An Efficient Operator-Splitting Method for the Eigenvalue Problem of the Monge-Amp\`{e}re Equation

We develop an efficient operator-splitting method for the eigenvalue problem of the Monge-Amp\`{e}re operator in the Aleksandrov sense. The backbone of our method relies on a convergent Rayleigh inverse iterative formulation proposed by Abedin and Kitagawa (Inverse iteration for the {M}onge-{A}mp{\`e}re eigenvalue problem, {\it Proceedings of the American Mathematical Society}, 148 (2020), no. 11, 4975-4886). Modifying the theoretical formulation, we develop an efficient algorithm for computing the eigenvalue and eigenfunction of the Monge-Amp\`{e}re operator by solving a constrained Monge-Amp\`{e}re equation during each iteration. Our method consists of four essential steps: (i) Formulate the Monge-Amp\`{e}re eigenvalue problem as an optimization problem with a constraint; (ii) Adopt an indicator function to treat the constraint; (iii) Introduce an auxiliary variable to decouple the original constrained optimization problem into simpler optimization subproblems and associate the resulting new optimization problem with an initial value problem; and (iv) Discretize the resulting initial-value problem by an operator-splitting method in time and a mixed finite element method in space. The performance of our method is demonstrated by several experiments. Compared to existing methods, the new method is more efficient in terms of computational cost and has a comparable rate of convergence in terms of accuracy

On the Numerical Solution of Nonlinear Eigenvalue Problems for the Monge-Amp\`{e}re Operator

In this article, we report the results we obtained when investigating the numerical solution of some nonlinear eigenvalue problems for the Monge-Amp\`{e}re operator $v\rightarrow \det \mathbf{D}^2 v$. The methodology we employ relies on the following ingredients: (i) A divergence formulation of the eigenvalue problems under consideration. (ii) The time discretization by operator-splitting of an initial value problem (a kind of gradient flow) associated with each eigenvalue problem. (iii) A finite element approximation relying on spaces of continuous piecewise affine functions. To validate the above methodology, we applied it to the solution of problems with known exact solutions: The results we obtained suggest convergence to the exact solution when the space discretization step $h\rightarrow 0$. We considered also test problems with no known exact solutions

A Note on time regularity of generalized Ornstein–Uhlenbeck processes with cylindrical stable noise

A necessary and sufficient condition of cadlag modification of Ornstein-Uhlenbeck process with cylindrical stable noise in a Hilbert space is given in this Note. Applying this result. some questions in Time irregularity of generalized Ornstein-Uhlenbeck processes [C. R. Acad. Sci. Paris, Ser. I 348 (2010) 273-276] and Structural properties of semilinear SPDEs driven by cylindrical stable process [Probab. Theory Related Fields 149 (2011) 97-137] are answered. (C) 2011 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.MathematicsSCI(E)0ARTICLE1-297-10035

Sustainability assessment of bioenergy from a global perspective: a review

Bioenergy, as a renewable energy resource, is expected to see significant development in the future. However, a key issue that will affect this trend is sustainability of bioenergy. There have been many studies on this topic, but mainly focusing on only one- or two-dimensions of the issue, and also with much of the literature directed at studies of European regions. To help understand the wider scope of bioenergy sustainability, this paper reviews a broad range of current research on the topic, and places the literature into a multi-dimensional framework covering the economic, environmental and ecological, social, and land-related aspects of bioenergy sustainability, as well as a geographical analysis of the areas for which the studies have been carried out. The review indicates that it is hard to draw an overall conclusion on the sustainability of bioenergy because of limited studies or contradictory results in some aspects. In addition, this review shows that crop-based bioenergy and forest bioenergy are seen as the main sources of bioenergy, and that most studies discuss the final utilization of bioenergy as being for electricity generation. Finally, research directions for future study are suggested, based on the literature reviewed here

Water use for shale gas extraction in the Sichuan Basin, China

This study investigates the use of water for extracting shale gas in the Sichuan Basin of China. Both net water use and water intensity (i.e., water use per unit of gas produced) of shale wells are estimated by applying a process-based life cycle inventory (LCI) model. The results show that the net water use and water intensity are around 24500 m3/well and 1.9 m3 water/104m3 gas respectively, and that the fracturing and completion stage of shale gas extraction accounts for the largest share in net water use. A comparison shows that China's water use for shale gas extraction is generally higher than that of other countries. By considering the predicted annual drilling activities in the Sichuan Basin, we find that the annual water demand for shale gas development is likely to be negligible compared to total regional water supply. However, considering the water demand for shale gas extraction and the water demand from other sectors may make water availability a significant concern for China's shale gas development in the future

Unleashing Mask: Explore the Intrinsic Out-of-Distribution Detection Capability

Out-of-distribution (OOD) detection is an indispensable aspect of secure AI when deploying machine learning models in real-world applications. Previous paradigms either explore better scoring functions or utilize the knowledge of outliers to equip the models with the ability of OOD detection. However, few of them pay attention to the intrinsic OOD detection capability of the given model. In this work, we generally discover the existence of an intermediate stage of a model trained on in-distribution (ID) data having higher OOD detection performance than that of its final stage across different settings, and further identify one critical data-level attribution to be learning with the atypical samples. Based on such insights, we propose a novel method, Unleashing Mask, which aims to restore the OOD discriminative capabilities of the well-trained model with ID data. Our method utilizes a mask to figure out the memorized atypical samples, and then finetune the model or prune it with the introduced mask to forget them. Extensive experiments and analysis demonstrate the effectiveness of our method. The code is available at: https://github.com/tmlr-group/Unleashing-Mask.Comment: accepted by ICML 202