Continuous, Semi-discrete, and Fully Discretized Navier-Stokes Equations

The Navier--Stokes equations are commonly used to model and to simulate flow phenomena. We introduce the basic equations and discuss the standard methods for the spatial and temporal discretization. We analyse the semi-discrete equations -- a semi-explicit nonlinear DAE -- in terms of the strangeness index and quantify the numerical difficulties in the fully discrete schemes, that are induced by the strangeness of the system. By analyzing the Kronecker index of the difference-algebraic equations, that represent commonly and successfully used time stepping schemes for the Navier--Stokes equations, we show that those time-integration schemes factually remove the strangeness. The theoretical considerations are backed and illustrated by numerical examples.Comment: 28 pages, 2 figure, code available under DOI: 10.5281/zenodo.998909, https://doi.org/10.5281/zenodo.99890

Minimal Curvature Trajectories: Riemannian Geometry Concepts for Model Reduction in Chemical Kinetics

In dissipative ordinary differential equation systems different time scales cause anisotropic phase volume contraction along solution trajectories. Model reduction methods exploit this for simplifying chemical kinetics via a time scale separation into fast and slow modes. The aim is to approximate the system dynamics with a dimension-reduced model after eliminating the fast modes by enslaving them to the slow ones via computation of a slow attracting manifold. We present a novel method for computing approximations of such manifolds using trajectory-based optimization. We discuss Riemannian geometry concepts as a basis for suitable optimization criteria characterizing trajectories near slow attracting manifolds and thus provide insight into fundamental geometric properties of multiple time scale chemical kinetics. The optimization criteria correspond to a suitable mathematical formulation of "minimal relaxation" of chemical forces along reaction trajectories under given constraints. We present various geometrically motivated criteria and the results of their application to three test case reaction mechanisms serving as examples. We demonstrate that accurate numerical approximations of slow invariant manifolds can be obtained.Comment: 22 pages, 18 figure

Differential-Algebraic Equations

Differential-Algebraic Equations (DAE) are today an independent field of research, which is gaining in importance and becoming of increasing interest for applications and mathematics itself. This workshop has drawn the balance after about 25 years investigations of DAEs and the research aims of the future were intensively discussed

Quantum State Estimation and Tracking for Superconducting Processors Using Machine Learning

Quantum technology has been rapidly growing; in particular, the experiments that have been performed with superconducting qubits and circuit QED have allowed us to explore the light-matter interaction at its most fundamental level. The study of coherent dynamics between two-level systems and resonator modes can provide insight into fundamental aspects of quantum physics, such as how the state of a system evolves while being continuously observed. To study such an evolving quantum system, experimenters need to verify the accuracy of state preparation and control since quantum systems are very fragile and sensitive to environmental disturbance. In this thesis, I look at these continuous monitoring and state estimation problems from a modern point of view. With the help of machine learning techniques, it has become possible to explore regimes that are not accessible with traditional methods: for example, tracking the state of a superconducting transmon qubit continuously with dynamics fast compared with the detector bandwidth. These results open up a new area of quantum state tracking, enabling us to potentially diagnose errors that occur during quantum gates. In addition, I investigate the use of supervised machine learning, in the form of a modified denoising autoencoder, to simultaneously remove experimental noise while encoding one and two-qubit quantum state estimates into a minimum number of nodes within the latent layer of a neural network. I automate the decoding of these latent representations into positive density matrices and compare them to similar estimates obtained via linear inversion and maximum likelihood estimation. Using a superconducting multiqubit chip, I experimentally verify that the neural network estimates the quantum state with greater fidelity than either traditional method. Furthermore, the network can be trained using only product states and still achieve high fidelity for entangled states. This simplification of the training overhead permits the network to aid experimental calibration, such as the diagnosis of multi-qubit crosstalk. As quantum processors increase in size and complexity, I expect automated methods such as those presented in this thesis to become increasingly attractive

Deep Clustering: A Comprehensive Survey

Cluster analysis plays an indispensable role in machine learning and data mining. Learning a good data representation is crucial for clustering algorithms. Recently, deep clustering, which can learn clustering-friendly representations using deep neural networks, has been broadly applied in a wide range of clustering tasks. Existing surveys for deep clustering mainly focus on the single-view fields and the network architectures, ignoring the complex application scenarios of clustering. To address this issue, in this paper we provide a comprehensive survey for deep clustering in views of data sources. With different data sources and initial conditions, we systematically distinguish the clustering methods in terms of methodology, prior knowledge, and architecture. Concretely, deep clustering methods are introduced according to four categories, i.e., traditional single-view deep clustering, semi-supervised deep clustering, deep multi-view clustering, and deep transfer clustering. Finally, we discuss the open challenges and potential future opportunities in different fields of deep clustering

