subject:(optimal mass transport)

University of Lund

1. Elvander, Filip. Modeling and Sampling of Spectrally Structured Signals.

Degree: 2020, University of Lund

URL: https://lup.lub.lu.se/record/ca35ac5b-ad2e-4e44-b01f-9232dd74664f ; https://portal.research.lu.se/ws/files/81904925/FilipElvanderPhD_full.pdf

► This thesis consists of five papers concerned with the modeling of stochastic signals, as well as deterministic signals in stochastic noise, exhibiting different kinds of… (more)

▼ This thesis consists of five papers concerned with the modeling of stochastic signals, as well as deterministic signals in stochastic noise, exhibiting different kinds of structure. This structure is manifested as the existence of finite-dimensional parameterizations, and/or in the geometry of the signals' spectral representations. The two first papers of the thesis, Papers A and B, consider the modeling of differences, or distances, between stochastic processes based on their second-order statistics, i.e., covariances. By relating the covariance structure of a stochastic process to spectral representations, it is proposed to quantify the dissimilarity between two processes in terms of the cost associated with morphing one spectral representation to the other, with the cost of morphing being defined in terms of the solutions to optimal mass transport problems. The proposed framework allows for modeling smooth changes in the frequency characteristics of stochastic processes, which is shown to yield interpretable and physically sensible predictions when used in applications of temporal and spatial spectral estimation. Also presented are efficient computational tools, allowing for the framework to be used in high-dimensional problems.Paper C considers the modeling of so-called inharmonic signals, i.e., signals that are almost, but not quite, harmonic. Such signals appear in many fields of signal processing, not least in audio. Inharmonicity may be interpreted as a deviation from a parametric structure, as well as from a particular spectral structure. Based on these views, as well as on a third, stochastic interpretation, Paper C proposes three different definitions of the concept of fundamental frequency for inharmonic signals, and studies the estimation theoretical implications of utilizing either of these definitions. Paper D then considers deliberate deviations from a parametric signal structure arising in spectroscopy applications. With the motivation of decreasing the computational complexity of parameter estimation, the paper studies the implications of estimating the parameters of the signal in a sequential fashion, starting out with a simplified model that is then refined step by step.Lastly, Paper E studies how parametric descriptions of signals can be leveraged as to design optimal, in an estimation theoretical sense, schemes for sampling or collecting measurements from the signal. By means of a convex program, samples are selected as to minimize bounds on estimator variance, allowing for efficiently measuring parametric signals, even when the parametrization is only partially known.

Subjects/Keywords: Probability Theory and Statistics; Signal Processing; spectral estimation; parameter estimation; optimal mass transport; covariance interpolation; misspecified models; inharmonicity

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APA (6^th Edition):

Elvander, F. (2020). Modeling and Sampling of Spectrally Structured Signals. (Doctoral Dissertation). University of Lund. Retrieved from https://lup.lub.lu.se/record/ca35ac5b-ad2e-4e44-b01f-9232dd74664f ; https://portal.research.lu.se/ws/files/81904925/FilipElvanderPhD_full.pdf

Chicago Manual of Style (16^th Edition):

Elvander, Filip. “Modeling and Sampling of Spectrally Structured Signals.” 2020. Doctoral Dissertation, University of Lund. Accessed March 07, 2021. https://lup.lub.lu.se/record/ca35ac5b-ad2e-4e44-b01f-9232dd74664f ; https://portal.research.lu.se/ws/files/81904925/FilipElvanderPhD_full.pdf.

MLA Handbook (7^th Edition):

Elvander, Filip. “Modeling and Sampling of Spectrally Structured Signals.” 2020. Web. 07 Mar 2021.

Vancouver:

Elvander F. Modeling and Sampling of Spectrally Structured Signals. [Internet] [Doctoral dissertation]. University of Lund; 2020. [cited 2021 Mar 07]. Available from: https://lup.lub.lu.se/record/ca35ac5b-ad2e-4e44-b01f-9232dd74664f ; https://portal.research.lu.se/ws/files/81904925/FilipElvanderPhD_full.pdf.

