AD-based perturbation methods for uncertainties and errors

International audienceThe progress of Automatic Differentiation ({\bf AD}) and its impact on perturbation methods is the object of this paper. AD studies show an important activity for developing methods addressing the management of modern CFD kernels, taking into account the language evolution, and intensive parallel computing. The evaluation of a posteriori error analysis and of resulting correctors will be addressed. Recents works in the AD-based contruction of second-derivatives for building reduced-order models based on a Taylor formula will be presented on the test case of a steady compressible flow around an aircraft

A high-performance open-source framework for multiphysics simulation and adjoint-based shape and topology optimization

The first part of this thesis presents the advances made in the Open-Source software SU2, towards transforming it into a high-performance framework for design and optimization of multiphysics problems. Through this work, and in collaboration with other authors, a tenfold performance improvement was achieved for some problems. More importantly, problems that had previously been impossible to solve in SU2, can now be used in numerical optimization with shape or topology variables. Furthermore, it is now exponentially simpler to study new multiphysics applications, and to develop new numerical schemes taking advantage of modern high-performance-computing systems. In the second part of this thesis, these capabilities allowed the application of topology optimiza- tion to medium scale fluid-structure interaction problems, using high-fidelity models (nonlinear elasticity and Reynolds-averaged Navier-Stokes equations), which had not been done before in the literature. This showed that topology optimization can be used to target aerodynamic objectives, by tailoring the interaction between fluid and structure. However, it also made ev- ident the limitations of density-based methods for this type of problem, in particular, reliably converging to discrete solutions. This was overcome with new strategies to both guarantee and accelerate (i.e. reduce the overall computational cost) the convergence to discrete solutions in fluid-structure interaction problems.Open Acces

09061 Abstracts Collection -- Combinatorial Scientific Computing

From 01.02.2009 to 06.02.2009, the Dagstuhl Seminar 09061 ``Combinatorial Scientific Computing \u27\u27 was held in Schloss Dagstuhl -- Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Sequence-to-sequence learning for machine translation and automatic differentiation for machine learning software tools

Cette thèse regroupe des articles d'apprentissage automatique et s'articule autour de deux thématiques complémentaires. D'une part, les trois premiers articles examinent l'application des réseaux de neurones artificiels aux problèmes du traitement automatique du langage naturel (TALN). Le premier article introduit une structure codificatrice-décodificatrice avec des réseaux de neurones récurrents pour traduire des segments de phrases de longueur variable. Le deuxième article analyse la performance de ces modèles de `traduction neuronale automatique' de manière qualitative et quantitative, tout en soulignant les difficultés posées par les phrases longues et les mots rares. Le troisième article s'adresse au traitement des mots rares et hors du vocabulaire commun en combinant des algorithmes de compression par dictionnaire et des réseaux de neurones récurrents. D'autre part, la deuxième partie de cette thèse fait abstraction de modèles particuliers de réseaux de neurones afin d'aborder l'infrastructure logicielle nécessaire à leur définition et entraînement. Les infrastructures modernes d'apprentissage profond doivent avoir la capacité d'exécuter efficacement des programmes d'algèbre linéaire et par tableaux, tout en étant capable de différentiation automatique (DA) pour calculer des dérivées multiples. Le premier article aborde les défis généraux posés par la conciliation de ces deux objectifs et propose la solution d'une représentation intermédiaire fondée sur les graphes. Le deuxième article attaque le même problème d'une manière différente: en implémentant un code source par bande dans un langage de programmation dynamique par tableau (Python et NumPy).This thesis consists of a series of articles that contribute to the field of machine learning. In particular, it covers two distinct and loosely related fields. The first three articles consider the use of neural network models for problems in natural language processing (NLP). The first article introduces the use of an encoder-decoder structure involving recurrent neural networks (RNNs) to translate from and to variable length phrases and sentences. The second article contains a quantitative and qualitative analysis of the performance of these `neural machine translation' models, laying bare the difficulties posed by long sentences and rare words. The third article deals with handling rare and out-of-vocabulary words in neural network models by using dictionary coder compression algorithms and multi-scale RNN models. The second half of this thesis does not deal with specific neural network models, but with the software tools and frameworks that can be used to define and train them. Modern deep learning frameworks need to be able to efficiently execute programs involving linear algebra and array programming, while also being able to employ automatic differentiation (AD) in order to calculate a variety of derivatives. The first article provides an overview of the difficulties posed in reconciling these two objectives, and introduces a graph-based intermediate representation that aims to tackle these difficulties. The second article considers a different approach to the same problem, implementing a tape-based source-code transformation approach to AD on a dynamically typed array programming language (Python and NumPy)

