Data-driven reduction strategies for Bayesian inverse problems

A persistent central challenge in computational science and engineering (CSE), with both national and global security implications, is the efficient solution of large-scale Bayesian inverse problems. These problems range from estimating material parameters in subsurface simulations to estimating phenomenological parameters in climate models. Despite recent progress, our ability to quantify uncertainties and solve large-scale inverse problems lags well behind our ability to develop the governing forward simulations. Inverse problems present unique computational challenges that are only magnified as we include larger observational data sets and demand higher-resolution parameter estimates. Even with the current state-of-the-art, solving deterministic large-scale inverse problems is prohibitively expensive. Large-scale uncertainty quantification (UQ), cast in the Bayesian inversion framework, is thus rendered intractable. To conquer these challenges, new methods that target the root causes of computational complexity are needed. In this dissertation, we propose data-driven strategies for overcoming this “curse of di- mensionality.” First, we address the computational complexity induced in large-scale inverse problems by high-dimensional observational data. We propose a randomized misfit approach (RMA), which uses random projections—quasi-orthogonal, information-preserving transformations—to map the high-dimensional data-misfit vector to a low-dimensional space. We provide the first theoretical explanation for why randomized misfit methods are successful in practice with a small reduced data-misfit dimension (n = O(1)). Next, we develop the randomized geostatistical approach (RGA) for Bayesian sub- surface inverse problems with high-dimensional data. We show that the RGA is able to resolve transient groundwater inverse problems with noisy observed data dimensions up to 107, whereas a comparison method fails due to out-of-memory errors. Finally, we address the solution of Bayesian inverse problems with spatially localized data. The motivation is CSE applications that would gain from high-fidelity estimation over a smaller data-local domain, versus expensive and uncertain estimation over the full simulation domain. We propose several truncated domain inversion methods using domain decomposition theory to build model-informed artificial boundary conditions. Numerical investigations of MAP estimation and sampling demonstrate improved fidelity and fewer partial differential equation (PDE) solves with our truncated methods.Computational Science, Engineering, and Mathematic

Diffraction from a wavelet point of view

Cataloged from PDF version of article.The system impulse response representing the Fresnel diffraction is shown to form a wavelet family of functions. The scale parameter of the wavelet family represents the depth (distance). This observation relates the diffraction-holography-related studies and the wavelet theory. The results may be used in various optical applications such as designing robust volume optical elements for optical signal processing and finding new formulations for optical inverse problems. The results also extend the wavelet concept to the nonbandpass family of functions with the implication of new applications in signal processing. The presented wavelet structure, for example, is a tool for space-depth analysis

IDENTIFICATION AND ESTIMATION OF NONPARAMETRIC STRUCTURAL

This paper concerns a new statistical approach to instrumental variables (IV) method for nonparametric structural models with additive errors. A general identifying condition of the model is proposed, based on richness of the space generated by marginal discretizations of joint density functions. For consistent estimation, we develop statistical regularization theory to solve a random Fredholm integral equation of the first kind. A\ minimal set of conditions are given for consistency of a general regularization method. Using an abstract smoothness condition, we derive some optimal bounds, given the accuracies of preliminary estimates, and show the convergence rates of various regularization methods, including (the ordinary/iterated/generalized) Tikhonov and Showalter's methods. An application of the general regularization theory is discussed with a focus on a kernel smoothing method. We show an exact closed form, as well as the optimal convergence rate, of the kernel IV estimates of various regularization methods. The finite sample properties of the estimates are investigated via a small-scale Monte Carlo experimentNonparametric Strucutral Models, IV estimation, Statistical inverse problems

Inverse Dynamics and Control for Nuclear Power Plants

A new nonlinear control technique was developed by reformulating one of the “inverse Problems” techniques in mathematics, namely the reconstruction problem. The theory identifies an important concept called inverse dynamics which is always a known property for systems already developed or designed. Accordingly, the paradigm is called “reconstructive inverse dynamics” (RID) control. The standard state-space representation of dynamic systems constitutes a sufficient foundation to derive an algebraic RID control law that provides solutions in one step computation. The existence of an inverse solution is guaranteed for a limited dynamic space. Outside the guaranteed range, existence depends on the nature of the system under consideration. Derivations include adaptive features to minimize the effects of modeling errors and measurement degradation on control performance. A comparative study is included to illustrate the relationship between the RID control and optimal control strategies. A set of performance factors were used to investigate the robustness against various uncertainties and the suitability for digital implementation in large scale-systems. All of the illustrations are based on computer simulations using nonlinear models. The simulation results indicate a significant improvement in robust control strategies. The control strategy can be implemented on-line by exploiting its algebraic design property. Three applications to nuclear reactor systems are presented. The objective is to investigate the merit of the RID control technique to improve nuclear reactor operations and increase plant availability. The first two applications include xenon induced power oscillations and feed water control in conventional light water reactors. The third application consists of an automatic control system design for the startup of the Experimental Breeder Reactor-II (EBR-II). The nonlinear dynamic models used in this analysis were previously validated against available plant data. The simulation results show that the RID technique has the potential to improve reactor control strategies significantly. Some of the observations include accurate xenon control, and rapid feed water maneuvers in pressurized water reactors, and successful automated startup of the EBR-II. The scope of the inverse dynamics approach is extended to incorporate artificial intelligence methods within a systematic strategy design procedure. Since the RID control law includes the dynamics of the system, its implementation may influence plant component and measurement design. The inverse dynamics concept is further studied in conjunction with artificial neural networks and expert systems to develop practical control tools

A New Approach to Flatness, Horizon and Late-time Accelerating Expansion Problems on the basis of Mach Principle

Based on the idea that the components of a cosmological metric may be determined by the total gravitational potential of the universe, the scalar field {\phi} = 1/G in the Jordan-Brans-Dicke (JBD) theory is introduced as evolving with the inverse square of the scale factor. Since the gravitational potential is related to the field {\phi} resulting from Mach principle and depends on time due to the expansion of space, this temporal evolution of the field should be in accord with the evolution of time and space intervals in the metric tensor. For the same reason, the time dependence of the field makes these comoving intervals relative for different points on the time axis. Thus, it has been shown that introduction of the cosmic gravitational potential as a time dependent scalar field which decreases with 1/a2 in the coordinate-transformed Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime, may resolve flatness, horizon and late-time accelerating expansion problems in the standard model of cosmology. The luminosity distance vs redshift data of Type Ia supernovae is in agreement with this approach.Comment: 7 page

Bayesian linear inverse problems in regularity scales

We obtain rates of contraction of posterior distributions in inverse problems defined by scales of smoothness classes. We derive abstract results for general priors, with contraction rates determined by Galerkin approximation. The rate depends on the amount of prior concentration near the true function and the prior mass of functions with inferior Galerkin approximation. We apply the general result to non-conjugate series priors, showing that these priors give near optimal and adaptive recovery in some generality, Gaussian priors, and mixtures of Gaussian priors, where the latter are also shown to be near optimal and adaptive. The proofs are based on general testing and approximation arguments, without explicit calculations on the posterior distribution. We are thus not restricted to priors based on the singular value decomposition of the operator. We illustrate the results with examples of inverse problems resulting from differential equations.Comment: 34 page