Council of Science Editors:

Elvander F. Modeling and Sampling of Spectrally Structured Signals. [Doctoral Dissertation]. University of Lund; 2020. Available from: https://lup.lub.lu.se/record/ca35ac5b-ad2e-4e44-b01f-9232dd74664f ; https://portal.research.lu.se/ws/files/81904925/FilipElvanderPhD_full.pdf

2. Wang, Yichen. Modeling, predicting, and guiding users' temporal behaviors.

Degree: PhD, Mathematics, 2018, Georgia Tech

URL: http://hdl.handle.net/1853/60208

► The increasing availability and granularity of temporal event data produced from user activities in online media, social networks and health informatics provide new opportunities and… (more)

Subjects/Keywords: Point processes; Hawkes processes; Survival analysis; Low-rank models; Mass transport; Fokker Planck equation; Stochastic optimal control; Reinforcement learning; Social network analysis; Information diffusion; Recommendation systems

…5.3.3 Mass Transport as s Banded Linear Ordinary Differential Equation… …Hybrid Inference Machine with Mass Transport Equation . . . 84 5.4 Applications and… …x29; = x|Hsi − ] using a mass transport equation. Panel (c) shows the… …transport of conditional mass at four different times (the initial probability mass φ̃(x… …Keywords: Point processes, Hawkes processes, Survival analysis, Low-rank models, Mass transport…

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APA (6^th Edition):

Wang, Y. (2018). Modeling, predicting, and guiding users' temporal behaviors. (Doctoral Dissertation). Georgia Tech. Retrieved from http://hdl.handle.net/1853/60208

Chicago Manual of Style (16^th Edition):

Wang, Yichen. “Modeling, predicting, and guiding users' temporal behaviors.” 2018. Doctoral Dissertation, Georgia Tech. Accessed March 07, 2021. http://hdl.handle.net/1853/60208.

MLA Handbook (7^th Edition):

Wang, Yichen. “Modeling, predicting, and guiding users' temporal behaviors.” 2018. Web. 07 Mar 2021.

Vancouver:

Wang Y. Modeling, predicting, and guiding users' temporal behaviors. [Internet] [Doctoral dissertation]. Georgia Tech; 2018. [cited 2021 Mar 07]. Available from: http://hdl.handle.net/1853/60208.

Council of Science Editors:

Wang Y. Modeling, predicting, and guiding users' temporal behaviors. [Doctoral Dissertation]. Georgia Tech; 2018. Available from: http://hdl.handle.net/1853/60208

KTH

3. Orrenius, Johan. Optimal mass transport: a viable alternative to copulas in financial risk modeling?.

Degree: Mathematics (Dept.), 2018, KTH

URL: http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-231829

►

Copulas as a description of joint probability distributions is today common when modeling financial risk. The optimal mass transport problem also describes dependence structures,… (more)

Subjects/Keywords: Optimal mass transport; Financial risk; Copulas; Probability Theory and Statistics; Sannolikhetsteori och statistik

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APA (6^th Edition):

Orrenius, J. (2018). Optimal mass transport: a viable alternative to copulas in financial risk modeling?. (Thesis). KTH. Retrieved from http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-231829

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^th Edition):

Orrenius, Johan. “Optimal mass transport: a viable alternative to copulas in financial risk modeling?.” 2018. Thesis, KTH. Accessed March 07, 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-231829.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^th Edition):

Orrenius, Johan. “Optimal mass transport: a viable alternative to copulas in financial risk modeling?.” 2018. Web. 07 Mar 2021.

Vancouver:

Orrenius J. Optimal mass transport: a viable alternative to copulas in financial risk modeling?. [Internet] [Thesis]. KTH; 2018. [cited 2021 Mar 07]. Available from: http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-231829.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Orrenius J. Optimal mass transport: a viable alternative to copulas in financial risk modeling?. [Thesis]. KTH; 2018. Available from: http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-231829

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

University of Minnesota

4. Farooq, Hamza. Geometric and Optimization Methods for Diffusion Magnetic Resonance Imaging.

Degree: PhD, Electrical/Computer Engineering, 2017, University of Minnesota

URL: http://hdl.handle.net/11299/216888

► This thesis presents novel mathematical and computational methods aimed at enhancing and improving brain tissue structural imaging techniques that are based on diffusion Magnetic Resonance… (more)

