A robust error estimator and a residual-free error indicator for reduced basis methods

The Reduced Basis Method (RBM) is a rigorous model reduction approach for solving parametrized partial differential equations. It identifies a low-dimensional subspace for approximation of the parametric solution manifold that is embedded in high-dimensional space. A reduced order model is subsequently constructed in this subspace. RBM relies on residual-based error indicators or {\em a posteriori} error bounds to guide construction of the reduced solution subspace, to serve as a stopping criteria, and to certify the resulting surrogate solutions. Unfortunately, it is well-known that the standard algorithm for residual norm computation suffers from premature stagnation at the level of the square root of machine precision. In this paper, we develop two alternatives to the standard offline phase of reduced basis algorithms. First, we design a robust strategy for computation of residual error indicators that allows RBM algorithms to enrich the solution subspace with accuracy beyond root machine precision. Secondly, we propose a new error indicator based on the Lebesgue function in interpolation theory. This error indicator does not require computation of residual norms, and instead only requires the ability to compute the RBM solution. This residual-free indicator is rigorous in that it bounds the error committed by the RBM approximation, but up to an uncomputable multiplicative constant. Because of this, the residual-free indicator is effective in choosing snapshots during the offline RBM phase, but cannot currently be used to certify error that the approximation commits. However, it circumvents the need for \textit{a posteriori} analysis of numerical methods, and therefore can be effective on problems where such a rigorous estimate is hard to derive

Optimally convergent hybridizable discontinuous Galerkin method for fifth-order Korteweg-de Vries type equations

We develop and analyze the first hybridizable discontinuous Galerkin (HDG) method for solving fifth-order Korteweg-de Vries (KdV) type equations. We show that the semi-discrete scheme is stable with proper choices of the stabilization functions in the numerical traces. For the linearized fifth-order equations, we prove that the approximations to the exact solution and its four spatial derivatives as well as its time derivative all have optimal convergence rates. The numerical experiments, demonstrating optimal convergence rates for both the linear and nonlinear equations, validate our theoretical findings

Hybrid Projection Methods for Large-scale Inverse Problems with Mixed Gaussian Priors

When solving ill-posed inverse problems, a good choice of the prior is critical for the computation of a reasonable solution. A common approach is to include a Gaussian prior, which is defined by a mean vector and a symmetric and positive definite covariance matrix, and to use iterative projection methods to solve the corresponding regularized problem. However, a main challenge for many of these iterative methods is that the prior covariance matrix must be known and fixed (up to a constant) before starting the solution process. In this paper, we develop hybrid projection methods for inverse problems with mixed Gaussian priors where the prior covariance matrix is a convex combination of matrices and the mixing parameter and the regularization parameter do not need to be known in advance. Such scenarios may arise when data is used to generate a sample prior covariance matrix (e.g., in data assimilation) or when different priors are needed to capture different qualities of the solution. The proposed hybrid methods are based on a mixed Golub-Kahan process, which is an extension of the generalized Golub-Kahan bidiagonalization, and a distinctive feature of the proposed approach is that both the regularization parameter and the weighting parameter for the covariance matrix can be estimated automatically during the iterative process. Furthermore, for problems where training data are available, various data-driven covariance matrices (including those based on learned covariance kernels) can be easily incorporated. Numerical examples from tomographic reconstruction demonstrate the potential for these methods

No-Reference Quality Assessment for 360-degree Images by Analysis of Multi-frequency Information and Local-global Naturalness

