Inverse Mappers for QCD Global Analysis

Inverse problems – using measured observations to determine unknown parameters – are well motivated but challenging in many scientific problems. Mapping parameters to observables is a well-posed problem with unique solutions, and therefore can be solved with differential equations or linear algebra solvers. However, the inverse problem requires backward mapping from observable to parameter space, which is often nonunique. Consequently, solving inverse problems is ill-posed and a far more challenging computational problem. Our motivated application in this dissertation is the inverse problems in nuclear physics that characterize the internal structure of the hadrons. We first present a machine learning framework called Variational Autoencoder Inverse Mapper (VAIM), as an autoencoder based neural network architecture to construct an effective “inverse function” that maps experimental data into QCFs. In addition to the well-known inverse problems challenges such as ill-posedness, an application specific issue is that the experimental data are observed on kinematics bins which are usually irregular and varying. To address this ill defined problem, we represent the observables together with their kinematics bins as an unstructured, high-dimensional point cloud. The point cloud representation is incorporated into the VAIM framework. Our new architecture point cloud-based VAIM (PCVAIM) enables the underlying deep neural networks to learn how the observables are distributed across kinematics. Next, we present our methods of extracting the leading twist Compton form factors (CFFs) from polarization observables. In this context, we extend VAIM framework to the Conditional -VAIM to extract the CFFs from the DVCS cross sections on several kinematics. Connected to this effort is a study of the effectiveness of incorporating physics knowledge into machine learning. We start this task by incorporating physics constraints to the forward problem of mapping the kinematics to the cross sections. First, we develop Physics Constrained Neural Networks (PCNNs) for Deeply Virtual Exclusive Scattering (DVCS) cross sections by integrating some of the physics laws such as the symmetry constraints of the cross sections. This provides us with an inception of incorporating physics rules into our inverse mappers which will one of the directions of our future research

Reconstruction of a source domain from boundary measurements

Inverse and ill-posed problems which consist of reconstructing the unknown support of a source from a single pair of exterior boundary Cauchy data are investigated. The underlying dependent variable, e.g. potential, temperature or pressure, may satisfy the Laplace, Poisson, Helmholtz or modified Helmholtz partial differential equations (PDEs). For constant coefficients, the solutions of these elliptic PDEs are sought as linear combinations of explicitly available fundamental solutions (free-space Greens functions), as in the method of fundamental solutions (MFS). Prior to this application of the MFS, the free-term inhomogeneity represented by the intensity of the source is removed by the method of particular solutions. The resulting transmission problem then recasts as that of determining the interface between composite materials. In order to ensure a unique solution, the unknown source domain is assumed to be star-shaped. This in turn enables its boundary to be parametrised by the radial coordinate, as a function of the polar or, spherical angles. The problem is nonlinear and the numerical solution which minimizes the gap between the measured and the computed data is achieved using the Matlab toolbox routine lsqnonlin which is designed to minimize a sum of squares starting from an initial guess and with no gradient required to be supplied by the user. Simple bounds on the variables can also be prescribed. Since the inverse problem is still ill-posed with respect to small errors in the data and possibly additional ill-conditioning introduced by the spectral feature of the MFS approximation, the least-squares functional which is minimized needs to be augmented with regularizing penalty terms on the MFS coefficients and on the radial function for a stable estimation of these couple of unknowns. Thorough numerical investigations are undertaken for retrieving regular and irregular shapes of the source support from both exact and noisy input data

On the filtering effect of iterative regularization algorithms for linear least-squares problems

Many real-world applications are addressed through a linear least-squares problem formulation, whose solution is calculated by means of an iterative approach. A huge amount of studies has been carried out in the optimization field to provide the fastest methods for the reconstruction of the solution, involving choices of adaptive parameters and scaling matrices. However, in presence of an ill-conditioned model and real data, the need of a regularized solution instead of the least-squares one changed the point of view in favour of iterative algorithms able to combine a fast execution with a stable behaviour with respect to the restoration error. In this paper we want to analyze some classical and recent gradient approaches for the linear least-squares problem by looking at their way of filtering the singular values, showing in particular the effects of scaling matrices and non-negative constraints in recovering the correct filters of the solution

Multitaper estimation on arbitrary domains

Multitaper estimators have enjoyed significant success in estimating spectral densities from finite samples using as tapers Slepian functions defined on the acquisition domain. Unfortunately, the numerical calculation of these Slepian tapers is only tractable for certain symmetric domains, such as rectangles or disks. In addition, no performance bounds are currently available for the mean squared error of the spectral density estimate. This situation is inadequate for applications such as cryo-electron microscopy, where noise models must be estimated from irregular domains with small sample sizes. We show that the multitaper estimator only depends on the linear space spanned by the tapers. As a result, Slepian tapers may be replaced by proxy tapers spanning the same subspace (validating the common practice of using partially converged solutions to the Slepian eigenproblem as tapers). These proxies may consequently be calculated using standard numerical algorithms for block diagonalization. We also prove a set of performance bounds for multitaper estimators on arbitrary domains. The method is demonstrated on synthetic and experimental datasets from cryo-electron microscopy, where it reduces mean squared error by a factor of two or more compared to traditional methods.Comment: 28 pages, 11 figure

Discretization of variational regularization in Banach spaces

Consider a nonlinear ill-posed operator equation $F(u)=y$ where $F$ is defined on a Banach space $X$. In general, for solving this equation numerically, a finite dimensional approximation of $X$ and an approximation of $F$ are required. Moreover, in general the given data \yd of $y$ are noisy. In this paper we analyze finite dimensional variational regularization, which takes into account operator approximations and noisy data: We show (semi-)convergence of the regularized solution of the finite dimensional problems and establish convergence rates in terms of Bregman distances under appropriate sourcewise representation of a solution of the equation. The more involved case of regularization in nonseparable Banach spaces is discussed in detail. In particular we consider the space of finite total variation functions, the space of functions of finite bounded deformation, and the $L^\infty$--space

Remarks on singular solutions of the Euler equations

We examine the blow-up claims of the incompressible Euler equations for two flows, the columnar eddies in the vicinity of stagnation, and a quasi-three-dimensional structure for illustrating oscillations and concentrations in shears. We assert that these finite-time singularities are not genuine.Comment: 7 page