Gaussian processes:iterative sparse approximations

In recent years there has been an increased interest in applying non-parametric methods to real-world problems. Significant research has been devoted to Gaussian processes (GPs) due to their increased flexibility when compared with parametric models. These methods use Bayesian learning, which generally leads to analytically intractable posteriors. This thesis proposes a two-step solution to construct a probabilistic approximation to the posterior. In the first step we adapt the Bayesian online learning to GPs: the final approximation to the posterior is the result of propagating the first and second moments of intermediate posteriors obtained by combining a new example with the previous approximation. The propagation of em functional forms is solved by showing the existence of a parametrisation to posterior moments that uses combinations of the kernel function at the training points, transforming the Bayesian online learning of functions into a parametric formulation. The drawback is the prohibitive quadratic scaling of the number of parameters with the size of the data, making the method inapplicable to large datasets. The second step solves the problem of the exploding parameter size and makes GPs applicable to arbitrarily large datasets. The approximation is based on a measure of distance between two GPs, the KL-divergence between GPs. This second approximation is with a constrained GP in which only a small subset of the whole training dataset is used to represent the GP. This subset is called the em Basis Vector, or BV set and the resulting GP is a sparse approximation to the true posterior. As this sparsity is based on the KL-minimisation, it is probabilistic and independent of the way the posterior approximation from the first step is obtained. We combine the sparse approximation with an extension to the Bayesian online algorithm that allows multiple iterations for each input and thus approximating a batch solution. The resulting sparse learning algorithm is a generic one: for different problems we only change the likelihood. The algorithm is applied to a variety of problems and we examine its performance both on more classical regression and classification tasks and to the data-assimilation and a simple density estimation problems.

A multi-agent system using iterative bidding mechanism to enhance manufacturing agility

The global market has become increasingly dynamic, unpredictable and customer-driven. This has led to rising rates of new product introduction and turbulent demand patterns across product mixes. As a result, manufacturing enterprises were facing mounting challenges to be agile and responsive to cope with market changes, so as to achieve the competitiveness of producing and delivering products to the market timely and cost-effectively. This paper introduces a currency-based iterative agent bidding mechanism to effectively and cost-efficiently integrate the activities associated with production planning and control, so as to achieve an optimised process plan and schedule. The aim is to enhance the agility of manufacturing systems to accommodate dynamic changes in the market and production. The iterative bidding mechanism is executed based on currency-like metrics; each operation to be performed is assigned with a virtual currency value and agents bid for the operation if they make a virtual profit based on this value. These currency values are optimised iteratively and so does the bidding process based on new sets of values. This is aimed at obtaining better and better production plans, leading to near-optimality. A genetic algorithm is proposed to optimise the currency values at each iteration. In this paper, the implementation of the mechanism and the test case simulation results are also discussed. © 2012 Elsevier Ltd. All rights reserved.

Typical case behaviour of spin systems in random graph and composite ensembles

This thesis includes analysis of disordered spin ensembles corresponding to Exact Cover, a multi-access channel problem, and composite models combining sparse and dense interactions. The satisfiability problem in Exact Cover is addressed using a statistical analysis of a simple branch and bound algorithm. The algorithm can be formulated in the large system limit as a branching process, for which critical properties can be analysed. Far from the critical point a set of differential equations may be used to model the process, and these are solved by numerical integration and exact bounding methods. The multi-access channel problem is formulated as an equilibrium statistical physics problem for the case of bit transmission on a channel with power control and synchronisation. A sparse code division multiple access method is considered and the optimal detection properties are examined in typical case by use of the replica method, and compared to detection performance achieved by interactive decoding methods. These codes are found to have phenomena closely resembling the well-understood dense codes. The composite model is introduced as an abstraction of canonical sparse and dense disordered spin models. The model includes couplings due to both dense and sparse topologies simultaneously. The new type of codes are shown to outperform sparse and dense codes in some regimes both in optimal performance, and in performance achieved by iterative detection methods in finite systems.

Iterative subspace analysis based on feature line distance

Nearest feature line-based subspace analysis is first proposed in this paper. Compared with conventional methods, the newly proposed one brings better generalization performance and incremental analysis. The projection point and feature line distance are expressed as a function of a subspace, which is obtained by minimizing the mean square feature line distance. Moreover, by adopting stochastic approximation rule to minimize the objective function in a gradient manner, the new method can be performed in an incremental mode, which makes it working well upon future data. Experimental results on the FERET face database and the UCI satellite image database demonstrate the effectiveness.

