A Reduced-Order Gauss-Newton Method for Nonlinear Problems Based on Compressed Sensing for PDE Applications

A global regularized Gauss-Newton (GN) method is proposed to obtain a zero residual for square nonlinear problems on an affine subspace built by wavelets, which allows reducing systems that arise from the discretization of nonlinear elliptic partial differential equations (PDEs) without performing a priori simulations. This chapter introduces a Petrov-Galerkin (PG) GN approach together with its standard assumptions that ensure retaining the q-quadratic rate of convergence. It also proposes a regularization strategy, which maintains the fast pace of convergence, to avoid singularities and high nonlinearities. It also includes a line-search method for achieving global convergence. The numerical results manifest the capability of the algorithm for reproducing the full-order model (FOM) essential features while decreasing the runtime by a significant magnitude. This chapter refers to a wavelet-based reduced-order model (ROM) as WROM, while PROM is the proper orthogonal decomposition (POD)-based counterpart. The authors also implemented the combination of WROM and PROM as a hybrid method referred herein as (HROM). Preliminary results with Bratu?s problem show that if the WROM could correctly reproduce the FOM behavior, then HROM can also reproduce that FOM accurately

Distributed-memory large deformation diffeomorphic 3D image registration

We present a parallel distributed-memory algorithm for large deformation diffeomorphic registration of volumetric images that produces large isochoric deformations (locally volume preserving). Image registration is a key technology in medical image analysis. Our algorithm uses a partial differential equation constrained optimal control formulation. Finding the optimal deformation map requires the solution of a highly nonlinear problem that involves pseudo-differential operators, biharmonic operators, and pure advection operators both forward and back- ward in time. A key issue is the time to solution, which poses the demand for efficient optimization methods as well as an effective utilization of high performance computing resources. To address this problem we use a preconditioned, inexact, Gauss-Newton- Krylov solver. Our algorithm integrates several components: a spectral discretization in space, a semi-Lagrangian formulation in time, analytic adjoints, different regularization functionals (including volume-preserving ones), a spectral preconditioner, a highly optimized distributed Fast Fourier Transform, and a cubic interpolation scheme for the semi-Lagrangian time-stepping. We demonstrate the scalability of our algorithm on images with resolution of up to $1024^3$ on the "Maverick" and "Stampede" systems at the Texas Advanced Computing Center (TACC). The critical problem in the medical imaging application domain is strong scaling, that is, solving registration problems of a moderate size of $256^3$---a typical resolution for medical images. We are able to solve the registration problem for images of this size in less than five seconds on 64 x86 nodes of TACC's "Maverick" system.Comment: accepted for publication at SC16 in Salt Lake City, Utah, USA; November 201

Distributed parameter and state estimation for wireless sensor networks

The research in distributed algorithms is linked with the developments of statistical inference in wireless sensor networks (WSNs) applications. Typically, distributed approaches process the collected signals from networked sensor nodes. That is to say, the sensors receive local observations and transmit information between each other. Each sensor is capable of combining the collected information with its own observations to improve performance. In this thesis, we propose novel distributed methods for the inference applications using wireless sensor networks. In particular, the efficient algorithms which are not computationally intensive are investigated. Moreover, we present a number of novel algorithms for processing asynchronous network events and robust state estimation. In the first part of the thesis, a distributed adaptive algorithm based on the component-wise EM method for decentralized sensor networks is investigated. The distributed component-wise Expectation-Maximization (EM) algorithm has been designed for application in a Gaussian density estimation. The proposed algorithm operates a component-wise EM procedure for local parameter estimation and exploit an incremental strategy for network updating, which can provide an improved convergence rate. Numerical simulation results have illustrated the advantages of the proposed distributed component-wise EM algorithm for both well-separated and overlapped mixture densities. The distributed component-wise EM algorithm can outperform other EM-based distributed algorithms in estimating overlapping Gaussian mixtures. In the second part of the thesis, a diffusion based EM gradient algorithm for density estimation in asynchronous wireless sensor networks has been proposed. Specifically, based on the asynchronous adapt-then-combine diffusion strategy, a distributed EM gradient algorithm that can deal with asynchronous network events has been considered. The Bernoulli model has been exploited to approximate the asynchronous behaviour of the network. Compared with existing distributed EM based estimation methods using a consensus strategy, the proposed algorithm can provide more accurate estimates in the presence of asynchronous networks uncertainties, such as random link failures, random data arrival times, and turning on or off sensor nodes for energy conservation. Simulation experiments have been demonstrated that the proposed algorithm significantly outperforms the consensus based strategies in terms of Mean-Square- Deviation (MSD) performance in an asynchronous network setting. Finally, the challenge of distributed state estimation in power systems which requires low complexity and high stability in the presence of bad data for a large scale network is addressed. A gossip based quasi-Newton algorithm has been proposed for solving the power system state estimation problem. In particular, we have applied the quasi-Newton method for distributed state estimation under the gossip protocol. The proposed algorithm exploits the Broyden- Fletcher-Goldfarb-Shanno (BFGS) formula to approximate the Hessian matrix, thus avoiding the computation of inverse Hessian matrices for each control area. The simulation results for IEEE 14 bus system and a large scale 4200 bus system have shown that the distributed quasi-Newton scheme outperforms existing algorithms in terms of Mean-Square-Error (MSE) performance with bad data

Numerical resolution of dynamic games

La teoría de Juegos estudia las interacciones estratégicas entre variosagentes (jugadores), donde cada uno toma una serie de decisiones buscando maximizaruna función objetivo (particular de cada jugador), pero cuyo resultado depende no sólode las decisiones de cada jugador sino también de las de los demás jugadores.Esta disciplina Matemática es muy utilizada en muchos campos y problemas deactualidad, pero tiene el inconveniente de que, incluso para problemas sencillos, enmuchas ocasiones no se pueden calcular soluciones explícitas y se tienen que aplicarmétodos numéricos.Estos métodos pueden ser costosos computacionalmente, sobre todo al aumentar elnúmero de jugadores, lo que hace muy importante el desarrollo de algoritmos eficientesque permitan calcular soluciones precisas en un tiempo razonable.Departamento de Matemática AplicadaMáster en Matemática

Registration of 3D Point Clouds and Meshes: A Survey From Rigid to Non-Rigid

Three-dimensional surface registration transforms multiple three-dimensional data sets into the same coordinate system so as to align overlapping components of these sets. Recent surveys have covered different aspects of either rigid or nonrigid registration, but seldom discuss them as a whole. Our study serves two purposes: 1) To give a comprehensive survey of both types of registration, focusing on three-dimensional point clouds and meshes and 2) to provide a better understanding of registration from the perspective of data fitting. Registration is closely related to data fitting in which it comprises three core interwoven components: model selection, correspondences and constraints, and optimization. Study of these components 1) provides a basis for comparison of the novelties of different techniques, 2) reveals the similarity of rigid and nonrigid registration in terms of problem representations, and 3) shows how overfitting arises in nonrigid registration and the reasons for increasing interest in intrinsic techniques. We further summarize some practical issues of registration which include initializations and evaluations, and discuss some of our own observations, insights and foreseeable research trends