▼ This thesis presents novel mathematical and computational methods aimed at enhancing and improving brain tissue structural imaging techniques that are based on diffusion Magnetic Resonance Imaging (dMRI). The most commonly used dMRI technique is Diffusion Tensor Imaging (DTI), which models water diffusion via a Gaussian pattern and estimates the corresponding covariance, also known as diffusion tensor. DTI forms the basis of brain structural connectivity methods like tractography and sub-cortical region parcellation, and thus provides useful markers for brain white matter integrity. Other, recently proposed dMRI techniques rely on modeling water diffusion in intra-axonal and extra-axonal spaces separately. Thereby, these so-called multi-compartment models hold the promise to provide detailed tissue microstructure information and to identify markers that may be specific to particular tissue development/diseases. In this thesis we address key mathematical challenges encountered by DTI, as well as by these newly proposed dMRI techniques, that pertain to recovering more detailed microstructure information. We begin by focusing on DTI and present novel geometrical methods to improve DTI analysis (Chapters 3, 4, and 5). In particular, (i) we utilize the mathematical theory of Optimal Mass Transport to improve brain parcellation by comparing sub-cortical regions connectivity profiles and compute their corresponding geometric ``average'' connectivity profiles, (ii) we introduce Ricci flow applied to diffusion tensor fields to enhance feature extraction, and finally (iii) we introduce a notion of discrete Ricci curvature in brain connectivity networks as a novel nodal measure to detect critical regions (nodes) of the structural brain networks. This notion of node curvature can be used to identify changes in brain network structure due to disease/development as it supplements information that can be obtained by other conventional network nodal measures. We then study multi-compartment dMRI models, and present a novel model fitting method to such tissue models (Chapter 6). Our proposed method is generic to all multicompartment models and enables for the first time dMRI-imaging in multiple fiber orientations and fiber-crossings situations. In addition to potential improvements in imaging technology, we hope that the advances presented in this work will contribute to the diagnosis and treatment of neurological disorders.

Subjects/Keywords: Brain Networks; Diffusion MRI; Diffusion Tensor Imaging; Graph curvature; Micro-structure Imaging; Optimal Mass Transport

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Farooq, H. (2017). Geometric and Optimization Methods for Diffusion Magnetic Resonance Imaging. (Doctoral Dissertation). University of Minnesota. Retrieved from http://hdl.handle.net/11299/216888

Chicago Manual of Style (16^th Edition):

Farooq, Hamza. “Geometric and Optimization Methods for Diffusion Magnetic Resonance Imaging.” 2017. Doctoral Dissertation, University of Minnesota. Accessed March 07, 2021. http://hdl.handle.net/11299/216888.

MLA Handbook (7^th Edition):

Farooq, Hamza. “Geometric and Optimization Methods for Diffusion Magnetic Resonance Imaging.” 2017. Web. 07 Mar 2021.

Vancouver:

Farooq H. Geometric and Optimization Methods for Diffusion Magnetic Resonance Imaging. [Internet] [Doctoral dissertation]. University of Minnesota; 2017. [cited 2021 Mar 07]. Available from: http://hdl.handle.net/11299/216888.

Council of Science Editors:

Farooq H. Geometric and Optimization Methods for Diffusion Magnetic Resonance Imaging. [Doctoral Dissertation]. University of Minnesota; 2017. Available from: http://hdl.handle.net/11299/216888

University of Minnesota

5. Chen, Yongxin. Modeling and control of collective dynamics: From Schroedinger bridges to optimal mass transport.

Degree: PhD, Mechanical Engineering, 2016, University of Minnesota

URL: http://hdl.handle.net/11299/182235

► We study modeling and control of collective dynamics. More specifically, we consider the problem of steering a particle system from an initial distribution to a… (more)

