682 research outputs found

    On a two-scale phasefield model for topology optimization

    Get PDF
    In this article, we consider a gradient flow stemming from a problem in two-scale topology optimization. We use the phase-field method, where a Ginzburg--Landau term with obstacle potential is added to the cost functional, which contains the usual compliance but also an additional contribution including a local volume constraint in a penalty term. The minimization of such an energy by its gradient-flow is analyzed in this paper. We use an regularization and discretization of the associated state-variable to show the existence of weak solutions to the considered system. Download Document

    Drift-diffusion models for innovative semiconductor devices and their numerical solution

    Get PDF
    We present charge transport models for novel semiconductor devices which may include ionic species as well as their thermodynamically consistent finite volume discretization

    A Copositive Framework for Analysis of Hybrid Ising-Classical Algorithms

    Full text link
    Recent years have seen significant advances in quantum/quantum-inspired technologies capable of approximately searching for the ground state of Ising spin Hamiltonians. The promise of leveraging such technologies to accelerate the solution of difficult optimization problems has spurred an increased interest in exploring methods to integrate Ising problems as part of their solution process, with existing approaches ranging from direct transcription to hybrid quantum-classical approaches rooted in existing optimization algorithms. While it is widely acknowledged that quantum computers should augment classical computers, rather than replace them entirely, comparatively little attention has been directed toward deriving analytical characterizations of their interactions. In this paper, we present a formal analysis of hybrid algorithms in the context of solving mixed-binary quadratic programs (MBQP) via Ising solvers. We show the exactness of a convex copositive reformulation of MBQPs, allowing the resulting reformulation to inherit the straightforward analysis of convex optimization. We propose to solve this reformulation with a hybrid quantum-classical cutting-plane algorithm. Using existing complexity results for convex cutting-plane algorithms, we deduce that the classical portion of this hybrid framework is guaranteed to be polynomial time. This suggests that when applied to NP-hard problems, the complexity of the solution is shifted onto the subroutine handled by the Ising solver

    On Quantum Speedups for Nonconvex Optimization via Quantum Tunneling Walks

    Get PDF
    Classical algorithms are often not effective for solving nonconvex optimization problems where local minima are separated by high barriers. In this paper, we explore possible quantum speedups for nonconvex optimization by leveraging the globalglobal effect of quantum tunneling. Specifically, we introduce a quantum algorithm termed the quantum tunneling walk (QTW) and apply it to nonconvex problems where local minima are approximately global minima. We show that QTW achieves quantum speedup over classical stochastic gradient descents (SGD) when the barriers between different local minima are high but thin and the minima are flat. Based on this observation, we construct a specific double-well landscape, where classical algorithms cannot efficiently hit one target well knowing the other well but QTW can when given proper initial states near the known well. Finally, we corroborate our findings with numerical experiments

    Theoretical Perspectives on Deep Learning Methods in Inverse Problems

    Get PDF
    In recent years, there have been significant advances in the use of deep learning methods in inverse problems such as denoising, compressive sensing, inpainting, and super-resolution. While this line of works has predominantly been driven by practical algorithms and experiments, it has also given rise to a variety of intriguing theoretical problems. In this paper, we survey some of the prominent theoretical developments in this line of works, focusing in particular on generative priors, untrained neural network priors, and unfolding algorithms. In addition to summarizing existing results in these topics, we highlight several ongoing challenges and open problems

