Guozhi Dong

Guozhi Dong, Bert Juettler, Otmar Scherzer et al.

We study Tikhonov regularization for solving ill--posed operator equations where the solutions are functions defined on surfaces. One contribution of this paper is an error analysis of Tikhonov regularization which takes into account perturbations of the surfaces, in particular when the surfaces are approximated by spline surfaces. Another contribution is that we highlight the analysis of regularization for functions with range in vector bundles over surfaces. We also present some practical applications, such as an inverse problem of gravimetry and an imaging problem for denoising vector fields on surfaces, and show the numerical verification.

Guozhi Dong, Otmar Scherzer

In this paper we derive nonlinear evolution equations associated with a class of non-convex energy functionals which can be used for correcting displacement errors in imaging data. We study properties of these filtering flows and provide experiments for correcting angular perturbations in tomographical data.

Kaifeng Yang, Guozhi Dong, Michael Affenzeller

Bayesian global optimization (BGO) is an efficient surrogate-assisted technique for problems involving expensive evaluations. A parallel technique can be used to parallelly evaluate the true-expensive objective functions in one iteration to boost the execution time. An effective and straightforward approach is to design an acquisition function that can evaluate the performance of a bath of multiple solutions, instead of a single point/solution, in one iteration. This paper proposes five alternatives of \emph{Probability of Improvement} (PoI) with multiple points in a batch (q-PoI) for multi-objective Bayesian global optimization (MOBGO), taking the covariance among multiple points into account. Both exact computational formulas and the Monte Carlo approximation algorithms for all proposed q-PoIs are provided. Based on the distribution of the multiple points relevant to the Pareto-front, the position-dependent behavior of the five q-PoIs is investigated. Moreover, the five q-PoIs are compared with the other nine state-of-the-art and recently proposed batch MOBGO algorithms on twenty bio-objective benchmarks. The empirical experiments on different variety of benchmarks are conducted to demonstrate the effectiveness of two greedy q-PoIs ($\kpoi_{\mbox{best}}$ and $\kpoi_{\mbox{all}}$) on low-dimensional problems and the effectiveness of two explorative q-PoIs ($\kpoi_{\mbox{one}}$ and $\kpoi_{\mbox{worst}}$) on high-dimensional problems with difficult-to-approximate Pareto front boundaries.

Guozhi Dong, Hailong Guo

This paper investigates gradient recovery schemes for data defined on discretized manifolds. The proposed method, parametric polynomial preserving recovery (PPPR), does not require the tangent spaces of the exact manifolds, and they have been assumed for some significant gradient recovery methods in the literature. Another advantage of PPPR is that superconvergence is guaranteed without the symmetric condition which has been asked in the existing techniques. There is also numerical evidence that the superconvergence by PPPR is high curvature stable, which distinguishes itself from the others. As an application, we show its capability of constructing an asymptotically exact \textit{a posteriori} error estimator. Several numerical examples on two-dimensional surfaces are presented to support the theoretical results and comparisons with existing methods are documented.

Guozhi Dong, Michael Hintermueller, Kostas Papafitsoros

Inspired by applications in optimal control of semilinear elliptic partial differential equations and physics-integrated imaging, differential equation constrained optimization problems with constituents that are only accessible through data-driven techniques are studied. A particular focus is on the analysis and on numerical methods for problems with machine-learned components. For a rather general context, an error analysis is provided, and particular properties resulting from artificial neural network based approximations are addressed. Moreover, for each of the two inspiring applications analytical details are presented and numerical results are provided.