The Hessian Riemannian flow and Newton's method for Effective Hamiltonians and Mather measures

Effective Hamiltonians arise in several problems, including homogenization of Hamilton--Jacobi equations, nonlinear control systems, Hamiltonian dynamics, and Aubry--Mather theory. In Aubry--Mather theory, related objects, Mather measures, are also of great importance. Here, we combine ideas from mean-field games with the Hessian Riemannian flow to compute effective Hamiltonians and Mather measures simultaneously. We prove the convergence of the Hessian Riemannian flow in the continuous setting. For the discrete case, we give both the existence and the convergence of the Hessian Riemannian flow. In addition, we explore a variant of Newton's method that greatly improves the performance of the Hessian Riemannian flow. In our numerical experiments, we see that our algorithms preserve the non-negativity of Mather measures and are more stable than {related} methods in problems that are close to singular. Furthermore, our method also provides a way to approximate stationary MFGs.Comment: 24 page

Two-scale homogenization of a stationary mean-field game

In this paper, we characterize the asymptotic behavior of a first-order stationary mean-field game (MFG) with a logarithm coupling, a quadratic Hamiltonian, and a periodically oscillating potential. This study falls into the realm of the homogenization theory, and our main tool is the two-scale convergence. Using this convergence, we rigorously derive the two-scale homogenized and the homogenized MFG problems, which encode the so-called macroscopic or effective behavior of the original oscillating MFG. Moreover, we prove existence and uniqueness of the solution to these limit problems.Comment: 36 page

A Mini-Batch Method for Solving Nonlinear PDEs with Gaussian Processes

Gaussian processes (GPs) based methods for solving partial differential equations (PDEs) demonstrate great promise by bridging the gap between the theoretical rigor of traditional numerical algorithms and the flexible design of machine learning solvers. The main bottleneck of GP methods lies in the inversion of a covariance matrix, whose cost grows cubically concerning the size of samples. Drawing inspiration from neural networks, we propose a mini-batch algorithm combined with GPs to solve nonlinear PDEs. The algorithm takes a mini-batch of samples at each step to update the GP model. Thus, the computational cost is allotted to each iteration. Using stability analysis and convexity arguments, we show that the mini-batch method steadily reduces a natural measure of errors towards zero at the rate of O(1/K + 1/M), where K is the number of iterations and M is the batch size. Numerical results show that smooth problems benefit from a small batch size, while less regular problems require careful sample selection for optimal accuracy.Comment: 19 pages, 4 figure

Sparse Gaussian processes for solving nonlinear PDEs

This article proposes an efficient numerical method for solving nonlinear partial differential equations (PDEs) based on sparse Gaussian processes (SGPs). Gaussian processes (GPs) have been extensively studied for solving PDEs by formulating the problem of finding a reproducing kernel Hilbert space (RKHS) to approximate a PDE solution. The approximated solution lies in the span of base functions generated by evaluating derivatives of different orders of kernels at sample points. However, the RKHS specified by GPs can result in an expensive computational burden due to the cubic computation order of the matrix inverse. Therefore, we conjecture that a solution exists on a ``condensed" subspace that can achieve similar approximation performance, and we propose a SGP-based method to reformulate the optimization problem in the ``condensed" subspace. This significantly reduces the computation burden while retaining desirable accuracy. The paper rigorously formulates this problem and provides error analysis and numerical experiments to demonstrate the effectiveness of this method. The numerical experiments show that the SGP method uses fewer than half the uniform samples as inducing points and achieves comparable accuracy to the GP method using the same number of uniform samples, resulting in a significant reduction in computational cost. Our contributions include formulating the nonlinear PDE problem as an optimization problem on a ``condensed" subspace of RKHS using SGP, as well as providing an existence proof and rigorous error analysis. Furthermore, our method can be viewed as an extension of the GP method to account for general positive semi-definite kernels.Comment: 31 page

Terephthalic acid–4,4′-bipyridine (2/1)

In the title compound, 2C8H6O4·C10H8N2, the 4,4′-bipyridine mol­ecule is located on an inversion centre. In the crystal structure, strong inter­molecular O—H⋯N hydrogen bonds between the terephthalic acid and 4,4′-bipyridine mol­ecules lead to the formation of chains with graph-set motif C 2 2(8) along the diagonal of the bc plane

catena-Poly[[bis­(p-toluene­sulfonato-κO)palladium(II)]bis­(μ-1,3-di-4-pyridylpropane-κ2 N:N′)]

In the title compound, [Pd(C7H7O3S)2(C13H14N2)2]n, the metal ion, located on a twofold rotation axis, exhibits a slightly distorted octa­hedral coordination environment, with bond angles that deviate by at most 2.2° from an ideal geometry, completed by two O atoms from two deprotonated p-toluene­sulfonic acid ligands and four N atoms from four 1,3-di-4-pyridylpropane ligands. One of the sulfonate O atoms is disordered over two positions [ratio 0.70 (5):0.30 (5)]

Forward-backward algorithm for functions with locally Lipschitz gradient: applications to mean field games

