Microstructure modeling and crystal plasticity parameter identification for predicting the cyclic mechanical behavior of polycrystalline metals

Computational homogenization permits to capture the influence of the microstructure on the cyclic mechanical behavior of polycrystalline metals. In this work we investigate methods to compute Laguerre tessellations as computational cells of polycrystalline microstructures, propose a new method to assign crystallographic orientations to the Laguerre cells and use Bayesian optimization to find suitable parameters for the underlying micromechanical model from macroscopic experiments

A review of nonlinear FFT-based computational homogenization methods

Since their inception, computational homogenization methods based on the fast Fourier transform (FFT) have grown in popularity, establishing themselves as a powerful tool applicable to complex, digitized microstructures. At the same time, the understanding of the underlying principles has grown, in terms of both discretization schemes and solution methods, leading to improvements of the original approach and extending the applications. This article provides a condensed overview of results scattered throughout the literature and guides the reader to the current state of the art in nonlinear computational homogenization methods using the fast Fourier transform

Electromagnetic scattering from thin tubular objects and an application in electromagnetic chirality

Asymptotic perturbation formulas characterize the effective behavior of waves as the volume of the scattering object tends to zero. In this work, wave propagation is described by time-harmonic Maxwell\u27s equations in free space and the corresponding scattering objects are thin tubular objects that feature a different electric permittivity and a different magnetic permeability than their surrounding medium. For this setting, we derive an asymptotic representation of the scattered electric field away from the thin tubular object and use the corresponding leading order term in a shape identification problem and for designing highly electromagnetically chiral objects. In inverse problems, the leading order term may be used to find the center curve of a thin wire that is supposed to emit a scattered field, which is reasonably close to a given measured field. For the optimal design of electromagnetically chiral structures, the representation formula provides an explicit formula for the leading order term of an asymptotic far field operator expansion. A chirality measure, usually requiring the far field operator, will now map aforementioned leading order term to a value between $0$ and $1$ dependent on the level of electromagnetic chirality of the thin tubular scatterer. This approximation greatly simplifies the challenge to maximize the chirality measure with respect to thin tubular objects. The fact that neither the evaluation of the leading order term nor the calculation of corresponding derivatives require a Maxwell system to be solved implies that the shape optimization scheme is highly efficient compared to shape optimization algorithms that use e.g. domain derivatives. In the visible range, the metallic nanowires obtained by our optimization scheme attain high values of electromagnetic chirality and even exceed those attained by traditional metallic helices

