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This book opens with a self-contained introduction to path integrals in Euclidean quantum mechanics and statistical mechanics, and moves on to cover lattice field theory, spin systems, gauge theories and more. Each chapter ends with illustrative problems.

We consider optimization problems on manifolds with equality and inequality constraints. A large body of work treats constrained optimization in Euclidean spaces. In this work, we consider extensions of existing algorithms from the Euclidean case to the Riemannian case. Thus, the variable lives on a known smooth manifold and is further constrained. In doing so, we exploit the growing literature on unconstrained Riemannian optimization. For the special case where the manifold is itself described by equality constraints, one could in principle treat the whole problem as a constrained problem in a Euclidean space. The main hypothesis we test here is whether it is sometimes better to exploit the geometry of the constraints, even if only for a subset of them. Specifically, this paper extends an augmented Lagrangian method and smoothed versions of an exact penalty method to the Riemannian case, together with some fundamental convergence results. Numerical experiments indicate some gains in computational efficiency and accuracy in some regimes for minimum balanced cut, non-negative PCA and k-means, especially in high dimensions.

We study the extension of the proximal gradient algorithm where only a stochastic gradient estimate is available and a relaxation step is allowed. We establish convergence rates for function values in the convex case, as well as almost sure convergence and convergence rates for the iterates under further convexity assumptions. Our analysis avoid averaging the iterates and error summability assumptions which might not be satisfied in applications, e.g. in machine learning. Our proofing technique extends classical ideas from the analysis of deterministic proximal gradient algorithms.

Recently, during the investigations on planetary oceans, Hirota-Satsuma-Ito-type models have been developed. In this paper, for a (2+1)-dimensional generalized variable-coefficient Hirota-Satsuma-Ito system describing the fluid dynamics of shallow-water waves in an open ocean, non-characteristic movable singular manifold and symbolic computation enable an oceanic auto-Bäcklund transformation with three sets of the oceanic solitonic solutions. The results rely on the oceanic variable coefficients in that system. Future oceanic observations might detect some nonlinear features predicted in this paper, and relevant oceanographic insights might be expected.

The present study is an attempt to establish relationship between the concentrations of particulate matter especially (PM2.5) and background meteorological parameters over Delhi, India with the help of statistical and correlative analysis. This work presents the evaluation of air quality in three different locations of Delhi. These locations were selected to fulfil the characteristics as residential, industrial and background locations and performed the analysis for pre and post covid-19, i.e. for 2019 and 2021. The outcome of the study shows that the meteorological parameters have significant influence on the PM2.5 concentration. It was also found that it has a seasonality with low concentration in the monsoon season, moderate in the pre-monsoon season and high during the winters and post-monsoon seasons. However, the statistical and correlative study shows a negative relation with the temperature during the winter, pre-monsoon and post-monsoon and has a positive correlation during the monsoon season. Similarly, it also has been observed that the concentration of PM2.5 shows strong negative correlation with temperature during the high humid conditions, i.e. when the relative humidity is above 50%. However, a weak correlation with ambient temperature has been established during the low humidity condition, i.e. below 50%. The overall study showed that the highest PM2.5 pollution has been observed at residential location followed by industrial and background. The study also concluded that the seasonal meteorology has a complex role in the PM2.5 concentration of the selected areas.

In this paper, we first present a sufficient condition(a variant) for the large deviation criteria of Budhiraja, Dupuis and Maroulas for functionals of Brownian motions. The sufficient condition is particularly more suitable for stochastic differential/partial differential equations with reflection. We then apply the sufficient condition to establish a large deviation principle for obstacle problems of quasi-linear stochastic partial differential equations. It turns out that the backward stochastic differential equations will also play an important role.

With respect to oceanic fluid dynamics, certain models have appeared, e.g., an extended time-dependent (3+1)-dimensional shallow water wave equation in an ocean or a river, which we investigate in this paper. Using symbolic computation, we find out, on one hand, a set of bilinear auto-Bäcklund transformations, which could connect certain solutions of that equation with other solutions of that equation itself, and on the other hand, a set of similarity reductions, which could go from that equation to a known ordinary differential equation. The results in this paper depend on all the oceanic variable coefficients in that equation.

In this paper, we are concerned with a non-asymptotic analysis of sampling algorithms used in nonconvex optimization. In particular, we obtain non-asymptotic estimates in Wasserstein-1 and Wasserstein-2 distances for a popular class of algorithms called Stochastic Gradient Langevin Dynamics (SGLD). In addition, the aforementioned Wasserstein-2 convergence result can be applied to establish a non-asymptotic error bound for the expected excess risk. Crucially, these results are obtained under a local Lipschitz condition and a local dissipativity condition where we remove the uniform dependence in the data stream. We illustrate the importance of this relaxation by presenting examples from variational inference and from index tracking optimization.

Unsteady cavitating flow often contains vapor structures with a wide range of different length scales, from micro-bubbles to large cavities, which issues a big challenge to precisely investigate its evolution mechanism by computational fluid dynamics (CFD) method. The present work reviews the development of simulation methods for cavitation, especially the emerging Euler-Lagrange approach. Additionally, the progress of the numerical investigation of hot and vital issues is discussed, including cavitation inception, cloud cavitation inner structure and its formation mechanism, cavitation erosion, and cavitation noise. It is indicated that the Euler-Lagrange method can determine cavitation inception point better. For cloud cavitation, the Euler-Lagrange method can reveal the source of microbubbles and their distribution law inside the shedding cloud. This method also has advantages and great potential in assessing cloud cavitation-induced erosion and noise. With the ever-growing demands of cavitation simulation accuracy in basic research and engineering applications, how to improve the Euler-Lagrange method’s stability and applicability is still an open problem. To further promote the application of this advanced CFD simulation technology in cavitation research, some key issues are to be solved and feasible suggestions are put forward for further work.

We investigate the problem of optimal transport in the so-called Kantorovich form, i.e. given two Radon measures on two compact sets, we seek an optimal transport plan which is another Radon measure on the product of the sets that has these two measures as marginals and minimizes a certain cost function. We consider quadratic regularization of the problem, which forces the optimal transport plan to be a square integrable function rather than a Radon measure. We derive the dual problem and show strong duality and existence of primal and dual solutions to the regularized problem. Then we derive two algorithms to solve the dual problem of the regularized problem: A Gauss–Seidel method and a semismooth quasi-Newton method and investigate both methods numerically. Our experiments show that the methods perform well even for small regularization parameters. Quadratic regularization is of interest since the resulting optimal transport plans are sparse, i.e. they have a small support (which is not the case for the often used entropic regularization where the optimal transport plan always has full measure).