▼ We study modeling and control of collective dynamics. More specifically, we consider the problem of steering a particle system from an initial distribution to a final one with minimum energy control during some finite time window. It turns out that this problem is closely related to Optimal Mass Transport (OMT) and the Schroedinger bridge problem (SBP). OMT is concerned with reallocating mass from a specified starting distribution to a final one while incurring minimum cost. The SBP, on the other hand, seeks a most likely density ow to reconcile two marginal distributions with a prior probabilistic model for the flow. Both of these problems can be reformulated as those of controlling a density flow that may represent either a model for the distribution of a collection of dynamical systems or, a model for the uncertainty of the state of single dynamical system. This thesis is concerned with extensions of and point of contact between these two subjects, OMT and SBP. The aim of the work is to provide theory and tools for modeling and control of collections of dynamical systems. The SBP can be seen as a stochastic counterpart of OMT and, as a result, OMT can be recovered as the limit of the SBP as the stochastic excitation vanishes. The link between these two problems gives rise to a novel and fast algorithm to compute solutions of OMT as a suitable limit of SBP. For the special case where the marginal distributions are Gaussian and the underlying dynamics linear, the solution to either problem can be expressed as linear state feedback and computed explicitly in closed form. A natural extension of the work in the thesis concerns OMT and the SBP on discrete spaces and graphs in particular. Along this line we develop a framework to schedule transportation of mass over networks. Control in this context amounts to selecting a transition mechanism that is consistent with initial and final marginal distributions. The SBP on graphs on the other hand can be viewed as an atypical stochastic control problem where, once again, the control consists in suitably modifying the prior transition mechanism. By taking the Ruelle-Bowen random walk as a prior, we obtain scheduling that tends to utilize all paths as uniformly as the topology allows. Effectively, a consequence of such a choice is reduced congestion and increased robustness. The paradigm of Schroedinger bridges as a mechanism for scheduling transport on networks can be adapted to weighted graphs. Thus, our approach may be used to design transportation plans that represent a suitable compromise between robustness and cost of transport.

Subjects/Keywords: control theory; optimal mass transport; Schroedinger bridge

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APA (6^th Edition):

Chen, Y. (2016). Modeling and control of collective dynamics: From Schroedinger bridges to optimal mass transport. (Doctoral Dissertation). University of Minnesota. Retrieved from http://hdl.handle.net/11299/182235

Chicago Manual of Style (16^th Edition):

Chen, Yongxin. “Modeling and control of collective dynamics: From Schroedinger bridges to optimal mass transport.” 2016. Doctoral Dissertation, University of Minnesota. Accessed March 07, 2021. http://hdl.handle.net/11299/182235.

MLA Handbook (7^th Edition):

Chen, Yongxin. “Modeling and control of collective dynamics: From Schroedinger bridges to optimal mass transport.” 2016. Web. 07 Mar 2021.

Vancouver:

Chen Y. Modeling and control of collective dynamics: From Schroedinger bridges to optimal mass transport. [Internet] [Doctoral dissertation]. University of Minnesota; 2016. [cited 2021 Mar 07]. Available from: http://hdl.handle.net/11299/182235.

Council of Science Editors:

Chen Y. Modeling and control of collective dynamics: From Schroedinger bridges to optimal mass transport. [Doctoral Dissertation]. University of Minnesota; 2016. Available from: http://hdl.handle.net/11299/182235

University of Oxford

6. Heng, Jeremy. On the use of transport and optimal control methods for Monte Carlo simulation.

Degree: PhD, 2016, University of Oxford

URL: https://ora.ox.ac.uk/objects/uuid:6cbc7690-ac54-4a6a-b235-57fa62e5b2fc ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.740797

► This thesis explores ideas from transport theory and optimal control to develop novel Monte Carlo methods to perform efficient statistical computation. The first project considers… (more)

▼ This thesis explores ideas from transport theory and optimal control to develop novel Monte Carlo methods to perform efficient statistical computation. The first project considers the problem of constructing a transport map between two given probability measures. In the Bayesian formalism, this approach is natural when one introduces a curve of probability measures connecting the prior to posterior by tempering the likelihood function. The main idea is to move samples from the prior using an ordinary differential equation (ODE), constructed by solving the Liouville partial differential equation (PDE) which governs the time evolution of measures along the curve. In this work, we first study the regularity solutions of Liouville equation should satisfy to guarantee validity of this construction. We place an emphasis on understanding these issues as it explains the difficulties associated with solutions that have been previously reported. After ensuring that the flow transport problem is well-defined, we give a constructive solution. However, this result is only formal as the representation is given in terms of integrals which are intractable. For computational tractability, we proposed a novel approximation of the PDE which yields an ODE whose drift depends on the full conditional distributions of the intermediate distributions. Even when the ODE is time-discretized and the full conditional distributions are approximated numerically, the resulting distribution of mapped samples can be evaluated and used as a proposal within Markov chain Monte Carlo and sequential Monte Carlo (SMC) schemes. We then illustrate experimentally that the resulting algorithm can outperform state-of-the-art SMC methods at a fixed computational complexity. The second project aims to exploit ideas from optimal control to design more efficient SMC methods. The key idea is to control the proposal distribution induced by a time-discretized Langevin dynamics so as to minimize the Kullback-Leibler divergence of the extended target distribution from the proposal. The optimal value functions of the resulting optimal control problem can then be approximated using algorithms developed in the approximate dynamic programming (ADP) literature. We introduce a novel iterative scheme to perform ADP, provide a theoretical analysis of the proposed algorithm and demonstrate that the latter can provide significant gains over state-of-the-art methods at a fixed computational complexity.