Components as coalgebras

In the tradition of mathematical modelling in physics and chemistry, constructive formal specification methods are based on the notion of a software model, understood as a state-based abstract machine which persists and evolves in time, according to a behavioural model capturing, for example, partiality or (different degrees of) nondeterminism. This can be identified with the more prosaic notion of a software component advocated by the software industry as ‘building block’ of large, often distributed, systems. Such a component typically encapsulates a number of services through a public interface which provides a limited access to a private state space, paying tribute to the nowadays widespread object-oriented programming principles. The tradition of communicating systems formal design, by contrast, has developed the notion of a process as an abstraction of the behavioural patterns of a computing system, deliberately ignoring the data and state aspects of software systems. Both processes and components are among the broad group of computing phenomena which are hardly definable (or simply not definable) algebraically, i.e., in terms of a complete set of constructors. Their semantics is essentially observational, in the sense that all that can be traced of their evolution is their interaction with the environment. Therefore, coalgebras, whose theory has recently witnessed remarkable developments, appear as a suitable modelling tool. The basic observation of category theory that universal constructions always come in pairs, has motivated research on the duality between algebras and coalgebras, which provides a bridge between models of static (constructive, data-oriented) and dynamical (observational, behaviour-oriented) systems. At the programming level, the intuitive symmetry between data and behaviour provides evidence of such a duality, in its canonical initial-final specialisation. This line of thought entails both definitional and proof principles, i.e., a basis for the development of program calculi directly based on (actually driven by) type specifications. Moreover, such properties can be expressed in terms of generic programming combinators which are used, not only to calculate programs, but also to program with. Framed in this context, this thesis addresses the following main themes: The investigation of a semantic model for (state-based) software components. These are regarded as concrete coalgebras for some Set endofunctors, with specified initial conditions, and organise themselves in a bicategorical setting. The model is able to capture both behavioural issues, which are usually left implicit in state-based specification methods, and interaction through structured data, which is usually a minor concern on process calculi. Two basic cases are considered entailing, respectively, a ‘functional’ and an ‘object-oriented’ shape for components. Both cases are parametrized by a model of behaviour, introduced as a strong (usually commutative) monad. The development of corresponding component calculi, also parametric on the behaviour model, which adds to the genericity of the approach. The study of processes and the ‘reconstruction’ of classical (CCS-like) process calculi on top of their representation as inhabitants of (the carriers of) final coalgebras, in an essentially pointfree, calculational style. An overall concern for genericity, in the sense that models and calculi for both components and processes are parametric on the behaviour model and the interaction discipline, respectively. The animation of both processes and components in CHARITY, a functional programming language entirely based on inductive and coinductive categorical data types. In particular this leads to the development of a process calculi interpreter parametric on the interaction discipline.PRAXIS XXI - Projecto LOGCAMP; POO11/IC-PME/II/S -Projecto KARMA; Fundação para a Ciência e Tecnologia; ALGORITMI Research Center

Customizable program analysis via generic abstract interpretation

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1999.Includes bibliographical references (p. 419-420).by Nathaniel D. Osgood.Ph.D