Relaxation of alternating iterative algorithms for the Cauchy problem associated with the modified Helmholtz equation

We propose two algorithms involving the relaxation of either the given Dirichlet data or the prescribed Neumann data on the over-specified boundary, in the case of the alternating iterative algorithm of ` 12 ` 12 `$12 `&12 `#12 `^12 `_12 `%12 `~12 *Kozlov91 applied to Cauchy problems for the modified Helmholtz equation. A convergence proof of these relaxation methods is given, along with a stopping criterion. The numerical results obtained using these procedures, in conjunction with the boundary element method (BEM), show the numerical stability, convergence, consistency and computational efficiency of the proposed methods.

An iterative method for the reconstruction of a stationary flow

In this article, an iterative algorithm based on the Landweber-Fridman method in combination with the boundary element method is developed for solving a Cauchy problem in linear hydrostatics Stokes flow of a slow viscous fluid. This is an iteration scheme where mixed well-posed problems for the stationary generalized Stokes system and its adjoint are solved in an alternating way. A convergence proof of this procedure is included and an efficient stopping criterion is employed. The numerical results confirm that the iterative method produces a convergent and stable numerical solution. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007

Polynomial function intervals for floating-point software verification

The focus of our work is the verification of tight functional properties of numerical programs, such as showing that a floating-point implementation of Riemann integration computes a close approximation of the exact integral. Programmers and engineers writing such programs will benefit from verification tools that support an expressive specification language and that are highly automated. Our work provides a new method for verification of numerical software, supporting a substantially more expressive language for specifications than other publicly available automated tools. The additional expressivity in the specification language is provided by two constructs. First, the specification can feature inclusions between interval arithmetic expressions. Second, the integral operator from classical analysis can be used in the specifications, where the integration bounds can be arbitrary expressions over real variables. To support our claim of expressivity, we outline the verification of four example programs, including the integration example mentioned earlier. A key component of our method is an algorithm for proving numerical theorems. This algorithm is based on automatic polynomial approximation of non-linear real and real-interval functions defined by expressions. The PolyPaver tool is our implementation of the algorithm and its source code is publicly available. In this paper we report on experiments using PolyPaver that indicate that the additional expressivity does not come at a performance cost when comparing with other publicly available state-of-the-art provers. We also include a scalability study that explores the limits of PolyPaver in proving tight functional specifications of progressively larger randomly generated programs. © 2014 Springer International Publishing Switzerland.

Using the max-sum algorithm for supply chain emergence in dynamic multiunit environments

Supply chain formation (SCF) is the process of determining the set of participants and exchange relationships within a network with the goal of setting up a supply chain that meets some predefined social objective. Many proposed solutions for the SCF problem rely on centralized computation, which presents a single point of failure and can also lead to problems with scalability. Decentralized techniques that aid supply chain emergence offer a more robust and scalable approach by allowing participants to deliberate between themselves about the structure of the optimal supply chain. Current decentralized supply chain emergence mechanisms are only able to deal with simplistic scenarios in which goods are produced and traded in single units only and without taking into account production capacities or input-output ratios other than 1:1. In this paper, we demonstrate the performance of a graphical inference technique, max-sum loopy belief propagation (LBP), in a complex multiunit unit supply chain emergence scenario which models additional constraints such as production capacities and input-to-output ratios. We also provide results demonstrating the performance of LBP in dynamic environments, where the properties and composition of participants are altered as the algorithm is running. Our results suggest that max-sum LBP produces consistently strong solutions on a variety of network structures in a multiunit problem scenario, and that performance tends not to be affected by on-the-fly changes to the properties or composition of participants.

The Perturbed Generalized Tikhonov's Algorithm

We work on the research of a zero of a maximal monotone operator on a real Hilbert space. Following the recent progress made in the context of the proximal point algorithm devoted to this problem, we introduce simultaneously a variable metric and a kind of relaxation in the perturbed Tikhonov’s algorithm studied by P. Tossings. So, we are led to work in the context of the variational convergence theory.

A PVT-Type Algorithm for Minimizing a Nonsmooth Convex Function

2000 Mathematics Subject Classification: 90C25, 68W10, 49M37.

A Fast Algorithm for Weber Problem on a Grid

AMS subject classification: 90B80.

Monte Carlo Algorithm for Solving Integral Equations with Polynomial Non-Linearity. Parallel Implementation

An iterative Monte Carlo algorithm for evaluating linear functionals of the solution of integral equations with polynomial non-linearity is proposed and studied. The method uses a simulation of branching stochastic processes. It is proved that the mathematical expectation of the introduced random variable is equal to a linear functional of the solution. The algorithm uses the so-called almost optimal density function. Numerical examples are considered. Parallel implementation of the algorithm is also realized using the package ATHAPASCAN as an environment for parallel realization.The computational results demonstrate high parallel efficiency of the presented algorithm and give a good solution when almost optimal density function is used as a transition density.

