Homogenization and Convergence Rates for Periodic Parabolic Equations with Highly Oscillating Potentials

This paper considers a family of second-order periodic parabolic equations with highly oscillating potentials, which have been considered many times for the time-varying potentials in stochastic homogenization. Following a standard two-scale expansions illusion, we can guess and succeed in determining the homogenized equation in different cases that the potentials satisfy the corresponding assumptions, based on suitable uniform estimates of the $L^2(0,T;H^1(\Omega))$-norm for the solutions. To handle the more singular case and obtain the convergence rates in $L^\infty(0,T;L^2(\Omega))$, we need to estimate the Hessian term as well as the t-derivative term more exactly, which may be depend on $\varepsilon$. The difficulty is to find suitable uniform estimates for the $L^2(0,T;H^1(\Omega))$-norm and suitable estimates for the higher order derivative terms

A New Perspective for the Rationality of Fossil Fuel Divestment - the Interaction between the Shifting of Capital Flow and Stranded Assets

In recent years, fossil fuel divestment has shown increasing popularity among individual and institutional investors. Investors adopt this strategy due to two types of consideration, which are moral and financial concerns. Out of moral concerns, investors want to shift the capital flow to help limit the fossil fuel companies’ business development and capital expansion, and out of financial concerns, investors want to avoid the financial risks coming from the stranded asset. However, the effectiveness of fossil fuel divestment as one of the climate actions is debated and challenged. The previous studies mainly focused on assessing the financial performance and carbon intensity of divestment or divest-reinvest portfolios to answer the question of “whether divestment can fulfill both financial and moral obligations”, and these studies received mixed findings. This study supports the rationality of fossil fuel divestment and reinvestment movements by supporting the two concepts, shifting of capital flow and stranded assets, which respectively relate to the moral and financial concerns. The direction of “causation” between these two concepts is also explored. The study focuses on the fossil fuel industry and the green energy industry, and the research sample selects 70 green energy companies from Nasdaq Clean Edge Green Energy Index (CELS) and 90 fossil fuel companies from Carbon Underground 200. Firstly, by applying the production theory and the Cobb-Douglas production function, two models are built with various factors of production or various financing methods with Ordinary Least Squares regression (OLS) on panel data of the two industries from 2013 - 2018. Secondly, the Granger Causation test is employed to explore the interaction between the various input factors of production and the industrial output, which supports the two concepts, shifting of capital flow and stranded asset. The interactions between these two concepts and the two concerns are also discussed. In addition, lagged OLS is employed to tackle the potential endogeneity issue in the regression model. The findings of this study are in line with previous studies. The descriptive statistics show that the green energy industry can be a growing industry, while the fossil fuel industry can be a mature and probably, declining industry. The transition to a low-carbon economy may accelerate this process. The regression identifies the factors which have significant influences on the output of the fossil fuel industry and the green energy industry. The Granger Causation test discovers the “bi-directional causality” (or “bi-directional feedback”) between the “market demand & industrial output” and the “various factors of production (model one) and various financing methods (model two)”, proving the two concepts of shifting of capital flow and stranded assets. Besides, the study explains the bi-directional interaction between these two concepts and finds that the shifting of capital flow precedes (contributes to, helps to predict) the stranded assets. At the same time, the stranded assets also precede (contributes to, helps to predict) the shifting of capital flow. This study provides support to the rationality of carbon divestment from a new perspective. Compared to previous research, this study is not to explore the effectiveness of divestment from the perspective of market performance data, but to support the effectiveness by explaining the mechanism of this strategy. Besides the academic value, the models built in this study also have practical values. The two models, which are respectively built with various factors of production and various financing sources, are referable for the low-carbon economy transition and the development of responsible investment products

You Only Propagate Once: Accelerating Adversarial Training via Maximal Principle

Deep learning achieves state-of-the-art results in many tasks in computer vision and natural language processing. However, recent works have shown that deep networks can be vulnerable to adversarial perturbations, which raised a serious robustness issue of deep networks. Adversarial training, typically formulated as a robust optimization problem, is an effective way of improving the robustness of deep networks. A major drawback of existing adversarial training algorithms is the computational overhead of the generation of adversarial examples, typically far greater than that of the network training. This leads to the unbearable overall computational cost of adversarial training. In this paper, we show that adversarial training can be cast as a discrete time differential game. Through analyzing the Pontryagin's Maximal Principle (PMP) of the problem, we observe that the adversary update is only coupled with the parameters of the first layer of the network. This inspires us to restrict most of the forward and back propagation within the first layer of the network during adversary updates. This effectively reduces the total number of full forward and backward propagation to only one for each group of adversary updates. Therefore, we refer to this algorithm YOPO (You Only Propagate Once). Numerical experiments demonstrate that YOPO can achieve comparable defense accuracy with approximately 1/5 ~ 1/4 GPU time of the projected gradient descent (PGD) algorithm. Our codes are available at https://https://github.com/a1600012888/YOPO-You-Only-Propagate-Once.Comment: Accepted as a conference paper at NeurIPS 201

Convergence Rates for the Stationary and Non-stationary Navier-Stokes Equations over Non-Lipschitz Boundaries