Towards a classification of continuity and on the emergence of generality

This dissertation has for its primary task the investigation, articulation, and comparison of a variety of concepts of continuity, as developed throughout the history of philosophy and a part of mathematics. It also motivates and aims to better understand some of the conceptual and historical connections between characterizations of the continuous, on the one hand, and ideas and commitments about what makes for generality (and universality), on the other. Many thinkers of the past have acknowledged the need for advanced science and philosophy to pass through the “labyrinth of the continuum” and to develop a sufficiently rich and precise model or description of the continuous; but it has been far less widely appreciated how the resulting description informs our ideas and commitments regarding how (and whether) things become general (or how we think about universality). The introduction provides some motivation for the project and gives some overview of the chapters. The first two chapters are devoted to Aristotle, as Aristotle’s Physics is arguably the foundational book on continuity. The first two chapters show that Aristotle\u27s efforts to understand and formulate a rich and demanding concept of the continuous reached across many of his investigations; in particular, these two chapters aim to better situate certain structural similarities and conceptual overlaps between his Posterior Analytics and his Physics, further revealing connections between the structure of demonstration or proof (the subject of logic and the sciences) and the structure of bodies in motion (the subject of physics and study of nature). This chapter also contributes to the larger narrative about continuity, where Aristotle emerges as one of the more articulate and influential early proponents of an account that aligns continuity with closeness or relations of nearness. Chapter 3 is devoted to Duns Scotus and Nicolas Oresme, and more generally, to the Medieval debate surrounding the “latitude of forms” or the “intension and remission of forms,” in which concerted efforts were made to re-focus attention onto the type of continuous motions mostly ignored by the tradition that followed in the wake of Aristotelian physics. In this context, the traditional appropriation of Aristotle’s thoughts on unity, contrariety, genera, forms, quantity and quality, and continuity is challenged in a number of important ways, reclaiming some of the largely overlooked insights of Aristotle into the intimate connections between continua and genera. By realizing certain of Scotus’s ideas concerning the intension and remission of qualities, Oresme initiates a radical transformation in the concept of continuity, and this chapter argues that Oresme’s efforts are best understood as an early attempt at freeing the concept of continuity from its ancient connection to closeness. Chapters 4 and 5 are devoted to unpacking and re-interpreting Spinoza’s powerful theory of what makes for the ‘oneness’ of a body in general and how ‘ones’ can compose to form ever more composite ‘ones’ (all the way up to Nature as a whole). Much of Spinoza reads like an elaboration on Oresme’s new model of continuity; however, the legacy of the Cartesian emphasis on local motion makes it difficult for Spinoza to give up on closeness altogether. Chapter 4 is dedicated to a closer look at some subtleties and arguments surrounding Descartes’ definition of local motion and ‘one body’, and Chapter 5 builds on this to develop Spinoza’s ideas about how the concept of ‘one body’ scales, in which context a number of far-reaching connections between continuity and generality are also unpacked. Chapter 6 leaves the realm of philosophy and is dedicated to the contributions to the continuitygenerality connection from one field of contemporary mathematics: sheaf theory (and, more generally, category theory). The aim of this chapter is to present something like a “tour” of the main philosophical contributions made by the idea of a sheaf to the specification of the concept of continuity (with particular regard for its connections to universality). The concluding chapter steps back and discusses a number of distinct characterizations of continuity in more abstract and synthetic terms, while touching on some of the corresponding representations of generality to which each such model gives rise. This chapter ends with a brief discussion of some of the arguments that have been deployed in the past to claim that continuity (or discreteness) is “better.

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 23rd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2020, which took place in Dublin, Ireland, in April 2020, and was held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020. The 31 regular papers presented in this volume were carefully reviewed and selected from 98 submissions. The papers cover topics such as categorical models and logics; language theory, automata, and games; modal, spatial, and temporal logics; type theory and proof theory; concurrency theory and process calculi; rewriting theory; semantics of programming languages; program analysis, correctness, transformation, and verification; logics of programming; software specification and refinement; models of concurrent, reactive, stochastic, distributed, hybrid, and mobile systems; emerging models of computation; logical aspects of computational complexity; models of software security; and logical foundations of data bases.

Matrix Polynomials and their Lower Rank Approximations

This thesis is a wide ranging work on computing a “lower-rank” approximation of a matrix polynomial using second-order non-linear optimization techniques. Two notions of rank are investigated. The first is the rank as the number of linearly independent rows or columns, which is the classical definition. The other notion considered is the lowest rank of a matrix polynomial when evaluated at a complex number, or the McCoy rank. Together, these two notions of rank allow one to compute a nearby matrix polynomial where the structure of both the left and right kernels is prescribed, along with the structure of both the infinite and finite eigenvalues. The computational theory of the calculus of matrix polynomial valued functions is developed and used in optimization algorithms based on second-order approximations. Special functions studied with a detailed error analysis are the determinant and adjoint of matrix polynomials. The unstructured and structured variants of matrix polynomials are studied in a very general setting in the context of an equality constrained optimization problem. The most general instances of these optimization problems are NP hard to approximate solutions to in a global setting. In most instances we are able to prove that solutions to our optimization problems exist (possibly at infinity) and discuss techniques in conjunction with an implementation to compute local minimizers to the problem. Most of the analysis of these problems is local and done through the Karush-Kuhn-Tucker optimality conditions for constrained optimization problems. We show that most formulations of the problems studied satisfy regularity conditions and admit Lagrange multipliers. Furthermore, we show that under some formulations that the second-order sufficient condition holds for instances of interest of the optimization problems in question. When Lagrange multipliers do not exist, we discuss why, and if it is reasonable to do so, how to regularize the problem. In several instances closed form expressions for the derivatives of matrix polynomial valued functions are derived to assist in analysis of the optimality conditions around a solution. From this analysis it is shown that variants of Newton’s method will have a local rate of convergence that is quadratic with a suitable initial guess for many problems. The implementations are demonstrated on some examples from the literature and several examples are cross-validated with different optimization formulations of the same mathematical problem. We conclude with a special application of the theory developed in this thesis is computing a nearby pair of differential polynomials with a non-trivial greatest common divisor, a non-commutative symbolic-numeric computation problem. We formulate this problem as finding a nearby structured matrix polynomial that is rank deficient in the classical sense