Subjects/Keywords: 519.5; Transport theory – Statistical methods; Optimal control; Normalizing constants; Sequential Monte Carlo; Approximate dynamic programming; Monte Carlo; Mass transport; Annealed importance sampling; Reinforcement learning; Markov chain Monte Carlo

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APA (6^th Edition):

Heng, J. (2016). On the use of transport and optimal control methods for Monte Carlo simulation. (Doctoral Dissertation). University of Oxford. Retrieved from https://ora.ox.ac.uk/objects/uuid:6cbc7690-ac54-4a6a-b235-57fa62e5b2fc ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.740797

Chicago Manual of Style (16^th Edition):

Heng, Jeremy. “On the use of transport and optimal control methods for Monte Carlo simulation.” 2016. Doctoral Dissertation, University of Oxford. Accessed March 07, 2021. https://ora.ox.ac.uk/objects/uuid:6cbc7690-ac54-4a6a-b235-57fa62e5b2fc ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.740797.

MLA Handbook (7^th Edition):

Heng, Jeremy. “On the use of transport and optimal control methods for Monte Carlo simulation.” 2016. Web. 07 Mar 2021.

Vancouver:

Heng J. On the use of transport and optimal control methods for Monte Carlo simulation. [Internet] [Doctoral dissertation]. University of Oxford; 2016. [cited 2021 Mar 07]. Available from: https://ora.ox.ac.uk/objects/uuid:6cbc7690-ac54-4a6a-b235-57fa62e5b2fc ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.740797.

Council of Science Editors:

Heng J. On the use of transport and optimal control methods for Monte Carlo simulation. [Doctoral Dissertation]. University of Oxford; 2016. Available from: https://ora.ox.ac.uk/objects/uuid:6cbc7690-ac54-4a6a-b235-57fa62e5b2fc ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.740797

University of Minnesota

7. Ning, Lipeng. Matrix-valued optimal mass transportation and its applications.

Degree: PhD, Electrical Engineering, 2013, University of Minnesota

URL: http://hdl.handle.net/11299/36765

► The subject of this thesis is the geometry of matrix-valued density functions. The main motivation is the need for quantitative measures to compare power spectral… (more)

Subjects/Keywords: Matrix-valued density function; Multivariate time-series; Optimal mass transport; Spectral analysis

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Ning, L. (2013). Matrix-valued optimal mass transportation and its applications. (Doctoral Dissertation). University of Minnesota. Retrieved from http://hdl.handle.net/11299/36765

Chicago Manual of Style (16^th Edition):

Ning, Lipeng. “Matrix-valued optimal mass transportation and its applications.” 2013. Doctoral Dissertation, University of Minnesota. Accessed March 07, 2021. http://hdl.handle.net/11299/36765.

MLA Handbook (7^th Edition):

Ning, Lipeng. “Matrix-valued optimal mass transportation and its applications.” 2013. Web. 07 Mar 2021.

Vancouver:

Ning L. Matrix-valued optimal mass transportation and its applications. [Internet] [Doctoral dissertation]. University of Minnesota; 2013. [cited 2021 Mar 07]. Available from: http://hdl.handle.net/11299/36765.