In this paper, we provide a generalization of the forward-backward splitting algorithm for minimizing the sum of a proper convex lower semicontinuous function and a differentiable convex function whose gradient satisfies a locally Lipschitztype condition. We prove the convergence of our method and derive a linear convergence rate when the differentiable function is locally strongly convex. We recover classical results in the case when the gradient of the differentiable function is globally Lipschitz continuous and an already known linear convergence rate when the function is globally strongly convex. We apply the algorithm to approximate equilibria of variational mean field game systems with local couplings. Compared with some benchmark algorithms to solve these problems, our numerical tests show similar performances in terms of the number of iterations but an important gain in the required computational time

The influence of local officials' promotion incentives on carbon emission in Yangtze River Delta, China

China's carbon emissions is heavily influenced by economic growth, which can be largely related to the local officials' promotion incentives. The current study was conducted to test the hypothesis that the influence of local officials' individual characteristics on carbon emissions was driven by the promotion incentives. Yangtze River Delta where carbon emissions accounted for around 13.95% of China's total emissions was selected as the research area. The multiple linear regression model was applied to determine the relationship between local officials' characteristics and total carbon emissions and carbon emissions from different sectors. The results indicated that local officials' promotion source, tenure and age significantly influenced the total carbon emission. Despite insignificance influence of officials' academic level on carbon emissions, the professional background in economics and management had a significant influence on carbon reduction. Our results indicated the importance of local officials' promotion incentives for carbon emission in China. Therefore, low carbon development should be included as an important part of official promotion system

Overtime work, job autonomy, and employees’ subjective well-being: Evidence from China

IntroductionChinese workers suffer more from overtime than in many countries. Excessive working hours can crowd out personal time and cause work-family imbalance, affecting workers’ subjective well-being. Meanwhile, self-determination theory suggests that higher job autonomy may improve the subjective well-being of employees.MethodsData came from the 2018 China Labor-force Dynamics Survey (CLDS 2018). The analysis sample consisted of 4,007 respondents. Their mean age was 40.71 (SD = 11.68), and 52.8% were males. This study adopted four measures of subjective well-being: happiness, life satisfaction, health status, and depression. Confirmation factor analysis was employed to extract the job autonomy factor. Multiple linear regression methods were applied to examine the relationship between overtime, job autonomy, and subjective well-being.ResultsOvertime hours showed weak association with lower happiness (β = −0.002, p < 0.01), life satisfaction (β = −0.002, p < 0.01), and health status (β = −0.002, p < 0.001). Job autonomy was positively related to happiness (β = 0.093, p < 0.01), life satisfaction (β = 0.083, p < 0.01). There was a significant negative correlation between involuntary overtime and subjective well-being. Involuntary overtime might decrease the level of happiness (β = −0.187, p < 0.001), life satisfaction (β = −0.221, p < 0.001), and health status (β = −0.129, p < 0.05) and increase the depressive symptoms (β = 1.157, p < 0.05).ConclusionWhile overtime had a minimal negative effect on individual subjective well-being, involuntary overtime significantly enlarged it. Improving individual’s job autonomy is beneficial for individual subjective well-being

Monitoring CO2 migration in a shallow sand aquifer using 3D crosshole electrical resistivity tomography

AbstractThree-dimensional (3D) crosshole electrical resistivity tomography (ERT) was used to monitor a pilot CO2 injection experiment at Vrøgum, western Denmark. The purpose was to evaluate the effectiveness of the ERT method for detection of small electrical conductivity (EC) changes during the first 2 days of CO2 injection in a shallow siliciclastic aquifer and to study the early-time behavior of a controlled small gaseous CO2 release. 45kg of CO2 was injected over a 50-h period at 9.85m depth. ERT data were collected using horizontal bipole-bipole (HBB) and vertical bipole-bipole (VBB) arrays. The combined HBB and VBB data sets were inverted using a difference inversion algorithm for cancellation of coherent noises and enhanced resolution of small changes. ERT detected the small bulk EC changes (<10%) from conductive dissolved CO2 and resistive gaseous CO2. The primary factors that control the migration of a CO2 plume consist of buoyancy of gaseous CO2, local heterogeneity, groundwater flow and external pressure exerted by the injector. The CO2 plume at the Vrøgum site migrated mostly upward due to buoyancy and it also skewed toward northeastern region by overcoming local groundwater flow. The conductive eastern part is more porous and becomes the preferential pathway for the CO2 plume, which was trapped within the slightly more porous glacial sand layer between 5m and 10m depths. The gaseous and dissolved CO2 plumes are collocated and grow in tandem for the first 24h and their opposite effects resulted in a small bulk EC increase. After raising the injection rate from 10g/min to 20g/min at the 24-h mark, the CO2 plume grew quickly. The bulk EC changes from ERT agreed partially with water sample EC and GPR data. The apparent disagreement between high CO2 gas saturation and prevailing positive bulk EC changes may be caused by limited and variable ERT resolution, low ERT sensitivity to resistive anomalies and uncalibrated CO2 gas saturation. ERT data show a broader CO2 plume while water sample EC had higher fine-scale variability. Our ERT electrode configuration can be optimized for more efficient data acquisition and better spatial resolution