Optimizing Techniques and Cramer-Rao Bound for Passive Source Location Estimation

This work is motivated by the problem of locating potential unstable areas in underground potash mines with better accuracy more consistently while introducing minimum extra computational load. It is important for both efficient mine design and safe mining activities, since these unstable areas may experience local, low-intensity earthquakes in the vicinity of an underground mine. The object of this thesis is to present localization algorithms that can deliver the most consistent and accurate estimation results for the application of interest. As the first step towards the goal, three most representative source localization algorithms given in the literature are studied and compared. A one-step energy based grid search (EGS) algorithm is selected to address the needs of the application of interest. The next step is the development of closed-form Cram´er-Rao bound (CRB) expressions. The mathematical derivation presented in this work deals with continuous signals using the Karhunen-Lo`eve (K-L) expansion, which makes the derivation applicable to non-stationary Gaussian noise problems. Explicit closed-form CRB expressions are presented only for stationary Gaussian noise cases using the spectrum representation of the signal and noise though. Using the CRB comparisons, two approaches are proposed to further improve the EGS algorithm. The first approach utilizes the corresponding analytic expression of the error estimation variance (EEV) given in [1] to derive an amplitude weight expression, optimal in terms of minimizing this EEV, for the case of additive Gaussian noise with a common spectrum interpretation across all the sensors. An alternate noniterative amplitude weighting scheme is proposed based on the optimal amplitude weight expression. It achieves the same performance with less calculation compared with the traditional iterative approach. The second approach tries to optimize the EGS algorithm in the frequency domain. An analytic frequency weighted EEV expression is derived using spectrum representation and the stochastic process theory. Based on this EEV expression, an integral equation is established and solved using the calculus of variations technique. The solution corresponds to a filter transfer function that is optimal in the sense that it minimizes this analytic frequency domain EEV. When various parts of the frequency domain EEV expression are ignored during the minimization procedure using Cauchy-Schwarz inequality, several different filter transfer functions result. All of them turn out to be well known classical filters that have been developed in the literature and used to deal with source localization problems. This demonstrates that in terms of minimizing the analytic EEV, they are all suboptimal, not optimal. Monte Carlo simulation is performed and shows that both amplitude and frequency weighting bring obvious improvement over the unweighted EGS estimator

A Riemannian low-rank method for optimization over semidefinite matrices with block-diagonal constraints

We propose a new algorithm to solve optimization problems of the form $\min f(X)$ for a smooth function $f$ under the constraints that $X$ is positive semidefinite and the diagonal blocks of $X$ are small identity matrices. Such problems often arise as the result of relaxing a rank constraint (lifting). In particular, many estimation tasks involving phases, rotations, orthonormal bases or permutations fit in this framework, and so do certain relaxations of combinatorial problems such as Max-Cut. The proposed algorithm exploits the facts that (1) such formulations admit low-rank solutions, and (2) their rank-restricted versions are smooth optimization problems on a Riemannian manifold. Combining insights from both the Riemannian and the convex geometries of the problem, we characterize when second-order critical points of the smooth problem reveal KKT points of the semidefinite problem. We compare against state of the art, mature software and find that, on certain interesting problem instances, what we call the staircase method is orders of magnitude faster, is more accurate and scales better. Code is available.Comment: 37 pages, 3 figure

Improving Mixed Variable Optimization of Computational and Model Parameters Using Multiple Surrogate Functions

This research focuses on reducing computational time in parameter optimization by using multiple surrogates and subprocess CPU times without compromising the quality of the results. This is motivated by applications that have objective functions with expensive computational times at high fidelity solutions. Applying, matching, and tuning optimization techniques at an algorithm level can reduce the time spent on unprofitable computations for parameter optimization. The objective is to recover known parameters of a flow property reference image by comparing to a template image that comes from a computational fluid dynamics simulation, followed by a numerical image registration and comparison process. Mixed variable pattern search and mesh adaptive direct search methods were applied using surrogate functions in the search step to produce solutions within a tolerance level of experimental observations. The surrogate functions are based on previous function values and computational times of those values. The use of multiple surrogates at each search step provides parameter selections that lead to improved solutions of an objective function evaluation with less computational time. Previously computed values for the objective function and computation time were used to compute a time cut-off parameter that allows termination during an objective function evaluation if the computational time exceeded a threshold or a divergent template image was created. This approach was tested using DACE and radial basis function surrogates within the NOMADm MATLAB® software. The numerical results are presented