Convergence of fully discrete schemes for diffusive dispersive conservation laws with discontinuous coefficient

We are concerned with fully-discrete schemes for the numerical approximation of diffusive-dispersive hyperbolic conservation laws with a discontinuous flux function in one-space dimension. More precisely, we show the convergence of approximate solutions, generated by the scheme corresponding to vanishing diffusive-dispersive scalar conservation laws with a discontinuous coefficient, to the corresponding scalar conservation law with discontinuous coefficient. Finally, the convergence is illustrated by several examples. In particular, it is delineated that the limiting solutions generated by the scheme need not coincide, depending on the relation between diffusion and the dispersion coefficients, with the classical Kruzkov-Oleinik entropy solutions, but contain nonclassical undercompressive shock waves.Comment: 38 Pages, 6 figure

The making of modern Indian diplomacy: a critique of Eurocentrism

Diplomacy is conventionally understood as an authentic European invention which was internationalized during colonialism. For Indians, the moment of colonial liberation was a false-dawn because the colonized had internalized a European logic and performed a European practice. Implicit in such a reading is the enduring centrality of Europe to understanding the logics of Indian diplomacy. The only contribution to diplomacy permitted of India is restricted to practice, to Indians adulterating pure, European, diplomacy. This Eurocentric discourse renders two possibilities impossible: that diplomacy may have Indian origins and that they offer un-theorised potentialities. These potentialities are the subject because combined they suggest that Indian diplomacy might move to a logic unknown to conventional approaches. However, what is first required is a conceptual space for this possibility, something, it is argued, civilizational analysis provides because its focus on continuities does not devalue transformational changes. Populating this conceptual space requires ascertaining empirically whether Indian diplomacy is indeed extra- European? It is why current practices are exposed and then placed in the context of the literature to reveal ruptures, what are termed controversies. The most significant, arguably, is the question of what is Indian diplomatic modernity? Resolving this controversy requires exploring not only the history of the revealed practices but also excavating the conceptual categories which produce them. The investigation therefore is not a history, but a genealogy for it identifies the present and then moves along two axes: tracing the origins of the bureaucratic apparatus and the rationales underpinning them. The genealogical moves made are dictated by the practitioners and practices themselves because the aim is not to theorize about the literature but to expose the rationalities which animate the practitioners of international politics today. The only means to actually verify if the identified mentalities do animate international politics is to demonstrate their impact on practice. It is why the project is argued empirically, in terms of the ‘stuff’ of IR

Solution of Physics-based Bayesian Inverse Problems with Deep Generative Priors

Inverse problems are notoriously difficult to solve because they can have no solutions, multiple solutions, or have solutions that vary significantly in response to small perturbations in measurements. Bayesian inference, which poses an inverse problem as a stochastic inference problem, addresses these difficulties and provides quantitative estimates of the inferred field and the associated uncertainty. However, it is difficult to employ when inferring vectors of large dimensions, and/or when prior information is available through previously acquired samples. In this paper, we describe how deep generative adversarial networks can be used to represent the prior distribution in Bayesian inference and overcome these challenges. We apply these ideas to inverse problems that are diverse in terms of the governing physical principles, sources of prior knowledge, type of measurement, and the extent of available information about measurement noise. In each case we apply the proposed approach to infer the most likely solution and quantitative estimates of uncertainty.Comment: Paper: 18 pages, 5 figures. Supplementary: 9 pages, 6 Figures, 2 Table

Iterative Surrogate Model Optimization (ISMO): An active learning algorithm for PDE constrained optimization with deep neural networks

We present a novel active learning algorithm, termed as iterative surrogate model optimization (ISMO), for robust and efficient numerical approximation of PDE constrained optimization problems. This algorithm is based on deep neural networks and its key feature is the iterative selection of training data through a feedback loop between deep neural networks and any underlying standard optimization algorithm. Under suitable hypotheses, we show that the resulting optimizers converge exponentially fast (and with exponentially decaying variance), with respect to increasing number of training samples. Numerical examples for optimal control, parameter identification and shape optimization problems for PDEs are provided to validate the proposed theory and to illustrate that ISMO significantly outperforms a standard deep neural network based surrogate optimization algorithm

Detecting troubled-cells on two-dimensional unstructured grids using a neural network

In a recent paper [Ray and Hesthaven, J. Comput. Phys. 367 (2018), pp 166-191], we proposed a new type of troubled-cell indicator to detect discontinuities in the numerical solutions of one-dimensional conservation laws. This was achieved by suitably training an articial neural network on canonical local solution structures for conservation laws. The proposed indicator was independent of problem-dependent parameters, giving it an advantage over existing limiter-based indicators. In the present paper, we extend this approach to train a similar network capable of detecting troubled-cells on two-dimensional unstructured grids. The proposed network has a smaller architecture compared to its one-dimensional predecessor, making it computationally efficient. Several numerical results are presented to demonstrate the performance of the new indicator

Non-intrusive reduced order modeling of unsteady flows using artificial neural networks with application to a combustion problem

A non-intrusive reduced-basis (RB) method is proposed for parametrized unsteady flows. A set of reduced basis functions are extracted from a collection of high-fidelity solutions via a proper orthogonal decomposition (POD), and the coefficients of the reduced basis functions are recovered by a feedforward neural network (NN). As a regression model of the RB method for unsteady flows, the neural network approximates the map between the time/parameter value and the projection coefficients of the high-fidelity solution onto the reduced space. The generation of the reduced basis and the training of the NN are accomplished in the offline stage, thus the RB solution of a new time/parameter value can be recovered via direct outputs of the NN in the online stage. Due to its non-intrusive nature, the proposed RB method, referred as the POD-NN, fully decouples the online stage and the high-fidelity scheme, and is thus able to provide fast and reliable solutions of complex unsteady flows. To test this assertion, the POD-NN method is applied to the reduced order modeling (ROM) of the quasi-one dimensional Continuously Variable Resonance Combustor (CVRC) flow. Numerical results demonstrate the efficiency and robustness of the POD-NN method

MATHICSE Technical Report: Controlling oscillations in high-order Discontinuous Galerkin schemes using artificial\ viscosity tuned by neural networks

High-order numerical solvers for conservation laws suer from Gibbs phenomenon close to discontinuities, leading to spurious oscillations and a detrimental effect on the solution accuracy. A possible strategy to reduce it comprises adding a suitable amount of artificial dissipation. Although several viscosity models have been proposed in the literature, their dependence on problem-dependent parameters often limits their performances. Motivated by the objective to construct a universal artificial viscosity method, we propose a new technique based on neural networks, integrated into a Runge-Kutta Discontinuous Galerkin solver. Numerical results are presented to demonstrate the performance of this network-based technique. We show that it is able both to guarantee optimal accuracy for smooth problems, and to accurately detect discontinuities, where dissipation has to be injected. A comparison with some classical models is carried out, showing the superior performance of the network-based model in capturing both complex and fine structures