Customization of the design of antenna using genetic algorithm and finite-difference time-domain technique

Antenna design is an iterative process in which structures are analyzed and changed to comply with certain performance parameters required. The classic approach starts with analyzing a "known" structure, obtaining the value of its performance parameter and changing this structure until the "target" value is achieved. This process relies on having an initial structure, which follows some known or "intuitive" patterns already familiar to the designer. The purpose of this research was to develop a method of designing UWB antennas. What is new in this proposal is that the design process is reversed: the designer will start with the target performance parameter and obtain a structure as the result of the design process. This method provided a new way to replicate and optimize existing performance parameters. The base of the method was the use of a Genetic Algorithm (GA) adapted to the format of the chromosome that will be evaluated by the Electromagnetic (EM) solver. For the electromagnetic study we used XFDTD™ program, based in the Finite-Difference Time-Domain technique. The programming portion of the method was created under the MatLab environment, which serves as the interface for converting chromosomes, file formats and transferring of data between the XFDTD™ and GA. A high level of customization had to be written into the code to work with the specific files generated by the XFDTD™ program. Two types of cost functions were evaluated; the first one seeking broadband performance within the UWB band, and the second one searching for curve replication of a reference geometry. The performance of the method was evaluated considering the speed provided by the computer resources used. Balance between accuracy, data file size and speed of execution was achieved by defining parameters in the GA code as well as changing the internal parameters of the XFDTD™ projects. The results showed that the GA produced geometries that were analyzed by the XFDTD™ program and changed following the search criteria until reaching the target value of the cost function. Results also showed how the parameters can change the search criteria and influence the running of the code to provide a variety of geometries.

Positron emission tomography (PET) tumor segmentation and quantification: Development of new algorithms

Tumor functional volume (FV) and its mean activity concentration (mAC) are the quantities derived from positron emission tomography (PET). These quantities are used for estimating radiation dose for a therapy, evaluating the progression of a disease and also use it as a prognostic indicator for predicting outcome. PET images have low resolution, high noise and affected by partial volume effect (PVE). Manually segmenting each tumor is very cumbersome and very hard to reproduce. To solve the above problem I developed an algorithm, called iterative deconvolution thresholding segmentation (IDTS) algorithm; the algorithm segment the tumor, measures the FV, correct for the PVE and calculates mAC. The algorithm corrects for the PVE without the need to estimate camera's point spread function (PSF); also does not require optimizing for a specific camera. My algorithm was tested in physical phantom studies, where hollow spheres (0.5-16 ml) were used to represent tumors with a homogeneous activity distribution. It was also tested on irregular shaped tumors with a heterogeneous activity profile which were acquired using physical and simulated phantom. The physical phantom studies were performed with different signal to background ratios (SBR) and with different acquisition times (1-5 min). The algorithm was applied on ten clinical data where the results were compared with manual segmentation and fixed percentage thresholding method called T50 and T60 in which 50% and 60% of the maximum intensity respectively is used as threshold. The average error in FV and mAC calculation was 30% and -35% for 0.5 ml tumor. The average error FV and mAC calculation were ~5% for 16 ml tumor. The overall FV error was ∼10% for heterogeneous tumors in physical and simulated phantom data. The FV and mAC error for clinical image compared to manual segmentation was around -17% and 15% respectively. In summary my algorithm has potential to be applied on data acquired from different cameras as its not dependent on knowing the camera's PSF. The algorithm can also improve dose estimation and treatment planning.^

Iterative Phase Optimization of Elementary Quantum Error Correcting Codes

Performing experiments on small-scale quantum computers is certainly a challenging endeavor. Many parameters need to be optimized to achieve high-fidelity operations. This can be done efficiently for operations acting on single qubits, as errors can be fully characterized. For multiqubit operations, though, this is no longer the case, as in the most general case, analyzing the effect of the operation on the system requires a full state tomography for which resources scale exponentially with the system size. Furthermore, in recent experiments, additional electronic levels beyond the two-level system encoding the qubit have been used to enhance the capabilities of quantum-information processors, which additionally increases the number of parameters that need to be controlled. For the optimization of the experimental system for a given task (e.g., a quantum algorithm), one has to find a satisfactory error model and also efficient observables to estimate the parameters of the model. In this manuscript, we demonstrate a method to optimize the encoding procedure for a small quantum error correction code in the presence of unknown but constant phase shifts. The method, which we implement here on a small-scale linear ion-trap quantum computer, is readily applicable to other AMO platforms for quantum-information processing.