Luis Tenorio

Prior information and uncertainty in inverse problems

Solving any inverse problem requires understanding the uncertainties in the data to know what it means to fit the data. We also need methods to incorporate data‐independent prior information to eliminate unreasonable models that fit the data. Both of these issues involve subtle choices that may significantly influence the results of inverse calculations. The specification of prior information is especially controversial. How does one quantify information? What does it mean to know something about a parameter a priori? In this tutorial we discuss Bayesian and frequentist methodologies that can be used to incorporate information into inverse calculations. In particular we show that apparently conservative Bayesian choices, such as representing interval constraints by uniform probabilities (as is commonly done when using genetic algorithms, for example) may lead to artificially small uncertainties. We also describe tools from statistical decision theory that can be used to characterize the performance of inv...

Statistical Regularization of Inverse Problems

In experimental sciences we often need to solve inverse problems. That is, we want to obtain information about the internal structure of a physical system from indirect noisy observations. Often the problem is not whether a solution exists; on the contrary, there are too many solutions that fit the data to a chosen tolerance level. The goal is to use prior information to determine a physically meaningful solution. Here, we present some of the basic questions that arise. We describe methods that can be used to find inversion estimates as well as ways to assess their performance.

Efficient automatic denoising of gravity gradiometry data

Gravity gradiometry data are prized for the high frequency information they provide. However, as any other geophysical data, gravity gradient measurements are contaminated by high-frequency noise. Separation of the high-frequency signal from noise is a crucial component of data processing. The separation can be performed in the frequency domain, which usually requires tuning filter parameters at each survey line to obtain optimal results. Because a modern gradiometry survey generates more data than a traditional gravity survey, such timeconsuming manual operations are not very practical. In addition, they may also introduce subjectivity into the process. To address this difficulty, we propose an automatic, data-adaptive 1D wavelet filtering technique specially designed to process gravity gradiometry data. The method is based on the thresholding of the wavelet coefficients to filter out high-frequency noise while preserving localized sharp signal features. We use an energy analysis across scales (specific for gravity gradiometry data) to select denoising thresholds and to identify sharp features of interest. We compare the proposed method with traditional Fourier-domain filters by applying them to synthetic data sets contaminated with either correlated or uncorrelated noise. The results demonstrate that the proposed filter is efficient and, when applied in the fully automated mode, produces results that are comparable to the best results achievable through frequencydomain filters. We further illustrate the method by applying it to a set of gravity gradiometry data acquired in the Gulf of Mexico and by characterizing the removed noise. Both synthetic and field examples show that the proposed method is an efficient and better alternative to other traditional frequency domain methods.

A simple but efficient algorithm for multiple-image deblurring

We consider the simultaneous deblurring of a set of noisy images whose point spread functions are different but known and spatially invariant, and with Gaussian noise. Currently available iterative algorithms that are typically used for this type of problem are computationally expensive, which makes their application for very large images impractical. We present a simple extension of a classical least-squares (LS) method where the multi-image deblurring is efficiently reduced to a computationally efficient single-image deblurring. In particular, we show that it is possible to remarkably improve the ill- conditioning of the LS problem by means of stable operations on the corresponding normal equations, which in turn speed up the convergence rate of the iterative algorithms. The performance and limitations of the method are analyzed through numerical simulations. Its connection with a column weighted least-squares approach is also considered in an appendix.

Numerical methods for A-optimal designs with a sparsity constraint for ill-posed inverse problems

We consider the problem of experimental design for linear ill-posed inverse problems. The minimization of the objective function in the classic A-optimal design is generalized to a Bayes risk minimization with a sparsity constraint. We present efficient algorithms for applications of such designs to large-scale problems. This is done by employing Krylov subspace methods for the solution of a subproblem required to obtain the experiment weights. The performance of the designs and algorithms is illustrated with a one-dimensional magnetotelluric example and an application to two-dimensional super-resolution reconstruction with MRI data.

