Estimating the trace of the matrix inverse by interpolating from the diagonal of an approximate inverse

A number of applications require the computation of the trace of a matrix that is implicitly available through a function. A common example of a function is the inverse of a large, sparse matrix, which is the focus of this paper. When the evaluation of the function is expensive, the task is computationally challenging because the standard approach is based on a Monte Carlo method which converges slowly. We present a different approach that exploits the pattern correlation, if present, between the diagonal of the inverse of the matrix and the diagonal of some approximate inverse that can be computed inexpensively. We leverage various sampling and fitting techniques to fit the diagonal of the approximation to the diagonal of the inverse. Depending on the quality of the approximate inverse, our method may serve as a standalone kernel for providing a fast trace estimate with a small number of samples. Furthermore, the method can be used as a variance reduction method for Monte Carlo in some cases. This is decided dynamically by our algorithm. An extensive set of experiments with various technique combinations on several matrices from some real applications demonstrate the potential of our method. (C) 2016 Published by Elsevier Inc

Modelling and inference for electromagnetic flow tomography

Electromagnetic flowmeters determine the bulk flow rate of an ohmic fluid in a pipe by measuring the voltage induced across the fluid by a transverse magnetic field. This thesis develops the theory of an electromagnetic flowmeter for groundwater aquifer applications. Electromagnetic flowmeters require slow, laminar flow for measurements of bulk flow to be accurate - even after calibration. In general, the measured voltage depends on the spatial distribution of the velocity of the fluid. Hence, determination of the velocity field is required in order to accurately measure the bulk flow rate in general flows. Accordingly, this thesis examines the inverse problem of electromagnetic flow tomography, which is the problem of determining the velocity field in a fluid from voltage measurements made at multiple locations. Electromagnetic flow tomography is a severely ill-posed linear inverse problem. The relationship between the flow and the potential induced across the fluid is described by the flowmeter equation - a boundary value problem in Poisson's equation, with the source due to the Faraday effect. A novel dipole-form of the flowmeter equation allows for analysis of spatial sensitivities. This boundary value problem is solved using Green's functions, derived by the method of images for the geometry of pipe cross-section and half-space. Computational implementation of the forward map uses a finite element method discretisation and assumes idealised point electrodes to simulate measurements. Analysis of the measurement kernel reveals extreme sensitivity to flow near the electrode locations, with low sensitivity to the majority of flow away from the electrodes. The resulting non-uniqueness in inverting the forward map implies that assumptions must be made about the spatial flow profile in order to make estimates of the bulk flow. This thesis examines a Bayesian formulation to this inverse problem, that includes a model for the forward map, and accounts for measurement noise and uncertainty in the velocity field. The Bayesian analysis of an inverse problem produces the posterior distribution, from which estimates of desired quantities may be calculated, along with uncertainties. In particular, prior modelling allows for exploring assumptions and representation of unknowns to determine potential biases. The resulting Bayesian model is a standard stochastic hierarchical model with hyperparameters to model modelling uncertainties such as the smoothness of the flow profile, or other effects. The flow is modelled as a Gaussian Markov random field and the hyperparameters are modelled using a Jefferys prior. The resulting model for the flow tomography inverse problem is a linear Gaussian model. Inference for this model is efficiently implemented using the recent marginal then conditional algorithm. That algorithm generates posterior samples by first using a Monte-Carlo Markov chain sampler for the low-dimensional marginal distribution over hyperparameters, then drawing from the full conditional distribution over the flow profile, which requires one solve of a linear equation. Posterior inference does not require the draw from the full Gaussian conditional, as moments of the Gaussian are available analytically. This method for computational Bayesian inference surpasses equivalent regularising methods in computational speed. To the best of this author's knowledge, this is the first time a Bayesian method has been used for analysing electromagnetic flow tomography. Measurements of the bulk flow in a pipe are computed using simulated data generated from physically sensible phantom flow profiles. Various geometries and electrode placements are examined, with different shapes and scales of phantom flow. The effect of using the fluid dynamics no-slip condition and changing hyperparameter values is also explored. Not surprisingly, increased number of electrodes increases the spatial flow profile resolution and accuracy of bulk flow estimates. The flow profile reconstructions and bulk flow estimates are more accurate for flow profiles which could easily be interpolated from values near electrode locations. Additionally, it is shown that there is an implicit scale in the system - the standard deviation of bulk flow correlated to the area of the pipe. The use of invasive measurements for the purpose of measuring groundwater flow is also investigated. Analysis in this thesis shows that measuring groundwater flow presents significant difficulty; the resolution of bulk flow from a realistic signal-to-noise ratio is several orders of magnitude larger than the expected bulk flow rate in unconfined aquifers