Quantum Neural Networks with Qutrits

Οι κβαντικοί υπολογιστές, εκμεταλλευόμενοι τις αρχές της κβαντικής μηχανικής, έχουν τη δυνατότητα να μεταμορφώσουν πολλούς τεχνολογικούς τομείς, χρησιμοποιώντας κβαντικά bit (qubits) που μπορούν να υπάρχουν σε υπέρθεση και εναγκαλισμό, επιτρέποντας, μεταξύ άλλων δυνατοτήτων, την παράλληλη αναζήτηση λύσεων. Πρόσφατες εξελίξεις στο κβαντικό υλικό επέτρεψαν την υλοποίηση πολυδιάστατων κβαντικών καταστάσεων σε νέες πλατφόρμες μικροκυκλωμάτων, προτείνοντας μια ακόμη ενδιαφέρουσα προσέγγιση. Η χρήση qudits, κβαντικών συστημάτων με υψηλότερες διάστασεις, προσφέρει αυξημένο χώρο για αναπαράστη πληροφορίας, αλλά επίσης πειραματικές υλοποιήσεις έχουν επιδείξει ανθεκτικότητα έναντι θορύβου και σφαλμάτων. Αυτό επισημαίνει περαιτέρω την θέση τους στο μέλλον του κβαντικού υπολογισμού. Σε αυτήν τη πτυχιακή, εξετάζεται η δυνατότητα των qutrits για την επίλυση προβλημάτων μηχανικής μάθησης σε κβαντικό υπολογιστή. Ο επεκταμένος χώρος καταστάσεων που προσφέρουν τα qutrits επιτρέπει πλουσιότερη αναπαράσταση δεδομένων. Για το σκοπό αυτό, χρησιμοποιώντας το μαθηματικό πλαίσιο του SU(3), εισάγεται η χρήση των πινάκων Gell-Mann για την κωδικοποίηση σε έναν 8-διάστατο χώρο. Αυτό εξοπλίζει τα συστήματα κβαντικού υπολογισμού με τη δυνατότητα επεξεργασίας και αναπαράστασης περισσότερων δεδομένων σε ένα μόνο qutrit. Η έρευνα επικεντρώνεται σε προβλήματα ταξινόμησης χρησιμοποιώντας qutrits, όπου διεξάγεται μια συγκριτική ανάλυση μεταξύ του προτεινόμενου χάρτη χαρακτηριστικών Gell-Mann, κυκλώματων που χρησιμοποιούν qubits και μοντέλων κλασσικής μηχανικής μάθησης. Επιπλέον, εξερευνούνται τεχνικές βελτιστοποίησης σε χώρους Hilbert υψηλών διαστάσεων, με σκοπό την αντιμετώπιση προκλήσεων, όπως τα vanishing gradients και το πρόβλημα των barren plateaus. Τέλος, καλύπτονται πρόσφατες εξελίξεις στον κβαντικό υλικό, με ειδική έμφαση σε συστήματα βασισμένα σε qutrits. Ο κύριος στόχος αυτής της πτυχιακής εργασίας είναι να εξετάσει τη δυνατότητα κωδικοποίησης Gell-Mann για προβλήματα ταξινόμησης, να αποδείξει την εφικτότητα της επέκτασης των χώρων Hilbert για εργασίες μηχανικής μάθησης και να ορίσει μια αξιόπιστη βάση για εργασία με γεωμετρικούς χάρτες χαρακτηριστικών. Αναλύωντας τις σχεδιαστικές επιλογές και πειραματικές διατάξεις λεπτομερώς, αυτή η έρευνα στοχεύει να συμβάλει στην ευρύτερη κατανόηση των δυνατοτήτων και των περιορισμών των συστημάτων με qutrits στο πλαίσιο της κβαντικής μηχανικής μάθησης, συνεισφέροντας στην πρόοδο του κβαντικού υπολογισμού και των εφαρμογών του σε πρακτικούς τομείς.Quantum computers, leveraging the principles of quantum physics, have the potential to revolutionize various domains by utilizing quantum bits (qubits) that can exist in superpositions and entanglement, allowing for parallel exploration of solutions. Recent advancements in quantum hardware have enabled the realization of high-dimensional quantum states on a chip-scale platform, proposing another potential avenue. The utilization of qudits, quantum systems with levels exceeding 2, not only offer increased information capacity, but also exhibit improved resilience against noise and errors. Experimental implementations have successfully showcased the potential of high-dimensional quantum systems in efficiently encoding complex quantum circuits, further highlighting their promise for the future of quantum computing. In this thesis, the potential of qutrits is explored to enhance machine learning tasks in quantum computing. The expanded state space offered by qutrits enables richer data representation, capturing intricate patterns and relationships. To this end, employing the mathematical framework of SU(3), the Gell-Mann feature map is introduced to encode information within an 8-dimensional space. This empowers quantum computing systems to process and represent larger amounts of data within a single qutrit. The primary focus of this thesis centers on classification tasks utilizing qutrits, where a comparative analysis is conducted between the proposed Gell-Mann feature map, well-established qubit feature maps, and classical machine learning models. Furthermore, optimization techniques within expanded Hilbert spaces are explored, addressing challenges such as vanishing gradients and barren plateaus landscapes. This work explores foundational concepts and principles in quantum computing and machine learning to ensure a solid understanding of the subject. It also highlights recent advancements in quantum hardware, specifically focusing on qutrit-based systems. The main objective is to explore the feasibility of the Gell-Mann encoding for multiclass classification in the SU(3) space, demonstrate the viability of expanded Hilbert spaces for machine learning tasks, and establish a robust foundation for working with geometric feature maps. By delving into the design considerations and experimental setups in detail, this research aims to contribute to the broader understanding of the capabilities and limitations of qutrit-based systems in the context of quantum machine learning, contributing to the advancement of quantum computing and its applications in practical domains

Introduction to Riemannian Geometry and Geometric Statistics: from basic theory to implementation with Geomstats