Council of Science Editors:

Ning L. Matrix-valued optimal mass transportation and its applications. [Doctoral Dissertation]. University of Minnesota; 2013. Available from: http://hdl.handle.net/11299/36765

8. Gallouët, Thomas. Transport optimal : régularité et applications : Optimal Transport : Regularity and applications.

Degree: Docteur es, Mathématiques, 2012, Lyon, École normale supérieure

URL: http://www.theses.fr/2012ENSL0797

►

Cette thèse comporte deux parties distinctes, toutes les deux liées à la théorie du transport optimal. Dans la première partie, nous considérons une variété riemannienne,… (more)

▼

Cette thèse comporte deux parties distinctes, toutes les deux liées à la théorie du transport optimal. Dans la première partie, nous considérons une variété riemannienne, deux mesures à densité régulière et un coût de transport, typiquement la distance géodésique quadratique et nous nous intéressons à la régularité de l’application de transport optimal. Le critère décisif à cette régularité s’avère être le signe du tenseur de Ma-Trudinger-Wang (MTW). Nous présentons tout d’abord une synthèse des travaux réalisés sur ce tenseur. Nous nous intéressons ensuite au lien entre la géométrie des lieux d’injectivité et le tenseur MTW. Nous montrons que dans de nombreux cas, la positivité du tenseur MTW implique la convexité des lieux d’injectivité. La deuxième partie de cette thèse est liée aux équations aux dérivées partielles. Certaines peuvent être considérées comme des flots gradients dans l’espace de Wasserstein W2. C’est le cas de l’équation de Keller-Segel en dimension 2. Pour cette équation nous nous intéressons au problème de quantification de la masse lors de l’explosion des solutions ; cette explosion apparaît lorsque la masse initiale est supérieure à un seuil critique Mc. Nous cherchons alors à montrer qu’elle consiste en la formation d’un Dirac de masse Mc. Nous considérons ici un modèle particulaire en dimension 1 ayant le même comportement que l’équation de Keller-Segel. Pour ce modèle nous exhibons des bassins d’attractions à l’intérieur desquels l’explosion se produit avec seulement le nombre critique de particules. Finalement nous nous intéressons au profil d’explosion : à l’aide d’un changement d’échelle parabolique nous montrons que la structure de l’explosion correspond aux points critiques d’une certaine fonctionnelle.

This thesis consists in two distinct parts both related to the optimal transport theory.The first part deals with the regularity of the optimal transport map. The key tool is the Ma-Trundinger-Wang tensor and especially its positivity. We first give a review of the known results about the MTW tensor. We then explore the geometrical consequences of the MTW tensor on the injectivity domain. We prove that in many cases the positivity of MTW implies the convexity of the injectivity domain. The second part is devoted to the behaviour of a Keller-Segel solution in the super critical case. In particular we are interested in the mass quantization problem: we wish to quantify the mass aggregated when the blow-up occurs. In order to study the behaviour of the solution we consider a particle approximation of a Keller-Segel type equation in dimension 1. We define this approximation using the gradient flow interpretation of the Keller-Segel equation and the particular structure of the Wasserstein space in dimension 1. We show two kinds of results; we first prove a stability theorem for the blow-up mechanism: we exhibit basins of attraction in which the solution blows up with only the critical number of particles. We then prove a rigidity theorem for the blow-up mechanism: thanks to a parabolic…

Advisors/Committee Members: Villani, Cédric (thesis director).

Subjects/Keywords: Transport optimal; Régularité; Ma-Trundinger-Wang; MTW; Coût; Variété riemannienne; Convexité; Domaine d'injectivité; Lipschitz; C-convexité; Keller-Segel; Quantification de la masse; Particules; 1D; Explosion; Wasserstein; Flot gradient; Espace métrique; Masse critique; Optimal transport; Regularity; Ma-Trundinger-Wang; MTW; Cost; Riemannian manifold; Convexity; Injectivity domain; Lipschitz continuous; C-convexity; Keller-Segel; Mass quantization; Particles; 1D; Blow-up; Wasserstein; Gradient flow; Metric space; Critical mass

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APA (6^th Edition):

Gallouët, T. (2012). Transport optimal : régularité et applications : Optimal Transport : Regularity and applications. (Doctoral Dissertation). Lyon, École normale supérieure. Retrieved from http://www.theses.fr/2012ENSL0797

Chicago Manual of Style (16^th Edition):

Gallouët, Thomas. “Transport optimal : régularité et applications : Optimal Transport : Regularity and applications.” 2012. Doctoral Dissertation, Lyon, École normale supérieure. Accessed March 07, 2021. http://www.theses.fr/2012ENSL0797.

MLA Handbook (7^th Edition):

Gallouët, Thomas. “Transport optimal : régularité et applications : Optimal Transport : Regularity and applications.” 2012. Web. 07 Mar 2021.