Modelling and Inference for Electromagnetic Flow Tomography

Electromagnetic flowmeters determine the bulk flow rate of an ohmic fluid in a pipe by measuring the voltage induced across the fluid by a transverse magnetic field. This thesis develops the theory of an electromagnetic flowmeter for groundwater aquifer applications. Electromagnetic flowmeters require slow, laminar flow for measurements of bulk flow to be accurate - even after calibration. In general, the measured voltage depends on the spatial distribution of the velocity of the fluid. Hence, determination of the velocity field is required in order to accurately measure the bulk flow rate in general flows. Accordingly, this thesis examines the inverse problem of electromagnetic flow tomography, which is the problem of determining the velocity field in a fluid from voltage measurements made at multiple locations. Electromagnetic flow tomography is a severely ill-posed linear inverse problem. The relationship between the flow and the potential induced across the fluid is described by the flowmeter equation - a boundary value problem in Poisson's equation, with the source due to the Faraday effect. A novel dipole-form of the flowmeter equation allows for analysis of spatial sensitivities. This boundary value problem is solved using Green's functions, derived by the method of images for the geometry of pipe cross-section and half-space. Computational implementation of the forward map uses a finite element method discretisation and assumes idealised point electrodes to simulate measurements. Analysis of the measurement kernel reveals extreme sensitivity to flow near the electrode locations, with low sensitivity to the majority of flow away from the electrodes. The resulting non-uniqueness in inverting the forward map implies that assumptions must be made about the spatial flow profile in order to make estimates of the bulk flow. This thesis examines a Bayesian formulation to this inverse problem, that includes a model for the forward map, and accounts for measurement noise and uncertainty in the velocity field. The Bayesian analysis of an inverse problem produces the posterior distribution, from which estimates of desired quantities may be calculated, along with uncertainties. In particular, prior modelling allows for exploring assumptions and representation of unknowns to determine potential biases. The resulting Bayesian model is a standard stochastic hierarchical model with hyperparameters to model modelling uncertainties such as the smoothness of the flow profile, or other effects. The flow is modelled as a Gaussian Markov random field and the hyperparameters are modelled using a Jefferys prior. The resulting model for the flow tomography inverse problem is a linear Gaussian model. Inference for this model is efficiently implemented using the recent marginal then conditional algorithm. That algorithm generates posterior samples by first using a Monte-Carlo Markov chain sampler for the low-dimensional marginal distribution over hyperparameters, then drawing from the full conditional distribution over the flow profile, which requires one solve of a linear equation. Posterior inference does not require the draw from the full Gaussian conditional, as moments of the Gaussian are available analytically. This method for computational Bayesian inference surpasses equivalent regularising methods in computational speed. To the best of this author's knowledge, this is the first time a Bayesian method has been used for analysing electromagnetic flow tomography. Measurements of the bulk flow in a pipe are computed using simulated data generated from physically sensible phantom flow profiles. Various geometries and electrode placements are examined, with different shapes and scales of phantom flow. The effect of using the fluid dynamics no-slip condition and changing hyperparameter values is also explored. Not surprisingly, increased number of electrodes increases the spatial flow profile resolution and accuracy of bulk flow estimates. The flow profile reconstructions and bulk flow estimates are more accurate for flow profiles which could easily be interpolated from values near electrode locations. Additionally, it is shown that there is an implicit scale in the system - the standard deviation of bulk flow correlated to the area of the pipe. The use of invasive measurements for the purpose of measuring groundwater flow is also investigated. Analysis in this thesis shows that measuring groundwater flow presents significant difficulty; the resolution of bulk flow from a realistic signal-to-noise ratio is several orders of magnitude larger than the expected bulk flow rate in unconfined aquifers