37 research outputs found
Interpolation Based Parametric Model Order Reduction
In this thesis, we consider model order reduction of parameter-dependent large-scale dynamical systems. The objective is to develop a methodology to reduce the order of the model and simultaneously preserve the dependence of the model on parameters. We use the balanced truncation method together with spline interpolation to solve the problem. The core of this method is to interpolate the reduced transfer function, based on the pre-computed transfer function at a sample in the parameter domain. Linear splines and cubic splines are employed here. The use of the latter, as expected, reduces the error of the method. The combination is proven to inherit the advantages of balanced truncation such as stability preservation and, based on a novel bound for the infinity norm of the matrix inverse, the derivation of error bounds. Model order reduction can be formulated in the projection framework. In the case of a parameter-dependent system, the projection subspace also depends on parameters. One cannot compute this parameter-dependent projection subspace, but has to approximate it by interpolation based on a set of pre-computed subspaces. It turns out that this is the problem of interpolation on Grassmann manifolds. The interpolation process is actually performed on tangent spaces to the underlying manifold. To do that, one has to invoke the exponential and logarithmic mappings which involve some singular value decompositions. The whole procedure is then divided into the offline and online stage. The computation time in the online stage is a crucial point. By investigating the formulation of exponential and logarithmic mappings and analyzing the structure of sums of singular value decompositions, we succeed to reduce the computational complexity of the online stage and therefore enable the use of this algorithm in real time
Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives
Part 2 of this monograph builds on the introduction to tensor networks and
their operations presented in Part 1. It focuses on tensor network models for
super-compressed higher-order representation of data/parameters and related
cost functions, while providing an outline of their applications in machine
learning and data analytics. A particular emphasis is on the tensor train (TT)
and Hierarchical Tucker (HT) decompositions, and their physically meaningful
interpretations which reflect the scalability of the tensor network approach.
Through a graphical approach, we also elucidate how, by virtue of the
underlying low-rank tensor approximations and sophisticated contractions of
core tensors, tensor networks have the ability to perform distributed
computations on otherwise prohibitively large volumes of data/parameters,
thereby alleviating or even eliminating the curse of dimensionality. The
usefulness of this concept is illustrated over a number of applied areas,
including generalized regression and classification (support tensor machines,
canonical correlation analysis, higher order partial least squares),
generalized eigenvalue decomposition, Riemannian optimization, and in the
optimization of deep neural networks. Part 1 and Part 2 of this work can be
used either as stand-alone separate texts, or indeed as a conjoint
comprehensive review of the exciting field of low-rank tensor networks and
tensor decompositions.Comment: 232 page
Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives
Part 2 of this monograph builds on the introduction to tensor networks and
their operations presented in Part 1. It focuses on tensor network models for
super-compressed higher-order representation of data/parameters and related
cost functions, while providing an outline of their applications in machine
learning and data analytics. A particular emphasis is on the tensor train (TT)
and Hierarchical Tucker (HT) decompositions, and their physically meaningful
interpretations which reflect the scalability of the tensor network approach.
Through a graphical approach, we also elucidate how, by virtue of the
underlying low-rank tensor approximations and sophisticated contractions of
core tensors, tensor networks have the ability to perform distributed
computations on otherwise prohibitively large volumes of data/parameters,
thereby alleviating or even eliminating the curse of dimensionality. The
usefulness of this concept is illustrated over a number of applied areas,
including generalized regression and classification (support tensor machines,
canonical correlation analysis, higher order partial least squares),
generalized eigenvalue decomposition, Riemannian optimization, and in the
optimization of deep neural networks. Part 1 and Part 2 of this work can be
used either as stand-alone separate texts, or indeed as a conjoint
comprehensive review of the exciting field of low-rank tensor networks and
tensor decompositions.Comment: 232 page
Recommended from our members
Widening the basin of convergence for the bundle adjustment type of problems in computer vision
Bundle adjustment is the process of simultaneously optimizing camera poses and 3D structure
given image point tracks. In structure-from-motion, it is typically used as the final refinement
step due to the nonlinearity of the problem, meaning that it requires sufficiently good
initialization. Contrary to this belief, recent literature showed that useful solutions can
be obtained even from arbitrary initialization for fixed-rank matrix factorization problems,
including bundle adjustment with affine cameras. This property of wide convergence basin of
high quality optima is desirable for any nonlinear optimization algorithm since obtaining good
initial values can often be non-trivial. The aim of this thesis is to find the key factor behind the
success of these recent matrix factorization algorithms and explore the potential applicability
of the findings to bundle adjustment, which is closely related to matrix factorization.