Deep Clustering and Deep Network Compression

The use of deep learning has grown increasingly in recent years, thereby becoming a much-discussed topic across a diverse range of fields, especially in computer vision, text mining, and speech recognition. Deep learning methods have proven to be robust in representation learning and attained extraordinary achievement. Their success is primarily due to the ability of deep learning to discover and automatically learn feature representations by mapping input data into abstract and composite representations in a latent space. Deep learning’s ability to deal with high-level representations from data has inspired us to make use of learned representations, aiming to enhance unsupervised clustering and evaluate the characteristic strength of internal representations to compress and accelerate deep neural networks.Traditional clustering algorithms attain a limited performance as the dimensionality in-creases. Therefore, the ability to extract high-level representations provides beneficial components that can support such clustering algorithms. In this work, we first present DeepCluster, a clustering approach embedded in a deep convolutional auto-encoder. We introduce two clustering methods, namely DCAE-Kmeans and DCAE-GMM. The DeepCluster allows for data points to be grouped into their identical cluster, in the latent space, in a joint-cost function by simultaneously optimizing the clustering objective and the DCAE objective, producing stable representations, which is appropriate for the clustering process. Both qualitative and quantitative evaluations of proposed methods are reported, showing the efficiency of deep clustering on several public datasets in comparison to the previous state-of-the-art methods.Following this, we propose a new version of the DeepCluster model to include varying degrees of discriminative power. This introduces a mechanism which enables the imposition of regularization techniques and the involvement of a supervision component. The key idea of our approach is to distinguish the discriminatory power of numerous structures when searching for a compact structure to form robust clusters. The effectiveness of injecting various levels of discriminatory powers into the learning process is investigated alongside the exploration and analytical study of the discriminatory power obtained through the use of two discriminative attributes: data-driven discriminative attributes with the support of regularization techniques, and supervision discriminative attributes with the support of the supervision component. An evaluation is provided on four different datasets.The use of neural networks in various applications is accompanied by a dramatic increase in computational costs and memory requirements. Making use of the characteristic strength of learned representations, we propose an iterative pruning method that simultaneously identifies the critical neurons and prunes the model during training without involving any pre-training or fine-tuning procedures. We introduce a majority voting technique to compare the activation values among neurons and assign a voting score to evaluate their importance quantitatively. This mechanism effectively reduces model complexity by eliminating the less influential neurons and aims to determine a subset of the whole model that can represent the reference model with much fewer parameters within the training process. Empirically, we demonstrate that our pruning method is robust across various scenarios, including fully-connected networks (FCNs), sparsely-connected networks (SCNs), and Convolutional neural networks (CNNs), using two public datasets.Moreover, we also propose a novel framework to measure the importance of individual hidden units by computing a measure of relevance to identify the most critical filters and prune them to compress and accelerate CNNs. Unlike existing methods, we introduce the use of the activation of feature maps to detect valuable information and the essential semantic parts, with the aim of evaluating the importance of feature maps, inspired by novel neural network interpretability. A majority voting technique based on the degree of alignment between a se-mantic concept and individual hidden unit representations is utilized to evaluate feature maps’ importance quantitatively. We also propose a simple yet effective method to estimate new convolution kernels based on the remaining crucial channels to accomplish effective CNN compression. Experimental results show the effectiveness of our filter selection criteria, which outperforms the state-of-the-art baselines.To conclude, we present a comprehensive, detailed review of time-series data analysis, with emphasis on deep time-series clustering (DTSC), and a founding contribution to the area of applying deep clustering to time-series data by presenting the first case study in the context of movement behavior clustering utilizing the DeepCluster method. The results are promising, showing that the latent space encodes sufficient patterns to facilitate accurate clustering of movement behaviors. Finally, we identify state-of-the-art and present an outlook on this important field of DTSC from five important perspectives

System- and Data-Driven Methods and Algorithms

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This two-volume handbook covers methods as well as applications. This first volume focuses on real-time control theory, data assimilation, real-time visualization, high-dimensional state spaces and interaction of different reduction techniques

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum

SCEE 2008 book of abstracts : the 7th International Conference on Scientific Computing in Electrical Engineering (SCEE 2008), September 28 – October 3, 2008, Helsinki University of Technology, Espoo, Finland

This report contains abstracts of presentations given at the SCEE 2008 conference.reviewe