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This thesis investigates the computation of piecewise polynomial solutions to ill- conditioned linear systems of equations when noise on the linear measurements is observed. Specifically, we enhance our understanding of and provide qualifications on when such ill-conditioned systems of equations can be solved to a satisfactory accuracy. We show that the conventional condition number of the coefficient matrix is not sufficiently informative in this regard and propose a more relevant conditioning measure that takes into account the decay rate of singular values. We also discuss interactions of several factors affecting the solvability of such systems, including the number of discontinuities in solutions, as well as the distribution of nonzero entries in sparse matrices. In addition, we construct and test an approach for computing piecewise polynomial solutions of highly ill-conditioned linear systems using a randomized, SVD-based truncation, and L1-norm regularization. The randomized truncation is a stabilization technique that helps reduce the cost of the traditional SVD truncation for large and severely ill-conditioned matrices. For L1-minimization, we apply a solver based on the Alternating Direction Method. Numerical results are presented to compare our approach that is faster and can solve larger problems, called RTL1 (randomized truncation L1-minimization), with a well-known solver PP-TSVD.

We discuss the pole placement problem for single-input or multi-input control models of the form _x=Ax+Bu. This is the problem of determining a linear state feedback of the formu=F xsuch that in the closed-loop system _x= (A+BF)x, the matrixA+BFhas a prescribed set of eigenvalues. We analyze the conditioning of this problem and show that it is an intrinsically ill-conditioned problem, and especially so when the system dimension is large. Thus even the best numerical methods for this problem may yield very bad results. On the other hand, we also discuss the question of whether one really needs to solve the pole placement problem. In most circum- stances what is really required is stabilization or that the poles are in a specified region of the complex plane. This related problem may have much better conditioning. We demonstrate this via the example of stabilization.

We discuss the pole placement problem for single-input or multi-input control models of the form _x=Ax+Bu. This is the problem of determining a linear state feedback of the formu=F xsuch that in the closed-loop system _x= (A+BF)x, the matrixA+BFhas a prescribed set of eigenvalues. We analyze the conditioning of this problem and show that it is an intrinsically ill-conditioned problem, and especially so when the system dimension is large. Thus even the best numerical methods for this problem may yield very bad results. On the other hand, we also discuss the question of whether one really needs to solve the pole placement problem. In most circum- stances what is really required is stabilization or that the poles are in a specified region of the complex plane. This related problem may have much better conditioning. We demonstrate this via the example of stabilization.

The first part of this dissertation focuses on matrix-covariate regression models. While they have been studied in many existing works, classical statistical and computational methods for the analysis of the regression coefficient estimation are highly affected by high dimensional matrix-valued covariates. To address these issues, we proposes a framework of matrix-covariate regression models based on a low-rank constraint and an additional regularization for structured signals, with considerations of models of both continuous and binary responses. In the second part, we examine a Mixture Multilayer Stochastic Block Model (MMLSBM), where layers can be grouped into sets of similar networks. Each group of networks is endowed with a unique Stochastic Block Model. The objective is to partition the multilayer network into clusters of similar layers and identify communities within those layers. We present an alternative approach called the Alternating Minimization Algorithm (ALMA), which aims to simultaneously recover the layer partition and estimate the matrices of connection probabilities for the distinct layers. In the last part, we demonstrates the effectiveness of the projected gradient descent algorithm. Firstly, its local convergence rate is independent of the condition number. Secondly, under conditions where the objective function is rank-2r restricted L-smooth and μ-strongly convex, with L/μ < 3, projected gradient descent with appropriate step size converges linearly to the solution. Moreover, a perturbed version of this algorithm effectively navigates away from saddle points, converging to an approximate solution or a second-order local minimizer across a wide range of step sizes. Furthermore, we establish that there are no spurious local minimizes in estimating asymmetric low-rank matrices when the objective function satisfies L/μ < 3.

