Kernel-based Image Reconstruction from Scattered Radon Data

Computerized tomography requires suitable numerical methods for the approximation of a bivariate function f from a finite set of discrete Radon data, each of whose data samples represents one line integral of f . In standard reconstruction methods, specific assumptions concerning the geometry of the Radon lines are usually made. In relevant applications of image reconstruction, however, such assumptions are often too restrictive. In this case, one would rather prefer to work with reconstruction methods allowing for arbitrary distributions of scattered Radon lines. This paper proposes a novel image reconstruction method for scattered Radon data, which combines kernel-based scattered data approximation with a well-adapted regularization of the Radon transform. This results in a very flexible numerical algorithm for image reconstruction, which works for arbitrary distributions of Radon lines. This is in contrast to the classical filtered back projection, which essentially relies on a regular distribution of the Radon lines, e.g. parallel beam geometry. The good performance of the kernel-based image reconstruction method is illustrated by numerical examples and comparisons

Augmenting Basis Sets by Normalizing Flows

Approximating functions by a linear span of truncated basis sets is a standard procedure for the numerical solution of differential and integral equations. Commonly used concepts of approximation methods are well-posed and convergent, by provable approximation orders. On the down side, however, these methods often suffer from the curse of dimensionality, which limits their approximation behavior, especially in situations of highly oscillatory target functions. Nonlinear approximation methods, such as neural networks, were shown to be very efficient in approximating high-dimensional functions. We investigate nonlinear approximation methods that are constructed by composing standard basis sets with normalizing flows. Such models yield richer approximation spaces while maintaining the density properties of the initial basis set, as we show. Simulations to approximate eigenfunctions of a perturbed quantum harmonic oscillator indicate convergence with respect to the size of the basis set.Comment: Corrected arXiv identifier for ref.

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semi-implicit SPH scheme for the shallow water equation

Computing excited states of molecules using normalizing flows

We present a new nonlinear variational framework for simultaneously computing ground and excited states of quantum systems. Our approach is based on approximating wavefunctions in the linear span of basis functions that are augmented and optimized \emph{via} composition with normalizing flows. The accuracy and efficiency of our approach are demonstrated in the calculations of a large number of vibrational states of the triatomic H$_2$S molecule as well as ground and several excited electronic states of prototypical one-electron systems including the hydrogen atom, the molecular hydrogen ion, and a carbon atom in a single-active-electron approximation. The results demonstrate significant improvements in the accuracy of energy predictions and accelerated basis-set convergence even when using normalizing flows with a small number of parameters. The present approach can be also seen as the optimization of a set of intrinsic coordinates that best capture the underlying physics within the given basis set