360-degree/omnidirectional images (OIs) have achieved remarkable attentions due to the increasing applications of virtual reality (VR). Compared to conventional 2D images, OIs can provide more immersive experience to consumers, benefitting from the higher resolution and plentiful field of views (FoVs). Moreover, observing OIs is usually in the head mounted display (HMD) without references. Therefore, an efficient blind quality assessment method, which is specifically designed for 360-degree images, is urgently desired. In this paper, motivated by the characteristics of the human visual system (HVS) and the viewing process of VR visual contents, we propose a novel and effective no-reference omnidirectional image quality assessment (NR OIQA) algorithm by Multi-Frequency Information and Local-Global Naturalness (MFILGN). Specifically, inspired by the frequency-dependent property of visual cortex, we first decompose the projected equirectangular projection (ERP) maps into wavelet subbands. Then, the entropy intensities of low and high frequency subbands are exploited to measure the multi-frequency information of OIs. Besides, except for considering the global naturalness of ERP maps, owing to the browsed FoVs, we extract the natural scene statistics features from each viewport image as the measure of local naturalness. With the proposed multi-frequency information measurement and local-global naturalness measurement, we utilize support vector regression as the final image quality regressor to train the quality evaluation model from visual quality-related features to human ratings. To our knowledge, the proposed model is the first no-reference quality assessment method for 360-degreee images that combines multi-frequency information and image naturalness. Experimental results on two publicly available OIQA databases demonstrate that our proposed MFILGN outperforms state-of-the-art approaches

Robust residual-guided iterative reconstruction for sparse-view CT in small animal imaging

Objective. We introduce a robust image reconstruction algorithm named residual-guided Golub–Kahan iterative reconstruction technique (RGIRT) designed for sparse-view computed tomography (CT), which aims at high-fidelity image reconstruction from a limited number of projection views. Approach. RGIRT utilizes an inner-outer dual iteration framework, with a flexible least square QR (FLSQR) algorithm implemented in the inner iteration and a restarted iterative scheme applied in the outer iteration. The inner FLSQR employs a flexible Golub–Kahan bidiagonalization method to reduce the size of the inverse problem, and a weighted generalized cross-validation method to adaptively estimate the regularization hyper-parameter. The inner iteration efficiently yields the intermediate reconstruction result, while the outer iteration minimizes the residual and refines the solution by using the result obtained from the inner iteration. Main results. The reconstruction performance of RGIRT is evaluated and compared to other reference methods (FBPConvNet, SART-TV, and FLSQR) using projection data from both numerical phantoms and real experimental Micro-CT data. The experimental findings, from testing various numbers of projection views and different noise levels, underscore the robustness of RGIRT. Meanwhile, theoretical analysis confirms the convergence of residual for our approach. Significance. We propose a robust iterative reconstruction algorithm for x-ray CT scans with sparse views, thereby shortening scanning time and mitigating excessive ionizing radiation exposure to small animals

Diesel degradation capability and environmental robustness of strain Pseudomonas aeruginosa WS02

Petroleum hydrocarbon (PHC) degrading bacteria have been frequently discovered. However, in practical application, a single species of PHC degrading bacterium with weak competitiveness may face environmental pressure and competitive exclusion due to the interspecific competition between petroleum-degrading bacteria as well as indigenous microbiota in soil, leading to a reduced efficacy or even malfunction. In this study, the diesel degradation ability and environmental robustness of an endophytic strain Pseudomonas aeruginosa WS02, were investigated. The results show that the cell membrane surface of WS02 was highly hydrophobic, and the strain secreted glycolipid surfactants. Genetic analysis results revealed that WS02 contained multiple metabolic systems and PHC degradation-related genes, indicating that this strain theoretically possesses the capability of oxidizing both alkanes and aromatic hydrocarbons. Gene annotation also showed many targets which coded for heavy metal resistant and metal transporter proteins. The gene annotation-based inference was confirmed by the experimental results: GC-MS analysis revealed that short chain PHCs (C10–C14) were completely degraded, and the degradation of PHCs ranging from C15–C22 were above 90% after 14 d in diesel-exposed culture; Heavy metal (Mn2+, Pb2+ and Zn2+) exposure was found to affect the growth of WS02 to some extent, but not its ability to degrade diesel, and the degradation efficiency was still maintained at 39–59%. WS02 also showed a environmental robustness along with PHC-degradation performance in the co-culture system with other bacterial strains as well as in the co-cultured system with the indigenous microbiota in soil fluid extracted from a PHC-contaminated site. It can be concluded that the broad-spectrum diesel degradation efficacy and great environmental robustness give P. aeruginosa WS02 great potential for application in the remediation of PHC-contaminated soil.<br/