont liées à des distances de transport entropiques optimales. Nous profitons également des progrès récents des techniques d’optimisation et des implémentations efficaces de l’algorithme Sinkhorn pour réaliser une étude numérique.

International audience; We address the “freeform optics” inverse problem of designing a reflector surface mapping a prescribed source distribution of light to a prescribed far field distribution, for a finite light source. When the finite source reduces to a point source, the light source distribution has support only on the optics ray directions. In this setting the inverse problem is well posed for arbitrary source and target probability distributions. It can be recast as an Optimal Transportation problem and has been studied both mathematically and nu-merically. We are not aware of any similar mathematical formulation in the finite source case: i.e. the source has an “´etendue” with support both in space and directions. We propose to leverage the well-posed variational formulation of the point source problem to build a smooth parameterization of the reflec-tor and the reflection map. Under this parameterization we can construct a smooth loss/misfit function to optimize for the best solution in this class of reflectors. Both steps, the parameterization and the loss, are related to Optimal Transportation distances. We also take advantage of recent progress in the numerical approximation and resolution of these mathematical objects to perform a numerical study.

Project Deliverable D4.2

Based on the multitude of industrial applications, benchmarks for model hierarchies will be created that will form a basis for the interdisciplinary research and for the training programme. These will be equipped with publically available data and will be used for training in modelling, model testing, reduced order modelling, error estimation, efficiency optimization in algorithmic approaches, and testing of the generated MSO/MOR software. The present document includes the description about the selection of (at least) eight benchmark cases of model hierarchies. The present document has been structured in three main parts to distinguish those contributions which are focused on coupling methods, model order reduction methods, and optimization methods., Project Deliverable (D5.1), Version 3.0

Based on the multitude of industrial applications, benchmarks for model hierarchies will be created that will form a basis for the interdisciplinary research and for the training programme. These will be equipped with publically available data and will be used for training in modelling, model testing, reduced order modelling, error estimation, efficiency optimization in algorithmic approaches, and testing of the generated MSO/MOR software. The present document includes a detailed description of the computer implementation of these benchmarks involving not only the required publically available data but also the used software packages, libraries and any other relevant information, which guarantee a fully reproducibility of the reported numerical results. The present document has been structured in three main parts to distinguish those contributions which are focused on coupling methods, model order reduction methods, and optimization methods., Project Deliverable (D5.2), Version 2.0

In the ROMSOC project, we develop specially-tailored algorithmic approaches to solve the practical problems of our industry partners. The mathematical fields involved in solving these problems are as diverse as the problems themselves. Thus, our project covers a very broad range within applied mathematics: from discrete optimization to inverse problems, from deep learning to optimal control. What all these fields have in common is that error estimation is an important topic in order to quantify the quality of a computed solution. Naturally, all of them their own error measures and different ways of computing and using them. This report summarizes the state of the art methods in the respective fields concerning error measures and their usefulness in solving real-world problems.

Based on the multitude of industrial applications, benchmarks for model hierarchies will be created that will form a basis for the interdisciplinary research and for the training programme. These will be equipped with publically available data and will be used for training in modelling, model testing, reduced order modelling, error estimation, efficiency optimization in algorithmic approaches, and testing of the generated MSO/MOR software. The present document includes a detailed description of the computer implementation of these benchmarks involving not only the required publically available data but also the used software packages, libraries and any other relevant information, which guarantee a fully reproducibility of the reported numerical results.

Based on the multitude of industrial applications, benchmarks for model hierarchies will be created that will form a basis for the interdisciplinary research and for the training programme. These will be equipped with publically available data and will be used for training in modelling, model testing, reduced order modelling, error estimation, efficiency optimization in algorithmic approaches, and testing of the generated MSO/MOR software. The present document includes the description about the selection of (at least) eight benchmark cases of model hierarchies.