In this paper, we consider the higher-order convergence rates for the 2D stationary and non-stationary Navier-Stokes Equations over highly oscillating periodic bumpy John domains with $C^{2}$ regularity in some neighborhood of the boundary point (0,0). For the stationary case and any $\gamma\in (0,1/2)$, using the variational equation satisfied by the solution and the correctors for the bumpy John domains obtained by Higaki, Prange and Zhuge \cite{higaki2021large,MR4619004} after correcting the values on the inflow/outflow boundaries $(\{0\}\cup\{1\})\times(0,1)$, we can obtain an $O(\varepsilon^{2-\gamma})$ approximation in $L^2$ for the velocity and an $O(\varepsilon^{2-\gamma})$ convergence rates in $L^1$ approximated by the so called Navier's wall laws, which generalized the results obtained by J\"{a}ger and Mikeli\'{c} \cite{MR1813101}. Moreover, for the non-stationary case, using the energy method, we can obtain an $O(\varepsilon^{2-\gamma}+\exp(-Ct))$ convergence rate for the velocity in $L_x^1$

Extension of First Passage Probability

In this paper, we consider the extension of first passage probability. First, we present the first, second, third, and generally k-th passage probability of a Markov Chain moving from one state to another state through step-by-step calculation and two other matrix-version methods. Similarly, we compute the first passage probability of a Markov Chain moving from one state to multiple states. In all discussions, we take into account the situations that one state moves to a different state and returns to itself. Also, we find the mean number of steps needed from one state to another state in a Markov Chain for the first, second, third, and generally k-th passage. Besides, we find the probability generating function for the number of steps. This makes the calculation of passage probabilities, mean and variance of passage steps, easier. Additionally, if we extend a discrete Markov Chain to its corresponding continuous Markov Process with the same transition probabilities and transition time in the form of an exponential distribution with parameter 1 between two states, we can obtain the mean time needed from one state to another state by Laplace Transforms, which is the same as with the discrete situation. Subsequently, we can calculate the variance of the time needed from one state to another state in the same way

Effects of clouds on aerosol and chemical species processing, production, and distribution in the boundary layer and upper troposphere

September 1998.Also issued as Yiping Zhang's dissertation (Ph.D.) -- Colorado State University, 1998.Includes bibliographical references.Clouds play important roles in boundary layer and tropospheric aerosol and chemical processes. This work addresses the aerosol and chemical species processing, production, and distribution through two important types of clouds: convective and stratocumulus clouds. A modeling study of the effects of convection on the transformation and redistribution of chemical species and evolution and redistribution of aerosol particles in the troposphere is presented. A two-mode, two-moment aerosol evolution model is coupled with a two­ dimensional, mixed-phase, two-moment microphysics, Eulerian cloud model and a sulfate cloud chemistry model [Kreidenweis et al., 1997; Taylor et al., 1997; Zhang et al., 1998] to examine the new particle formation mechanism and the importance of different pathways for aqueous sulfate production. In the simulations, the complexation of CH20 with S(IV) is found to be of minor importance in most of the model cloud, compared with the oxidation of S(IV) by H202 and 03, while Fe (III)-catalyzed oxidation plays an important role in aqueous sulfate production. Significant S02 is convectively transported to the mid-to­ upper troposphere, where it is oxidized to gas-phase H2S04. After cloud processing, cloud condensation nuclei (CCN) particles are removed by precipitation and graupel to form a CCN-depleted region above cloud top and in the cold and humidified cloud outflow region. The new particle formation in the mid- to upper- troposphere interacts with cloud processing and transport of chemical species and aerosol particles and produces a peak of small particle concentration in the outflow region. The model results suggest that both small aerosols and aerosol precursors can be transported into the mid- to upper- troposphere by convective clouds, affecting vertical profiles of aerosol concentrations. The sensitivity of the S(VI) and aerosol production, S02 and aerosol redistribution to variations in the initial chemical and aerosol conditions and several model parameters are also examined. A trajectory ensemble model (TEM) is used to investigate stratocumulus processing of gases and C CN in the boundary layer. The fully coupled aqueous chemistry/ cloud microphysics model (Feingold et al., 1998; Zhang et al., 1998] is driven by a set of boundary layer parcel trajectories derived from a large eddy simulation model to study the effects of variations in the initial chemical fields and initial aerosol number concentration on chemical heterogeneity, broadening of the CCN and drop spectra, effective drop radius, and differences in the overall fractional conversion between the TEM and a single parcel experiencing mean conditions in a stratocumulus-capped marine boundary layer. It is found that the TEM offers a more representative method of describing the stratocumulus processing of aerosol and gases than does a single parcel model. In the base case simulation, the 03 oxidation rate averaged over all parcels is larger than the H202 oxidation rate, whereas the volume-mean cloudwater pH might suggest that H202 oxidation dominates. The liquid water-weighted pH generally increases with increasing drop size, to a peak pH. The drop size at this peak corresponds to the minimum in S(VI) concentration and is located near the mode of the drop mass distribution. However, the pH dependence on drop size at larger cloud drop sizes is affected by the initial chemical conditions. Aqueous chemistry contributes to the broadening of the drop size distribution, but the magnitude of the broadening depends on the initial aerosol and chemical conditions. In cases where more mass is added onto large particles in the tail of the initial CCN spectrum, the broadening of the drop spectrum is most evident, and may even trigger the collision coalescence process and drizzle formation in stratocumulus clouds. The change in initial CCN number concentration has the most prominent effect on the effective drop radius.Sponsored by NSF under grant ATM 9305684; DOE-ARM under grant DE-FG03-95ER61958; and DOC-NOAA under grant NA67RJ0152