Recursive POD expansion for reaction-diffusion equation

This paper focuses on the low-dimensional representation of multivariate functions. We study a recursive POD representation, based upon the use of the power iterate algorithm to recursively expand the modes retained in the previous step. We obtain general error estimates for the truncated expansion, and prove that the recursive POD representation provides a quasi-optimal approximation in L2 norm. We also prove an exponential rate of convergence, when applied to the solution of the reaction-diffusion partial differential equation. Some relevant numerical experiments show that the recursive POD is computationally more accurate than the Proper Generalized Decomposition for multivariate functions. We also recover the theoretical exponential convergence rate for the solution of the reaction-diffusion equation

Computer Science for Continuous Data:Survey, Vision, Theory, and Practice of a Computer Analysis System

Building on George Boole's work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied "his" machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics

When Deep Learning Meets Polyhedral Theory: A Survey

In the past decade, deep learning became the prevalent methodology for predictive modeling thanks to the remarkable accuracy of deep neural networks in tasks such as computer vision and natural language processing. Meanwhile, the structure of neural networks converged back to simpler representations based on piecewise constant and piecewise linear functions such as the Rectified Linear Unit (ReLU), which became the most commonly used type of activation function in neural networks. That made certain types of network structure \unicode{x2014}such as the typical fully-connected feedforward neural network\unicode{x2014} amenable to analysis through polyhedral theory and to the application of methodologies such as Linear Programming (LP) and Mixed-Integer Linear Programming (MILP) for a variety of purposes. In this paper, we survey the main topics emerging from this fast-paced area of work, which bring a fresh perspective to understanding neural networks in more detail as well as to applying linear optimization techniques to train, verify, and reduce the size of such networks

Gratings: Theory and Numeric Applications

International audienceThe book containes 11 chapters written by an international team of specialist in electromagnetic theory, numerical methods for modelling of light diffraction by periodic structures having one-, two-, or three-dimensional periodicity, and aiming numerous applications in many classical domains like optical engineering, spectroscopy, and optical telecommunications, together with newly born fields such as photonics, plasmonics, photovoltaics, metamaterials studies, cloaking, negative refraction, and super-lensing. Each chapter presents in detail a specific theoretical method aiming to a direct numerical application by university and industrial researchers and engineers

Gratings: Theory and Numeric Applications, Second Revisited Edition

International audienceThe second Edition of the Book contains 13 chapters, written by an international team of specialist in electromagnetic theory, numerical methods for modelling of light diffraction by periodic structures having one-, two-, or three-dimensional periodicity, and aiming numerous applications in many classical domains like optical engineering, spectroscopy, and optical telecommunications, together with newly born fields such as photonics, plasmonics, photovoltaics, metamaterials studies, cloaking, negative refraction, and super-lensing. Each chapter presents in detail a specific theoretical method aiming to a direct numerical application by university and industrial researchers and engineers.In comparison with the First Edition, we have added two more chapters (ch.12 and ch.13), and revised four other chapters (ch.6, ch.7, ch.10, and ch.11

Man-portable power generation devices : product design and supporting algorithms

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2006.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 351-380).A methodology for the optimal design and operation of microfabricated fuel cell systems is proposed and algorithms for relevant optimization problems are developed. The methodology relies on modeling, simulation and optimization at three levels of modeling detail. The first class of optimization problems considered are parametric mixed-integer linear programs and the second class are bilevel programs with nonconvex inner and outer programs; no algorithms exist currently in the open literature for the global solution of either problem in the form considered here. Microfabricated fuel cell systems are a promising alternative to batteries for manportable power generation. These devices are potential consumer products that comprise a more or less complex chemical process, and can therefore be considered chemical products. With current computational possibilities and available algorithms it is impossible to solve for the optimal design and operation in one step since the devices considered involve complex geometries, multiple scales, time-dependence and parametric uncertainty. Therefore, a methodology is presented based on decomposition into three levels of modeling detail, namely system-level models for process synthesis,(cont.) intermediate fidelity models for optimization of sizes and operation, and detailed, computational fluid dynamics models for geometry improvement. Process synthesis, heat integration and layout considerations are addressed through the use of lumped algebraic models, general enough to be independent of detailed design choices, such as reactor configuration and catalyst choice. Through the use of simulation and parametric mixed-integer optimization the most promising process structures along with idealized layouts are selected among thousands of alternatives. At the intermediate fidelity level space-distributed models are used, which allow optimization of unit sizes and operation for a given process structure without the need to specify a detailed geometry. The resulting models involve partial differential-algebraic equations and dynamic optimization is employed as the solution technique. Finally, the use of detailed two- and three-dimensional computational fluid dynamics facilitates geometrical improvements as well as the derivation and validation of modeling assumptions that are employed in the system-level and intermediate fidelity models. Steady-state case studies are presented assuming a constant power demand;(cont.) the methodology can be also applied to transient considerations and the case of variable power demand. Parametric programming provides the solution of an optimization problem, the data of which depend on one or many unknown real-valued parameters, for each possible value of the parameter(s). In this thesis mixed-integer linear programs are considered, i.e., optimization programs with affine functions involving real- and integervalued variables. In the first part the multiparametric cost-vector case is considered, i.e., an arbitrary finite number of parameters is allowed, that influence only the coefficients of the objective function. The extension of a well-known algorithm for the single-parameter case is presented, and the algorithm behavior is illustrated on simple examples with two parameters. The optimality region of a given basis is a polyhedron in the parameter space, and the algorithm relies on progressively constructing these polyhedra and solving mixed-integer linear programs at their vertices. Subsequently, two algorithmic alternatives are developed, one based on the identification of optimality regions, and one on branch-and-bound. In the second part the single-parameter general case is considered,(cont.) i.e., a single parameter is allowed that can simultaneously influence the coefficients of the objective function, the right-hand side of the constraints, and also the coefficients of the matrix. Two algorithms for mixed-integer linear programs are proposed. The first is based on branch-and-bound on the integer variables, solving a parametric linear program at each node, and the second is based on decomposition of the parametric optimization problem into a series of mixed-integer linear and mixed-integer nonlinear optimization problems. For the parametric linear programs an improvement of a literature algorithm for the solution of linear programs based on rational operations is presented and an alternative based on predictor-continuation is proposed. A set of test problems is introduced and numerical results for these test problems are discussed. The algorithms are then applied to case studies from the man-portable power generation. Finally extensions to the nonlinear case are discussed and an example from chemical equilibrium is analyzed. Bilevel programs are hierarchical programs where an outer program is constrained by an embedded inner program.(cont.) Here the co-operative formulation of inequality constrained bilevel programs involving real-valued variables and nonconvex functions in both the inner and outer programs is considered. It is shown that previous literature proposals for the global solution of such programs are not generally valid for nonconvex inner programs and several consequences of nonconvexity in the inner program are identified. Subsequently, a bounding algorithm for the global solution is presented. The algorithm is rigorous and terminates finitely to a solution that satisfies e-optimality in the inner and outer programs. For the lower bounding problem, a relaxed program, containing the constraints of the inner and outer programs augmented by a parametric upper bound on the optimal solution function of the inner program, is solved to global optimality. For the case that the inner program satisfies a constraint qualification, a heuristic for tighter lower bounds is presented based on the KKT necessary conditions of the inner program. The upper bounding problem is based on probing the solution obtained in the lower bounding procedure. Branching and probing are not required for convergence but both have potential advantages.(cont.) Three branching heuristics are described and analyzed. A set of test problems is introduced and numerical results for these test problems and for literature examples are presented.by Alexander Mitsos.Ph.D