International audienceAs data is a predominant resource in applications, Riemannian geometry is a natural framework to model and unify complex nonlinear sources of data.However, the development of computational tools from the basic theory of Riemannian geometry is laborious.The work presented here forms one of the main contributions to the open-source project geomstats, that consists in a Python package providing efficient implementations of the concepts of Riemannian geometry and geometric statistics, both for mathematicians and for applied scientists for whom most of the difficulties are hidden under high-level functions. The goal of this monograph is two-fold. First, we aim at giving a self-contained exposition of the basic concepts of Riemannian geometry, providing illustrations and examples at each step and adopting a computational point of view. The second goal is to demonstrate how these concepts are implemented in Geomstats, explaining the choices that were made and the conventions chosen. The general concepts are exposed and specific examples are detailed along the text.The culmination of this implementation is to be able to perform statistics and machine learning on manifolds, with as few lines of codes as in the wide-spread machine learning tool scikit-learn. We exemplify this with an introduction to geometric statistics

Recent Experiences in Multidisciplinary Analysis and Optimization, part 2

The papers presented at the NASA Symposium on Recent Experiences in Multidisciplinary Analysis and Optimization held at NASA Langley Research Center, Hampton, Virginia, April 24 to 26, 1984 are given. The purposes of the symposium were to exchange information about the status of the application of optimization and the associated analyses in industry or research laboratories to real life problems and to examine the directions of future developments

Machine Learning and Its Application to Reacting Flows

This open access book introduces and explains machine learning (ML) algorithms and techniques developed for statistical inferences on a complex process or system and their applications to simulations of chemically reacting turbulent flows. These two fields, ML and turbulent combustion, have large body of work and knowledge on their own, and this book brings them together and explain the complexities and challenges involved in applying ML techniques to simulate and study reacting flows. This is important as to the world’s total primary energy supply (TPES), since more than 90% of this supply is through combustion technologies and the non-negligible effects of combustion on environment. Although alternative technologies based on renewable energies are coming up, their shares for the TPES is are less than 5% currently and one needs a complete paradigm shift to replace combustion sources. Whether this is practical or not is entirely a different question, and an answer to this question depends on the respondent. However, a pragmatic analysis suggests that the combustion share to TPES is likely to be more than 70% even by 2070. Hence, it will be prudent to take advantage of ML techniques to improve combustion sciences and technologies so that efficient and “greener” combustion systems that are friendlier to the environment can be designed. The book covers the current state of the art in these two topics and outlines the challenges involved, merits and drawbacks of using ML for turbulent combustion simulations including avenues which can be explored to overcome the challenges. The required mathematical equations and backgrounds are discussed with ample references for readers to find further detail if they wish. This book is unique since there is not any book with similar coverage of topics, ranging from big data analysis and machine learning algorithm to their applications for combustion science and system design for energy generation

Machine Learning and Its Application to Reacting Flows

This open access book introduces and explains machine learning (ML) algorithms and techniques developed for statistical inferences on a complex process or system and their applications to simulations of chemically reacting turbulent flows. These two fields, ML and turbulent combustion, have large body of work and knowledge on their own, and this book brings them together and explain the complexities and challenges involved in applying ML techniques to simulate and study reacting flows. This is important as to the world’s total primary energy supply (TPES), since more than 90% of this supply is through combustion technologies and the non-negligible effects of combustion on environment. Although alternative technologies based on renewable energies are coming up, their shares for the TPES is are less than 5% currently and one needs a complete paradigm shift to replace combustion sources. Whether this is practical or not is entirely a different question, and an answer to this question depends on the respondent. However, a pragmatic analysis suggests that the combustion share to TPES is likely to be more than 70% even by 2070. Hence, it will be prudent to take advantage of ML techniques to improve combustion sciences and technologies so that efficient and “greener” combustion systems that are friendlier to the environment can be designed. The book covers the current state of the art in these two topics and outlines the challenges involved, merits and drawbacks of using ML for turbulent combustion simulations including avenues which can be explored to overcome the challenges. The required mathematical equations and backgrounds are discussed with ample references for readers to find further detail if they wish. This book is unique since there is not any book with similar coverage of topics, ranging from big data analysis and machine learning algorithm to their applications for combustion science and system design for energy generation