Low-Thrust Lyapunov to Lyapunov and Halo to Halo with L2L^2-Minimization

In this work, we develop a new method to design energy minimum low-thrust missions (L2-minimization). In the Circular Restricted Three Body Problem, the knowledge of invariant manifolds helps us initialize an indirect method solving a transfer mission between periodic Lyapunov orbits. Indeed, using the PMP, the optimal control problem is solved using Newton-like algorithms finding the zero of a shooting function. To compute a Lyapunov to Lyapunov mission, we first compute an admissible trajectory using a heteroclinic orbit between the two periodic orbits. It is then used to initialize a multiple shooting method in order to release the constraint. We finally optimize the terminal points on the periodic orbits. Moreover, we use continuation methods on position and on thrust, in order to gain robustness. A more general Halo to Halo mission, with different energies, is computed in the last section without heteroclinic orbits but using invariant manifolds to initialize shooting methods with a similar approach

Transferts interplanétaires à faible consommation utilisant les propriétés du problème restreint des trois corps

The first objective of this work is to understand the dynamical properties of the circular restricted three body problem in order to use them to design low consumption missions for spacecrafts with a low thrust engine. A fundamental property is the existence of invariant manifolds associated with periodic orbits around Lagrange points. Following the Interplanetary Transport Network concept, invariant manifolds are very useful to design spacecraft missions because they are gravitational currents. A large part of this work is devoted to designing a numerical method that performs an optimal transfer between invariant manifolds. The cost we want to minimize is the L1-norm of the control which is equivalent to minimizing the consumption of the engines. We also consider the L2-norm of the control which is easier to minimize numerically. The numerical methods are indirect ones coupled with different continuations on the thrust, on the cost, and on the final state, to provide robustness. These methods are based on the application of the Pontryagin Maximum Principal. The algorithms developed in this work allow for the design of real life missions such as missions between the realms of libration points. The basic idea is to initialize a multiple shooting method with an admissible trajectory that contains controlled parts (local transfers) and uncontrolled parts following the natural dynamics (invariant manifolds). The methods developed here are efficient and fast (less than a few minutes to obtain the whole optimal trajectory). Finally, we develop a hybrid method, with both direct and indirect methods, to adjust the position of the matching points on the invariant manifolds for missions with large energy gaps. The gradient of the value function is given by the values of the costates at the matching points and does not require any additional computation. Hence, the implementation of the gradient descent is easy.Le premier objectif de cette thèse est de bien comprendre les propriétés de la dynamique du problème circulaire restreint des trois corps et de les utiliser pour calculer des missions pour satellites pourvus de moteurs à faible poussée. Une propriété fondamentale est l'existence de variétés invariantes associées à des orbites périodiques autour des points de Lagrange. En suivant l'idée de l'Interplanetary Transport Network, la connaissance et le calcul des variétés invariantes, comme courants gravitationnels, sont cruciaux pour le design de missions spatiales. Une grande partie de ce travail de thèse est consacrée au développement de méthodes numériques pour calculer le transfert entre variétés invariantes de façon optimale. Le coût que l'on cherche alors à minimiser est la norme L1 du contrôle car elle est équivalente à minimiser la consommation des moteurs. On considère aussi la norme L2 du contrôle car elle est, numériquement, plus facile à minimiser. Les méthodes numériques que nous utilisons sont des méthodes indirectes rendues plus robustes par des méthodes de continuation sur le coût, sur la poussée, et sur l'état final. La mise en œuvre de ces méthodes repose sur l'application du Principe du Maximum de Pontryagin. Les algorithmes développés dans ce travail permettent de calculer des missions réelles telles que des missions entre des voisinages des points de Lagrange. L'idée principale est d'initialiser un tir multiple avec une trajectoire admissible composée de parties contrôlées (des transferts locaux) et de parties non-contrôlées suivant la dynamique libre (les variétés invariantes). Les méthodes mises au point ici, sont efficaces et rapides puisqu'il suffit de quelques minutes pour obtenir la trajectoire optimale complète. Enfin, on développe une méthode hybride, avec à la fois des méthodes directes et indirectes, qui permettent d'ajuster la positions des points de raccord sur les variétés invariantes pour les missions à grandes variations d'énergie. Le gradient de la fonction valeur est donné par les valeurs des états adjoints aux points de raccord et donc ne nécessite pas de calculs supplémentaire. Ainsi, l'implémentation de algorithme du gradient est aisée