Mathematical modeling is a key component of various disciplines in science and engineering. A mathematical model which represents important behavior of a real system can be used as a substitute for the real process for many analysis and synthesis tasks. The performance of model based techniques, e.g. system analysis, computer simulation, controller design, sensor development, state filtering, product monitoring, and process optimization, is highly dependent on the quality of the model used. Therefore, it is very important to be able to develop an accurate model from available experimental data. Parameter estimation is usually formulated as an optimization problem where the parameter estimate is computed by minimizing the discrepancy between the model prediction and the experimental data. If a simple model and a large amount of data are available then the estimation problem is frequently well-posed and a small error in data fitting automatically results in an accurate model. However, this is not always the case. If the model is complex and only sparse and noisy data are available, then the estimation problem is often ill-conditioned and good data fitting does not ensure accurate model predictions. Many challenges that can often be neglected for estimation involving simple models need to be carefully considered for estimation problems involving complex models. To obtain a reliable and accurate estimate from sparse and noisy data, a set of techniques is developed by addressing the challenges encountered in estimation of complex models, including (1) model analysis and simplification which identifies the important sources of uncertainty and reduces the model complexity; (2) experimental design for collecting information-rich data by setting optimal experimental conditions; (3) regularization of estimation problem which solves the ill-conditioned large-scale optimization problem by reducing the number of parameters; (4) nonlinear estimation and filtering which fits the data by various estimation and filtering algorithms; (5) model verification by applying statistical hypothesis test to the prediction error. The developed methods are applied to different types of models ranging from models found in the process industries to biochemical networks, some of which are described by ordinary differential equations with dozens of state variables and more than a hundred parameters.

In the thesis, a calibration method for positions of each accelerometer in an Inertial Sensor Array (IMU) sensor array is designed and implemented. In order to model the motion of the sensor array in the real world, we build up a state space model. Based on the model we use, the problem is to estimate the parameters within the state space model. In this thesis, this problem is solved using Maximum Likelihood (ML) framework and two methods are implemented and analyzed. One is based on Expectation Maximization (EM) and the other is to optimize the cost function directly using Gradient Descent (GD). In the EM algorithm, an ill-conditioned problem exists in the M step, which degrades the performance of the algorithm especially when the initial error is small, and the final Mean Square Error (MSE) curve will diverge in this case. The EM algorithm with enough data samples works well when the initial error is large. In the Gradient Descent method, a reformulation of the problem avoids the ill-conditioned problem. After the parameter estimation part, we analyze the MSE curve of these parameters through the Monte Carlo Simulation. The final MSE curves show that the Gradient Descent based method is more robust in handling the numerical issues of the parameter estimation. The Gradient Descent method is also robust to noise level based on the simulation result. / I denna rapport utvecklas och implementeras en kalibreringsmethod för att skatta positionen för en grupp av accelerometrar placerade i en så kallad IMU sensor array. För att beskriva rörelsen för hela sensorgruppen, härleds en dynamisk tillståndsmodell. Problemställningen är då att skatta parametrarna i tillståndsmodellen. Detta löses med hjälp av Maximum Likelihood-metoden (ML) där två stycken algoritmer implementeras och analyseras. En baseras på Expectation Maximization (EM) och i den andra optimeras kostnadsfunktionen direkt med gradientsökning. I EM-algoritmen uppstår ett illa konditionerat delproblem i M-steget, vilket försämrar algoritmens prestanda, speciellt när det initiala felet är litet. Den resulterande MSE-kurvan kommer att avvika i detta fall. Däremot fungerar EM-algoritmen väl när antalet datasampel är tillräckligt och det initiala felet är större. I gradientsökningsmetoden undviks konditioneringsproblemen med hjälp av en omformulering. Slutligen analyseras medelkvadratfelet (MSE) för parameterskattningarna med hjälp av Monte Carlo-simulering. De resulterande MSE-kurvorna visar att gradientsökningsmetoden är mer robust mot numeriska problem, speciellt när det initiala felet är litet. Simuleringarna visar även att gradientsökning är robust mot brus.