    Adaptive Automated Machine Learning

    Get PDF
    The ever-growing demand for machine learning has led to the development of automated machine learning (AutoML) systems that can be used off the shelf by non-experts. Further, the demand for ML applications with high predictive performance exceeds the number of machine learning experts and makes the development of AutoML systems necessary. Automated Machine Learning tackles the problem of finding machine learning models with high predictive performance. Existing approaches incorporating deep learning techniques assume that all data is available at the beginning of the training process (offline learning). They configure and optimise a pipeline of preprocessing, feature engineering, and model selection by choosing suitable hyperparameters in each model pipeline step. Furthermore, they assume that the user is fully aware of the choice and, thus, the consequences of the underlying metric (such as precision, recall, or F1-measure). By variation of this metric, the search for suitable configurations and thus the adaptation of algorithms can be tailored to the user’s needs. With the creation of a vast amount of data from all kinds of sources every day, our capability to process and understand these data sets in a single batch is no longer viable. By training machine learning models incrementally (i.ex. online learning), the flood of data can be processed sequentially within data streams. However, if one assumes an online learning scenario, where an AutoML instance executes on evolving data streams, the question of the best model and its configuration remains open. In this work, we address the adaptation of AutoML in an offline learning scenario toward a certain utility an end-user might pursue as well as the adaptation of AutoML towards evolving data streams in an online learning scenario with three main contributions: 1. We propose a System that allows the adaptation of AutoML and the search for neural architectures towards a particular utility an end-user might pursue. 2. We introduce an online deep learning framework that fosters the research of deep learning models under the online learning assumption and enables the automated search for neural architectures. 3. We introduce an online AutoML framework that allows the incremental adaptation of ML models. We evaluate the contributions individually, in accordance with predefined requirements and to state-of-the- art evaluation setups. The outcomes lead us to conclude that (i) AutoML, as well as systems for neural architecture search, can be steered towards individual utilities by learning a designated ranking model from pairwise preferences and using the latter as the target function for the offline learning scenario; (ii) architectual small neural networks are in general suitable assuming an online learning scenario; (iii) the configuration of machine learning pipelines can be automatically be adapted to ever-evolving data streams and lead to better performances

    Lattice Boltzmann Methods for Partial Differential Equations

    Get PDF
    Lattice Boltzmann methods provide a robust and highly scalable numerical technique in modern computational fluid dynamics. Besides the discretization procedure, the relaxation principles form the basis of any lattice Boltzmann scheme and render the method a bottom-up approach, which obstructs its development for approximating broad classes of partial differential equations. This work introduces a novel coherent mathematical path to jointly approach the topics of constructability, stability, and limit consistency for lattice Boltzmann methods. A new constructive ansatz for lattice Boltzmann equations is introduced, which highlights the concept of relaxation in a top-down procedure starting at the targeted partial differential equation. Modular convergence proofs are used at each step to identify the key ingredients of relaxation frequencies, equilibria, and moment bases in the ansatz, which determine linear and nonlinear stability as well as consistency orders of relaxation and space-time discretization. For the latter, conventional techniques are employed and extended to determine the impact of the kinetic limit at the very foundation of lattice Boltzmann methods. To computationally analyze nonlinear stability, extensive numerical tests are enabled by combining the intrinsic parallelizability of lattice Boltzmann methods with the platform-agnostic and scalable open-source framework OpenLB. Through upscaling the number and quality of computations, large variations in the parameter spaces of classical benchmark problems are considered for the exploratory indication of methodological insights. Finally, the introduced mathematical and computational techniques are applied for the proposal and analysis of new lattice Boltzmann methods. Based on stabilized relaxation, limit consistent discretizations, and consistent temporal filters, novel numerical schemes are developed for approximating initial value problems and initial boundary value problems as well as coupled systems thereof. In particular, lattice Boltzmann methods are proposed and analyzed for temporal large eddy simulation, for simulating homogenized nonstationary fluid flow through porous media, for binary fluid flow simulations with higher order free energy models, and for the combination with Monte Carlo sampling to approximate statistical solutions of the incompressible Euler equations in three dimensions

    Integrality and cutting planes in semidefinite programming approaches for combinatorial optimization

    Get PDF
    Many real-life decision problems are discrete in nature. To solve such problems as mathematical optimization problems, integrality constraints are commonly incorporated in the model to reflect the choice of finitely many alternatives. At the same time, it is known that semidefinite programming is very suitable for obtaining strong relaxations of combinatorial optimization problems. In this dissertation, we study the interplay between semidefinite programming and integrality, where a special focus is put on the use of cutting-plane methods. Although the notions of integrality and cutting planes are well-studied in linear programming, integer semidefinite programs (ISDPs) are considered only recently. We show that manycombinatorial optimization problems can be modeled as ISDPs. Several theoretical concepts, such as the Chvátal-Gomory closure, total dual integrality and integer Lagrangian duality, are studied for the case of integer semidefinite programming. On the practical side, we introduce an improved branch-and-cut approach for ISDPs and a cutting-plane augmented Lagrangian method for solving semidefinite programs with a large number of cutting planes. Throughout the thesis, we apply our results to a wide range of combinatorial optimization problems, among which the quadratic cycle cover problem, the quadratic traveling salesman problem and the graph partition problem. Our approaches lead to novel, strong and efficient solution strategies for these problems, with the potential to be extended to other problem classes
    • …
    corecore