LuaLaTeX pour les non sorciers, deux exemples

International audienc

Convergence analysis of adaptive DIIS algorithms with application to electronic ground state calculations

This paper deals with a general class of algorithms for the solution of fixed-point problems that we refer to as \emph{Anderson--Pulay acceleration}. This family includes the DIIS technique and its variant sometimes called commutator-DIIS, both introduced by Pulay in the 1980s to accelerate the convergence of self-consistent field procedures in quantum chemistry, as well as the related Anderson acceleration which dates back to the 1960s, and the wealth of techniques they have inspired. Such methods aim at accelerating the convergence of any fixed-point iteration method by combining several iterates in order to generate the next one at each step. This extrapolation process is characterised by its \emph{depth}, i.e. the number of previous iterates stored, which is a crucial parameter for the efficiency of the method. It is generally fixed to an empirical value. In the present work, we consider two parameter-driven mechanisms to let the depth vary along the iterations. In the first one, the depth grows until a certain nondegeneracy condition is no longer satisfied; then the stored iterates (save for the last one) are discarded and the method "restarts". In the second one, we adapt the depth continuously by eliminating at each step some of the oldest, less relevant, iterates. In an abstract and general setting, we prove under natural assumptions the local convergence and acceleration of these two adaptive Anderson--Pulay methods, and we show that one can theoretically achieve a superlinear convergence rate with each of them. We then investigate their behaviour in quantum chemistry calculations. These numerical experiments show that both adaptive variants exhibit a faster convergence than a standard fixed-depth scheme, and require on average less computational effort per iteration. This study is complemented by a review of known facts on the DIIS, in particular its link with the Anderson acceleration and some multisecant-type quasi-Newton methods.Comment: Final version to appear in ESAIM:M2A

Transfer Between Invariant Manifolds: From Impulse Transfer to Low-Thrust Transfer

International audienceIn this work, a new robust and fast method is developed to perform transfers that minimize fuel consumption between two invariant manifolds of periodic orbits in the circular restricted three-body problem. The method starts with an impulse transfer between two invariant manifolds to build an optimal control problem. This allows to choose an adequate fixed transfer time. Using the Pontryagin maximum principle, the resolution of the problem is formulated as that of finding the zero of a shooting function (indirect method). The algorithm couples different kinds of continuations (on cost, final state, and thrust) to improve robustness and to initialize the solver. The efficiency of the method is illustrated with numerical examples. Finally, the influence of the transfer time is studied numerically thanks to a continuation on this parameter, and it checks that, when transfer duration goes to zero, the control converges to the impulse transfer that it started with. It shows the robustness of the method and establishes a mathematical link between the two problems

Sulcal morphology in Alzheimer's disease: an effective marker of diagnosis and cognition

Julien Lagarde and Fabian Corlier contributed equally.International audienceMeasuring the morphology of brain sulci has been recently proposed as a novel imaging approach in AD. We aimed to investigate the relevance of such an approach in AD, by exploring its (1) clinical relevance in comparison with traditional imaging methods, (2) relationship with amyloid deposition, (3) association with cognitive functions. Here, 51 patients (n=32 MCI/mild dementia-AD, n=19 moderate/severe dementia-AD) diagnosed according to clinical-biological criteria (CSF biomarkers and amyloid-PET) and 29 controls (with negative amyloid-PET) underwent neuropsychological and 3T-MRI examinations. Mean sulcal width (SW) & mean cortical thickness around the sulcus (CT-S) were automatically measured. We found higher SW and lower CT-S in AD patients than in controls. These differences were more pronounced at later stages of the disease and provided the best diagnostic accuracies among the imaging markers. Correlations were not found between CT-S or SW and amyloid deposition but between specific cognitive functions and regional CT-S/SW in key associated regions. Sulcal morphology is a good supporting diagnosis tool, that reflects the main cognitive impairments in AD. It could be considered as a good surrogate marker to evaluate the efficacy of new drugs

L'imagerie par résonance magnétique pour le diagnostic précoce de la maladie d'Alzheimer

