An enhanced and highly efficient semi-implicit combined Lagrange multiplier approach with preserving original energy law for dissipative systems

Recently, a new Lagrange multiplier approach was introduced by Cheng, Liu and Shen in \cite{cheng2020new}, which has been broadly used to solve various challenging phase field problems. To design original energy stable schemes, they have to solve a nonlinear algebraic equation to determine the introduced Lagrange multiplier, which can be computationally expensive, especially for large-scale and long-time simulations involving complex nonlinear terms. This paper presents an essential improved technique to modify this issue, which can be seen as a semi-implicit combined Lagrange multiplier approach. In general, the new constructed schemes keep all the advantages of the Lagrange multiplier method and significantly reduce the computation costs. Besides, the new proposed BDF2 scheme dissipates the original energy, as opposed to a modified energy for the classical Lagrange multiplier approach in \cite{cheng2020new}. We further construct high-order BDF$k$ schemes based on the new proposed approach. In addition, we establish a general framework for extending our constructed method to dissipative systems. Finally several examples have been presented to demonstrate the effectiveness of the proposed approach

High-efficiency and positivity-preserving stabilized SAV methods for gradient flows

The scalar auxiliary variable (SAV)-type methods are very popular techniques for solving various nonlinear dissipative systems. Compared to the semi-implicit method, the baseline SAV method can keep a modified energy dissipation law but doubles the computational cost. The general SAV approach does not add additional computation but needs to solve a semi-implicit solution in advance, which may potentially compromise the accuracy and stability. In this paper, we construct a novel first- and second-order unconditional energy stable and positivity-preserving stabilized SAV (PS-SAV) schemes for $L^2$ and $H^{-1}$ gradient flows. The constructed schemes can reduce nearly half computational cost of the baseline SAV method and preserve its accuracy and stability simultaneously. Meanwhile, the introduced auxiliary variable is always positive while the baseline SAV cannot guarantee this positivity-preserving property. Unconditionally energy dissipation laws are derived for the proposed numerical schemes. We also establish a rigorous error analysis of the first-order scheme for the Allen-Cahn type equation in $l^{\infty}(0,T; H^1(\Omega) )$ norm. In addition we propose an energy optimization technique to optimize the modified energy close to the original energy. Several interesting numerical examples are presented to demonstrate the accuracy and effectiveness of the proposed methods

From text to effectiveness:Quantifying green industrial policies in China

The evolution of green industrial policy in China is deeply embedded within a unique political, economic, cultural, and social milieu. The intricacies and complexities inherent in this context present challenges for quantitative policy research. In the existing literature, the study of thematic clustering of green industry policy instruments and their effectiveness measures in large-scale policy text repositories remains a relatively unexplored area. To fill this research gap, this study provides an analysis of China's green industrial policies from 1994 to 2022. By adopting the Latent Dirichlet Allocation (LDA) topic model, implicit and optimal themes are identified, and key policy instruments are systematically and comprehensively parsed. Based on this, the cutting-edge Policy Modeling Consistency-Text Encoder (PMC-TE) index model was used to conduct an in-depth quantitative evaluation of various policy tools. The findings reveal 15 policy instruments from 7891 Chinese green industry policy documents. Notably, technical support, information resource demonstration project policies emerged as the most frequently deployed and efficacious. Conversely, government procurement and service outsourcing policies, while showing incremental growth year-on-year, lagged in utilization and effectiveness. These policy instruments can be broadly categorized into three primary archetypes: supply-type, demand-type, and environment-type. The supply-type instruments dominate, while environment-type instruments remain underutilized. Additionally, this study delineates the four chronological phases of China's green industrial policy evolution, illuminating China's strategic pivots-transitioning from resource allocation, to market dynamics, and ultimately to environmental governance trajectories

Blow-Up of Solutions to a Novel Two-Component Rod System

We consider a novel two-component rod system which is closely connected to the shallow water theory. The present work is mainly concerned with the blow-up mechanism of strong solutions; we establish new conditions in view of some special classes of initial value to guarantee finite time blow-up of solutions

Harmonizing Output Imbalance for semantic segmentation on extremely-imbalanced input data

Semantic segmentation is a high level computer vision task that assigns a label for each pixel of an image. It is challenging to deal with extremely-imbalanced data in which the ratio of target pixels to background pixels is lower than 1:1000. Such severe input imbalance leads to output imbalance for poor model training. This paper considers three issues for extremely-imbalanced data: inspired by the region-based Dice loss, an implicit measure for the output imbalance is proposed, and an adaptive algorithm is designed for guiding the output imbalance hyperparameter selection; then it is generalized to distribution-based loss for dealing with output imbalance; and finally a compound loss with our adaptive hyperparameter selection algorithm can keep the consistency of training and inference for harmonizing the output imbalance. With four popular deep architectures on our private dataset from three different input imbalance scales and three public datasets, extensive experiments demonstrate the competitive/promising performance of the proposed method.Comment: 18 pages, 13 figures, 2 appendixe