Vancouver:

Gallouët T. Transport optimal : régularité et applications : Optimal Transport : Regularity and applications. [Internet] [Doctoral dissertation]. Lyon, École normale supérieure; 2012. [cited 2021 Mar 07]. Available from: http://www.theses.fr/2012ENSL0797.

Council of Science Editors:

Gallouët T. Transport optimal : régularité et applications : Optimal Transport : Regularity and applications. [Doctoral Dissertation]. Lyon, École normale supérieure; 2012. Available from: http://www.theses.fr/2012ENSL0797

9. Rehman, Tauseef ur. Efficient numerical method for solution of L² optimal mass transport problem.

Degree: PhD, Electrical and Computer Engineering, 2010, Georgia Tech

URL: http://hdl.handle.net/1853/33891

► In this thesis, a novel and efficient numerical method is presented for the computation of the L² optimal mass transport mapping in two and three… (more)

▼ In this thesis, a novel and efficient numerical method is presented for the computation of the L² optimal mass transport mapping in two and three dimensions. The method uses a direct variational approach. A new projection to the constraint technique has been formulated that can yield a good starting point for the method as well as a second order accurate discretization to the problem. The numerical experiments demonstrate that the algorithm yields accurate results in a relatively small number of iterations that are mesh independent. In the first part of the thesis, the theory and implementation details of the proposed method are presented. These include the reformulation of the Monge-Kantorovich problem using a variational approach and then using a consistent discretization in conjunction with the "discretize-then-optimize" approach to solve the resulting discrete system of differential equations. Advanced numerical methods such as multigrid and adaptive mesh refinement have been employed to solve the linear systems in practical time for even 3D applications. In the second part, the methods efficacy is shown via application to various image processing tasks. These include image registration and morphing. Application of (OMT) to registration is presented in the context of medical imaging and in particular image guided therapy where registration is used to align multiple data sets with each other and with the patient. It is shown that an elastic warping methodology based on the notion of mass transport is quite natural for several medical imaging applications where density can be a key measure of similarity between different data sets e.g. proton density based imagery provided by MR. An application is also presented of the two dimensional optimal mass transport algorithm to compute diffeomorphic correspondence maps between curves for geometric interpolation in an active contour based visual tracking application. Advisors/Committee Members: Tannenbaum, Allen (Committee Chair), Haber, Eldad (Committee Co-Chair), Biros, George (Committee Member), Shamma, Jeff (Committee Member), Skrinjar, Oskar (Committee Member), Yezzi, Anthony (Committee Member).

Subjects/Keywords: Optimal mass transport; Image registration; Image morphing; Active contour tracking; Mass transfer; Differential equations, Linear Numerical solutions

…Registration Warp Velocity Measurement . . . . . . . . . . 72 6.3.2 Optimal Mass Transport Error… …the optimal mass transport solver with final performance ratios reaching well beyond an… …82 17 18 19 26 27 xii SUMMARY Optimal Mass Transport (OMT) is an… …numerical method for the computation of the L2 optimal mass transport mapping in two and three… …present an application of two dimensional optimal mass transport algorithm to compute…

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APA (6^th Edition):

Rehman, T. u. (2010). Efficient numerical method for solution of L² optimal mass transport problem. (Doctoral Dissertation). Georgia Tech. Retrieved from http://hdl.handle.net/1853/33891

Chicago Manual of Style (16^th Edition):

Rehman, Tauseef ur. “Efficient numerical method for solution of L² optimal mass transport problem.” 2010. Doctoral Dissertation, Georgia Tech. Accessed March 07, 2021. http://hdl.handle.net/1853/33891.

MLA Handbook (7^th Edition):

Rehman, Tauseef ur. “Efficient numerical method for solution of L² optimal mass transport problem.” 2010. Web. 07 Mar 2021.

Vancouver:

Rehman Tu. Efficient numerical method for solution of L² optimal mass transport problem. [Internet] [Doctoral dissertation]. Georgia Tech; 2010. [cited 2021 Mar 07]. Available from: http://hdl.handle.net/1853/33891.

Council of Science Editors:

Rehman Tu. Efficient numerical method for solution of L² optimal mass transport problem. [Doctoral Dissertation]. Georgia Tech; 2010. Available from: http://hdl.handle.net/1853/33891

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