International audienceBackground: Previous studies analyzed the ability of hippocampal volumes (HV) to differentiate Alzheimer's disease (AD) from frontotemporal dementia (FTD). However, these studies did not include patients selected according to clinico-biological criteria, using pathophysiological biomarkers. Objective: To analyze the effectiveness of hippocampal volumetric measures to distinguish AD from behavioral variant FTD (bvFTD), using strict inclusion criteria based on clinical and pathophysiological markers. Methods: Seventy-two participants were included: 31 AD patients with predominant and progressive episodic memory deficits associated with typical AD cerebrospinal fluid (CSF) profile and/or positive amyloid imaging (PET with 11C-labeled Pittsburgh Compound B [PiB]), 26 bvFTD patients diagnosed according to consensual clinical criteria and with no AD CSF profile, and 15 healthy controls without amyloid retention on PiB-PET exam. HV were segmented with an automated method and were normalized to total intracranial volume (nHV). Results: Significant reductions in HV were found in both AD and bvFTD patients compared with controls, but there were no significant difference between AD and bvFTD patients. Mean nHV distinguished normal controls from either AD or bvFTD with high sensitivity (80.6% and 76.9%, respectively) and specificity (93.3% for both), but it was inefficient in differentiating AD from bvFTD (9.7% specificity). There was no difference in the clinical and neuropsychological profiles according to HV in bvFTD and AD patients. Conclusions: When considered alone, measures of HV are not good markers to differentiate AD from bvFTD. Hippocampal sclerosis associated with FTD may explain the high degree of overlap in nHV between both groups

L'imagerie par résonance magnétique pour le diagnostic précoce de la maladie d'Alzheimer

International audienceBackground: Previous studies analyzed the ability of hippocampal volumes (HV) to differentiate Alzheimer's disease (AD) from frontotemporal dementia (FTD). However, these studies did not include patients selected according to clinico-biological criteria, using pathophysiological biomarkers. Objective: To analyze the effectiveness of hippocampal volumetric measures to distinguish AD from behavioral variant FTD (bvFTD), using strict inclusion criteria based on clinical and pathophysiological markers. Methods: Seventy-two participants were included: 31 AD patients with predominant and progressive episodic memory deficits associated with typical AD cerebrospinal fluid (CSF) profile and/or positive amyloid imaging (PET with 11C-labeled Pittsburgh Compound B [PiB]), 26 bvFTD patients diagnosed according to consensual clinical criteria and with no AD CSF profile, and 15 healthy controls without amyloid retention on PiB-PET exam. HV were segmented with an automated method and were normalized to total intracranial volume (nHV). Results: Significant reductions in HV were found in both AD and bvFTD patients compared with controls, but there were no significant difference between AD and bvFTD patients. Mean nHV distinguished normal controls from either AD or bvFTD with high sensitivity (80.6% and 76.9%, respectively) and specificity (93.3% for both), but it was inefficient in differentiating AD from bvFTD (9.7% specificity). There was no difference in the clinical and neuropsychological profiles according to HV in bvFTD and AD patients. Conclusions: When considered alone, measures of HV are not good markers to differentiate AD from bvFTD. Hippocampal sclerosis associated with FTD may explain the high degree of overlap in nHV between both groups

Sulcal morphology as a new imaging marker for the diagnosis of early onset Alzheimer's disease

International audienceWe investigated the utility of sulcal width measures in the diagnosis of Alzheimer's disease (AD). Sixty-six biologically confirmed AD patients (positive amyloid positron emission tomography [PET] and/or AD cerebrospinal fluid profile) were contrasted to 35 controls with negative amyloid PET. Patients were classified into prodromal or dementia stages as well as into late onset (LOAD, n = 31) or early onset (EOAD, n = 35) subgroups according to their age of onset. An automated method was used to calculate sulcal widths and hippocampal volumes (HV). In EOAD, the greatest ability to differentiate patients from age-matched controls, regardless of severity, was displayed by sulcal width of the temporoparietal cortex. In this region, diagnosis accuracy was better than the HV, especially at prodromal stage. In LOAD, HV provided the best discrimination power from age-matched controls. In conclusion, sulcal width measures are better markers than the HV for identifying prodromal AD in patients aged <65 years. In contrast, in older patients, the risk of over-diagnosis from using only sulcal enlargement is important