Zixi Hu

Zixi Hu, Zhewei Yao, Jinglai Li

Many scientific and engineering problems require to perform Bayesian inferences for unknowns of infinite dimension. In such problems, many standard Markov Chain Monte Carlo (MCMC) algorithms become arbitrary slow under the mesh refinement, which is referred to as being dimension dependent. To this end, a family of dimensional independent MCMC algorithms, known as the preconditioned Crank-Nicolson (pCN) methods, were proposed to sample the infinite dimensional parameters. In this work we develop an adaptive version of the pCN algorithm, where the covariance operator of the proposal distribution is adjusted based on sampling history to improve the simulation efficiency. We show that the proposed algorithm satisfies an important ergodicity condition under some mild assumptions. Finally we provide numerical examples to demonstrate the performance of the proposed method.

Qingping Zhou, Zixi Hu, Zhewei Yao et al.

The preconditioned Crank-Nicolson (pCN) method is a Markov Chain Monte Carlo (MCMC) scheme, specifically designed to perform Bayesian inferences in function spaces. Unlike many standard MCMC algorithms, the pCN method can preserve the sampling efficiency under the mesh refinement, a property referred to as being dimension independent. In this work we consider an adaptive strategy to further improve the efficiency of pCN. In particular we develop a hybrid adaptive MCMC method: the algorithm performs an adaptive Metropolis scheme in a chosen finite dimensional subspace, and a standard pCN algorithm in the complement space of the chosen subspace. We show that the proposed algorithm satisfies certain important ergodicity conditions. Finally with numerical examples we demonstrate that the proposed method has competitive performance with existing adaptive algorithms.

Jeffrey J. Donatelli, Miaoqi Chu, Zixi Hu et al.

Coherent surface scattering imaging (CSSI) is an emerging experimental technique uniquely suited to probing the structure of thin nanostructures. In these experiments, a specimen is placed on a substrate, and a series of X-ray diffraction patterns is collected at grazing incidence angles as the specimen is rotated. However, reconstructing the specimen's 3D structure from the data is challenging due to dynamical scattering effects induced by the experimental geometry and the lack of direct phase measurements. Specifically, the data involves nonuniformly sampled Fourier-transform values of the specimen density, and failure to effectively address this nonuniformity can lead to errors or degraded performance. Here we introduce a mathematical inversion framework that combines iterative-projection-based phasing techniques with new fast nonuniform Fourier inversion methods to efficiently reconstruct isolated 3D structures from their CSSI rotation-series data. We also analyze the theoretical properties of CSSI reconstruction to derive requirements on experimental parameters and characterize solution uniqueness. We validate our approach using CSSI data simulated from a conical Siemens star and a porous medium, demonstrating that high-resolution 3D structures can be reconstructed even in the presence of significant dynamical scattering, from data collected at as few as one or two incident angles. More broadly, the presented nonuniform reconstruction framework provides a foundation for solving challenging generalizations of the phase problem in which measurements involve nonlinear combinations of nonuniformly sampled Fourier values.

Kai Rothauge, Zhewei Yao, Zixi Hu et al.

We regard pre-trained residual networks (ResNets) as nonlinear systems and use linearization, a common method used in the qualitative analysis of nonlinear systems, to understand the behavior of the networks under small perturbations of the input images. We work with ResNet-56 and ResNet-110 trained on the CIFAR-10 data set. We linearize these networks at the level of residual units and network stages, and the singular value decomposition is used in the stability analysis of these components. It is found that most of the singular values of the linearizations of residual units are 1 and, in spite of the fact that the linearizations depend directly on the activation maps, the singular values differ only slightly for different input images. However, adjusting the scaling of the skip connection or the values of the weights in a residual unit has a significant impact on the singular value distributions. Inspection of how random and adversarial perturbations of input images propagate through the network reveals that there is a dramatic jump in the magnitude of adversarial perturbations towards the end of the final stage of the network that is not present in the case of random perturbations. We attempt to gain a better understanding of this phenomenon by projecting the perturbations onto singular vectors of the linearizations of the residual units.