29 research outputs found
A Parametric Modal Decomposition for Fluid Mechanics Problems
Modal decomposition is widely used to simplify complex
fluid mechanics problems
by splitting the data set of the problem into different modes, which collect the main
effects of the parameters involved in the problem. This data set is collected in a
matrix form. Nevertheless, in case of the presence of more than two parameters
affecting the system, a tensor multilinear modal decomposition should be applied.
In this thesis, a brief development of matrix and tensor decomposition is addressed.
In addition, multilinear modal decomposition is applied to perform a parametric
study of an experimental
flapping wing problem, where a large number of parameters
can influence the system.
Finally, it is obtained that the multilinear modal decomposition is a powerful method
to simplify and extract the most significant features of a system. Since this method
is only an experimental and mathematical method, it allows to extract the most
relevant behaviors of each parameter without the necessity of entering into too much
detail in the field of
fluid mechanics.Ingeniería Aeroespacia
Tensor Decompositions for Signal Processing Applications From Two-way to Multiway Component Analysis
The widespread use of multi-sensor technology and the emergence of big
datasets has highlighted the limitations of standard flat-view matrix models
and the necessity to move towards more versatile data analysis tools. We show
that higher-order tensors (i.e., multiway arrays) enable such a fundamental
paradigm shift towards models that are essentially polynomial and whose
uniqueness, unlike the matrix methods, is guaranteed under verymild and natural
conditions. Benefiting fromthe power ofmultilinear algebra as theirmathematical
backbone, data analysis techniques using tensor decompositions are shown to
have great flexibility in the choice of constraints that match data properties,
and to find more general latent components in the data than matrix-based
methods. A comprehensive introduction to tensor decompositions is provided from
a signal processing perspective, starting from the algebraic foundations, via
basic Canonical Polyadic and Tucker models, through to advanced cause-effect
and multi-view data analysis schemes. We show that tensor decompositions enable
natural generalizations of some commonly used signal processing paradigms, such
as canonical correlation and subspace techniques, signal separation, linear
regression, feature extraction and classification. We also cover computational
aspects, and point out how ideas from compressed sensing and scientific
computing may be used for addressing the otherwise unmanageable storage and
manipulation problems associated with big datasets. The concepts are supported
by illustrative real world case studies illuminating the benefits of the tensor
framework, as efficient and promising tools for modern signal processing, data
analysis and machine learning applications; these benefits also extend to
vector/matrix data through tensorization. Keywords: ICA, NMF, CPD, Tucker
decomposition, HOSVD, tensor networks, Tensor Train
Quasioptimality of maximum-volume cross interpolation of tensors
We consider a cross interpolation of high-dimensional arrays in the tensor train format. We prove that the maximum-volume choice of the interpolation sets provides the quasioptimal interpolation accuracy, that differs from the best possible accuracy by the factor which does not grow exponentially with dimension. For nested interpolation sets we prove the interpolation property and propose greedy cross interpolation algorithms. We justify the theoretical results and measure speed and accuracy of the proposed algorithm with numerical experiments
Dynamic Control Flow in Large-Scale Machine Learning
Many recent machine learning models rely on fine-grained dynamic control flow
for training and inference. In particular, models based on recurrent neural
networks and on reinforcement learning depend on recurrence relations,
data-dependent conditional execution, and other features that call for dynamic
control flow. These applications benefit from the ability to make rapid
control-flow decisions across a set of computing devices in a distributed
system. For performance, scalability, and expressiveness, a machine learning
system must support dynamic control flow in distributed and heterogeneous
environments.
This paper presents a programming model for distributed machine learning that
supports dynamic control flow. We describe the design of the programming model,
and its implementation in TensorFlow, a distributed machine learning system.
Our approach extends the use of dataflow graphs to represent machine learning
models, offering several distinctive features. First, the branches of
conditionals and bodies of loops can be partitioned across many machines to run
on a set of heterogeneous devices, including CPUs, GPUs, and custom ASICs.
Second, programs written in our model support automatic differentiation and
distributed gradient computations, which are necessary for training machine
learning models that use control flow. Third, our choice of non-strict
semantics enables multiple loop iterations to execute in parallel across
machines, and to overlap compute and I/O operations.
We have done our work in the context of TensorFlow, and it has been used
extensively in research and production. We evaluate it using several real-world
applications, and demonstrate its performance and scalability.Comment: Appeared in EuroSys 2018. 14 pages, 16 figure
The Tensor Networks Anthology: Simulation techniques for many-body quantum lattice systems
We present a compendium of numerical simulation techniques, based on tensor
network methods, aiming to address problems of many-body quantum mechanics on a
classical computer. The core setting of this anthology are lattice problems in
low spatial dimension at finite size, a physical scenario where tensor network
methods, both Density Matrix Renormalization Group and beyond, have long proven
to be winning strategies. Here we explore in detail the numerical frameworks
and methods employed to deal with low-dimension physical setups, from a
computational physics perspective. We focus on symmetries and closed-system
simulations in arbitrary boundary conditions, while discussing the numerical
data structures and linear algebra manipulation routines involved, which form
the core libraries of any tensor network code. At a higher level, we put the
spotlight on loop-free network geometries, discussing their advantages, and
presenting in detail algorithms to simulate low-energy equilibrium states.
Accompanied by discussions of data structures, numerical techniques and
performance, this anthology serves as a programmer's companion, as well as a
self-contained introduction and review of the basic and selected advanced
concepts in tensor networks, including examples of their applications.Comment: 115 pages, 56 figure
Computing Large-Scale Matrix and Tensor Decomposition with Structured Factors: A Unified Nonconvex Optimization Perspective
The proposed article aims at offering a comprehensive tutorial for the
computational aspects of structured matrix and tensor factorization. Unlike
existing tutorials that mainly focus on {\it algorithmic procedures} for a
small set of problems, e.g., nonnegativity or sparsity-constrained
factorization, we take a {\it top-down} approach: we start with general
optimization theory (e.g., inexact and accelerated block coordinate descent,
stochastic optimization, and Gauss-Newton methods) that covers a wide range of
factorization problems with diverse constraints and regularization terms of
engineering interest. Then, we go `under the hood' to showcase specific
algorithm design under these introduced principles. We pay a particular
attention to recent algorithmic developments in structured tensor and matrix
factorization (e.g., random sketching and adaptive step size based stochastic
optimization and structure-exploiting second-order algorithms), which are the
state of the art---yet much less touched upon in the literature compared to
{\it block coordinate descent} (BCD)-based methods. We expect that the article
to have an educational values in the field of structured factorization and hope
to stimulate more research in this important and exciting direction.Comment: Final Version; to appear in IEEE Signal Processing Magazine; title
revised to comply with the journal's rul