Statistics of Parameter Estimates: A Concrete Example ∗

Most mathematical models include parameters that need to be determined from measurements. The estimated values of these parameters and their uncertainties depend on assumptions made about noise levels, models, or prior knowledge. But what can we say about the validity of such estimates, and the influence of these assumptions? This paper is concerned with methods to address these questions, and for didactic purposes it is written in the context of a concrete nonlinear parameter estimation problem. We will use the results of a physical experiment conducted by Allmaras et al. at Texas A&M University [M. Allmaras et al., SIAM Rev., 55 (2013), pp. 149--167] to illustrate the importance of validation procedures for statistical parameter estimation. We describe statistical methods and data analysis tools to check the choices of likelihood and prior distributions, and provide examples of how to compare Bayesian results with those obtained by non-Bayesian methods based on different types of assumptions. We explain...

On optimal detection of point sources in CMB maps

Point-source contamination in high-precision Cosmic Microwave Background (CMB) maps severely aects the precision of cosmological parameter estimates. Among the methods that have been proposed for source detection, the family of pseudo-filters optimizes a measure of signal-to-noise and amplitude-scale relation. In this paper we show that these filters are in fact only restrictive cases of a more general class of matched filters that optimize signal-to-noise ratio and that have, in general, better source detection capabilities, especially for lower amplitude sources. These conclusions are confirmed by some numerical experiments.

Estimation of regularization parameters in multiple-image deblurring

We consider the estimation of the regularization parameter for the simultaneous deblurring of multiple noisy images via Tikhonov regularization. We approach the problem in three ways. We first reduce the problem to a single-image deblurring for which the regularization parameter can be estimated through a classic generalized cross-validation (GCV) method. A mod- ification of this function is used for correcting the undersmoothing typical of the original technique. With a second method, we minimize an average least-squares fit to the images and define a new GCV function. In the last approach, we use the classi- cal GCV on a single higher-dimensional image obtained by concatenating all the images into a single vector. With a reliable estimator of the regularization parameter, one can fully exploit the excellent computational characteristics typical of direct de- blurring methods, which, especially for large images, makes them competitive with the more flexible but much slower iterative algorithms. The performance of the techniques is analyzed through numerical experiments. We find that under the independent homoscedastic and Gaussian assumptions made on the noise, the three approaches provide almost identical results with the first single image providing the practical advantage that no new software is required and the same image can be used with other deblurring algorithms.

Front Range Aggregates Optimizes Feeder Movements at Its Quarry

Front-end loaders extract sand and gravel (aggregate) from a pit and haul it to a feeder, which releases the aggregate onto a conveyor belt that is connected to a stockpile; the material is subsequently distributed to a processing plant. As mining progresses, the mining frontier moves farther away from the feeder, increasing loader cycle time. In turn, plant managers add loaders to maintain production rates. Eventually, the feeder must be moved closer to the mining frontier. Such a move requires shutting down production so that a crew can move the feeder. Historically, because a feeder movement did not occur until all loaders were in operation, such feeder movements overtaxed the loaders and lacked advance warning. We present a model to determine how often the feeder should be moved to the mining frontier. A shortest-path algorithm can quickly solve our model to minimize feeder movement and loader cycle-time costs. This model revolutionizes how aggregate companies, specifically Front Range Aggregates, pla...

Digital deblurring of CMB maps: Performance and efficient implementation

Digital deblurring of images is an important problem that arises in multifrequency observations of the Cosmic Microwave Background (CMB) where, because of the width of the point spread functions (PSF), maps at different frequencies suff er ad ifferent loss of spatial resolution. Deblurring is useful for various reasons: first, it helps to restore high frequency components lost through the smoothing effect of the instruments PSF; second, emissions at various frequencies observed with different resolutions can be better studied on a comparable resolution; third, some map-based component separation algorithms require maps with similar level of degradation. Because of computational efficiency, deblurring is usually done in the frequency domain. But this approach has some limitations as it requires spatial invariance of the PSF, stationarity of the noise, and is not flexible in the selection of more appropriate boundary conditions. Deblurring in real space is more flexible but usually not used because of its high computational cost. In this paper (the first in a series on the subject) we present new algorithms that allow the use of real space deblurring techniques even for very large images. In particular, we consider the use of Tikhonov deblurring of noisy maps with applications to PLANCK. We provide details for efficient implementations of the algorithms. Their performance is tested on Gaussian and non-Gaussian simulated CMB maps, and PSFs with both circular and elliptical symmetry. Matlab code is made available.