The thesis begins by unifying a handful of matrix factorization algorithms and comparing
similarities and differences between them. The theoretical analysis shows that the set
of successful algorithms actually stems from the same root of the optimization method
called variable projection (VarPro). The investigation then extends to address why VarPro
outperforms the joint optimization technique, which is widely used in computer vision. This
algorithmic comparison of these methods yields a larger unification, leading to a conclusion
that VarPro benefits from an unequal trust region assumption between two matrix factors.
The thesis then explores ways to incorporate VarPro to bundle adjustment problems
using projective and perspective cameras. Unfortunately, the added nonlinearity causes
a substantial decrease in the convergence basin of VarPro, and therefore a bootstrapping
strategy is proposed to bypass this issue. Experimental results show that it is possible to
yield feasible metric reconstructions and pose estimations from arbitrary initialization given
relatively clean point tracks, taking one step towards initialization-free structure-from-motion.Microsoft
Toshiba Research Europ
Flow control and sensing using data-driven reduced-order modeling
Transfer operators, such as the Koopman operator, are driving a paradigm shift in how we perform data-driven modeling of fluid flows. Approximations of the Koopman operator provide linear representations even for strongly nonlinear flows, which enables the application of standard linear methods for estimation and control under realistic flow conditions. In the past decade, we have witnessed several breakthroughs in obtaining low-dimensional approximations of the Koopman operators, providing a tractable reduced-order model for complex fluid flows using data from numerical simulations or experiments. In this thesis, we leverage these recent developments in operator-theoretic modeling of fluid flows and provide data-driven solutions to the flow control and sensing problems. The contributions of this thesis can be divided into three parts.
In the first part, we introduce a novel method, low-rank Dynamic Mode Decomposition (lrDMD), for data-driven reduced-order modeling of fluid flows. While existing data-driven modeling methods fit an endomorphic linear function on a low-dimensional subspace, lrDMD approximates flow dynamics using a linear map between different subspaces. We show that this approach leads to the design of better reduced-order feedback controllers. We formulate a rank-constrained matrix optimization problem and propose two complementary methods to solve the problem. lrDMD outperforms existing methods in feedback control and optimal actuator placement.
In the second part, we present a completely data-driven framework for sensor placement in fluid flows. This framework can be applied in conjunction with any reduced-order modeling technique that constructs a linear model for the flow dynamics. We formulate an optimization problem that minimizes the trace of a data-driven approximation of the estimation error covariance matrix, where the estimates are provided by a Kalman filter. We propose an efficient adjoint-based gradient descent method to solve the optimization problem. We show that sensors placed using our method lead to better performance in important applications, such as flow estimation and control, compared to existing data-driven sensor placement methods.
