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The aim of this text is to elucidate the oscillating patterns (see C. Cheverry [Res. Rep. Math. (2018)]) which are generated in a toroidal plasma by a strong external magnetic field and a nonzero electric field. It is also to justify and then study new modulation equations which are valid for longer times than before. Oscillating coherent structures are induced by the collective motions of charged particles which satisfy a system of ODEs implying a large parameter, the gyrofrequency \varepsilon ^{-1} \gg 1. By exploiting the properties of underlying integrable systems, we can complement the KAM picture (see G. Benettin and P. Sempio [Nonlinearity 7 (1994), pp. 281–303]; M. Braun [SIAM Rev. 23 (1981), pp. 61–93]) and go beyond the classical results about gyrokinetics (see M. Bostan [Multiscale Model. Simul. 8 (2010), pp. 1923–1957]; A. J. Brizard and T. S. Hahm [Rev. Modern Phys. 79 (2007), pp. 421–468]). The purely magnetic situation was addressed by C. Cheverry [Comm. Math. Phys. 338 (2015), pp. 641–703; J. Differential Equations 262 (2017), pp. 2987–3033]. We are concerned here with the numerous additional difficulties due to the influence of a nonzero electric field. [ABSTRACT FROM AUTHOR]

We consider the non-symmetric coupling of finite and boundary elements to solve second-order nonlinear partial differential equations defined in unbounded domains. We present a novel condition that ensures that the associated semi-linear form induces a strongly monotone operator, keeping track of the dependence on the linear combination of the interior domain equation with the boundary integral one. We show that an optimal ellipticity condition, relating the nonlinear operator to the contraction constant of the shifted double-layer integral operator, is guaranteed by choosing a particular linear combination. These results generalize those obtained by Of and Steinbach [Is the one-equation coupling of finite and boundary element methods always stable?, ZAMM Z. Angew. Math. Mech.93 (2013), 6–7, 476–484] and [On the ellipticity of coupled finite element and one-equation boundary element methods for boundary value problems, Numer. Math.127 (2014), 3, 567–593], and by Steinbach [A note on the stable one-equation coupling of finite and boundary elements, SIAM J. Numer. Anal.49 (2011), 4, 1521–1531], where the simple sum of the two coupling equations has been considered. Numerical examples confirm the theoretical results on the sharpness of the presented estimates. [ABSTRACT FROM AUTHOR]

Recently proposed numerical algorithms for solving high-dimensional nonlinear partial differential equations (PDEs) based on neural networks have shown their remarkable performance. We review some of them and study their convergence properties. The methods rely on probabilistic representation of PDEs by backward stochastic differential equations and their iterated time discretization. Our proposed algorithm, called the deep backward multistep (MDBDP) scheme, is a machine learning version of the least square multistep-forward dynamic programming scheme of Gobet and Turkedjiev [Math. Comp., 85, 2016, pp. 1359--1391]. It estimates simultaneously by backward induction the solution and its gradient by neural networks through sequential minimizations of suitable quadratic loss functions that are performed by stochastic gradient descent. Our main theoretical contribution is to provide an approximation error analysis of the MDBDP scheme as well as the deep splitting scheme for semilinear PDEs designed by Beck, Becker, Cheridito, Jentzen, and Neufeld [SIAM J. Sci. Comput., 43 (2021), pp. A3135--A3154]. We also supplement the error analysis of the deep learning backward dynamic programming (DBDP) scheme of Hur\'e, Pham, and Warin [Math. Comp., 89 (2020), pp. 1547--1580]. This yields notably convergence rate in terms of the number of neurons for a class of deep Lipschitz continuous GroupSort neural networks when the PDE is linear in the gradient of the solution for the MDBDP scheme, and in the semilinear case for the DBDP scheme. We illustrate our results with some numerical tests that are compared with some other machine learning algorithms in the literature. [ABSTRACT FROM AUTHOR]

We propose a continuous-time version of the adaptive robust methodology introduced in T. R. Bielecki et al. [SIAM J. Control Optim., 57 (2019), pp. 925-946]. An agent solves a stochastic control problem where the underlying uncertainty follows a jump-diffusion process and the agent does not know the drift parameters of the process. The agent considers a set of alternative measures to make the control problem robust to model misspecification and employs a continuous-time estimator to learn the value of the unknown parameters to make the control problem adaptive to the arrival of new information. We use measurable selection theorems to prove the dynamic programming principle of the adaptive robust problem and show that the value function of the agent is characterized by a nonlinear partial differential equation. As an example, we derive the optimal adaptive robust strategy for an agent who acquires a large amount of shares in an order driven market and illustrate the financial performance of the execution strategy. [ABSTRACT FROM AUTHOR]

The Kuramoto model is a system of nonlinear differential equations that is often used to model synchronization between coupled oscillators in a network. A particular form of synchronization is phase-locking whereby the oscillators rotate at a common frequency with fixed angle differences. It has been observed that the oscillators will phase-lock if their natural frequencies have small differences, and conversely, that they won't if the natural frequencies have large differences. In [Bronski and Ferguson, SIAM J. Appl. Dyn. Syst., 17 (2018), pp. 128--156] Bronski and the author gave upper and lower bounds for the volume of the set of natural frequencies for which the Kuramoto model exhibits phase-locking. This was done under the assumption that any two oscillators influence each other with equal strength. In this paper the author generalizes these upper and lower bounds by removing this assumption. Similar to [Bronski and Ferguson, SIAM J. Appl. Dyn. Syst., 17 (2018), pp. 128--156], where the upper and lower bounds are sums over spanning trees of the network, our generalized upper and lower bounds are sums over certain directed subgraphs of the network. In particular, our lower bound and one of our upper bounds are sums over divergent directed spanning trees. We also relate our results to observations by Poignard, Pereira, Pade in [SIAM J. Appl. Math., 78 (2018), pp. 372--394] and Pade and Pereira in [APS March Meeting Abstracts, 2015, M45.011] on how changes in the network can affect synchronization. [ABSTRACT FROM AUTHOR]

In this paper, a new mass-based numerical method is developed using the notion of Forestier-Coste & Mancini (Forestier-Coste & Mancini 2012, SIAM J. Sci. Comput. 34, B840-B860. (doi:10.1137/110847998)) for solving a one-dimensional aggregation population balance equation. The existing scheme requires a large number of grids to predict both moments and number density function accurately, making it computationally very expensive. Therefore, a mass-based finite volume is developed which leads to the accurate prediction of different integral properties of number distribution functions using fewer grids. The new mass-based and existing finite volume schemes are extended to solve simultaneous aggregation-growth and aggregation-nucleation problems. To check the accuracy and efficiency, the mass-based formulation is compared with the existing method for two kinds of benchmark kernels, namely analytically solvable and practical oriented kernels. The comparison reveals that the mass-based method computes both number distribution functions and moments more accurately and efficiently than the existing method. [ABSTRACT FROM AUTHOR]