In the third and final part, we propose a new method of interface tracking and reconstruction in multiphase flows using shadowgraphs or back-lit imaging data. First, we show that while traditional modeling methods provide valuable information about the spatio-temporal structure of flow instabilities, they are not able to resolve spatial or temporal discontinuities, such as the liquid-gas interface, in the data. To remedy this, we propose a two-step approach, using data-driven modeling techniques in conjunction with optical flow methods, that preserves sharp interfaces and provides reliable reconstruction and short-time prediction of the flow. We apply our method to an experimental liquid jet with a co-axial air-blast atomizer using back-lit imaging. Our method is able to accurately reconstruct and predict the flow while preserving the sharpness of the liquid-gas interface
Machine Learning, Low-Rank Approximations and Reduced Order Modeling in Computational Mechanics
The use of machine learning in mechanics is booming. Algorithms inspired by developments in the field of artificial intelligence today cover increasingly varied fields of application. This book illustrates recent results on coupling machine learning with computational mechanics, particularly for the construction of surrogate models or reduced order models. The articles contained in this compilation were presented at the EUROMECH Colloquium 597, « Reduced Order Modeling in Mechanics of Materials », held in Bad Herrenalb, Germany, from August 28th to August 31th 2018. In this book, Artificial Neural Networks are coupled to physics-based models. The tensor format of simulation data is exploited in surrogate models or for data pruning. Various reduced order models are proposed via machine learning strategies applied to simulation data. Since reduced order models have specific approximation errors, error estimators are also proposed in this book. The proposed numerical examples are very close to engineering problems. The reader would find this book to be a useful reference in identifying progress in machine learning and reduced order modeling for computational mechanics
Numerical and variational aspects of mesh parameterization and editing
A surface parameterization is a smooth one-to-one mapping between the surface and a parametric domain. Typically, surfaces with disk topology are mapped onto the plane and genus-zero surfaces onto the sphere. As any attempt to flatten a non-trivial surface onto the plane will inevitably induce a certain amount of distortion, the main concern of research on this topic is to minimize parametric distortion. This thesis aims at presenting a balanced blend of mathematical rigor and engineering intuition to address the challenges raised by the mesh parameterization problem. We study the numerical aspects of mesh parameterization in the light of parallel developments in both mathematics and engineering. Furthermore, we introduce the concept of quasi-harmonic maps for reducing distortion in the fixed boundary case and extend it to both the free boundary and the spherical case. Thinking of parameterization in a more general sense as the construction of one or several scalar fields on a surface, we explore the potential of this construction for mesh deformation and surface matching. We propose an \u27;on-surface parameterization\u27; for guiding the deformation process and performing surface matching. A direct harmonic interpolation in the quaternion domain is also shown to give promising results for deformation transfer.Eine FlĂ€chenparameterisierung ist eine globale bijektive Abbildung zwischen der FlĂ€che und einem zugehörigen parametrischen Gebiet. Gewöhnlich werden FlĂ€chen mit scheibenförmiger Topologie auf eine Kreisscheibe und FlĂ€chen mit Genus Null auf eine SphĂ€re abgebildet. Das Hauptinteresse der Forschung an diesem Thema ist die Minimierung der parametrischen Verzerrung, die unweigerlich bei jedem Versuch, eine nicht triviale FlĂ€che ĂŒber einer Ebene zu parameterisieren, erzeugt wird. Diese Arbeit strebt zur Behandlung des Parametrisierungsproblems eine ausgeglichene Mischung zwischen mathematischer PrĂ€zision und ingenieurwissenschaftlicher Intuition an. Wir behandeln dabei die numerischen Aspekte des Parameterisierungsproblems im Hinblick auf die aktuellen parallelen Entwicklungen in der Mathematik und den Ingenieurwissenschaften. Weiterhin fĂŒhren wir das Konzept der quasi-harmonischen Abbildungen ein, um die Verzerrung bei gegebenen Randbedingungen zu verringern. AnschlieĂend verallgemeinern wir dieses Konzept auf den sphĂ€rischen Fall und auf den Fall mit freien Randbedingungen. Durch allgemeinere Betrachtung der Parameterisierung als Konstruktion eines oder mehrerer skalarer Felder auf einer FlĂ€che ergibt sich ein neuer Ansatz zur Netzdeformation und der Erzeugung von FlĂ€chenkorrespondenzen. Wir stellen eine \u27;on-surface parameterization\u27; vor, welche den Deformationsprozess leitet und FlĂ€chenkorrespondenzen erstellt. DarĂŒber hinaus zeigt eine direkte harmonische Interpolation in der DomĂ€ne der Quaternionen auch vielversprechende Resultate fĂŒr die Ăbertragung von Deformationen