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We introduce a finite volume scheme to solve isotropic 3-wave kinetic equations. We test our numerical solution against theoretical results concerning the long time behavior of the energy and observe that our solutions verify the energy cascade phenomenon. Up to our knowledge, this is the first numerical scheme that could capture the long time asymptotic behavior of solutions to isotropic 3-wave kinetic equations, where the energy cascade can be observed. Our numerical energy cascade rates are in good agreement with the theoretical one obtained by Soffer and Tran. Our finite volume algorithm relies on a new identity, that allows one to reduce the number of terms needed to be approximated in the collision operators.

This paper considers the numerical treatment of the time-dependent Gross-Pitaevskii equation. In order to conserve the time invariants of the equation as accurately as possible, we propose a Crank-Nicolson-type time discretization that is combined with a suitable generalized finite element discretization in space. The space discretization is based on the technique of Localized Orthogonal Decompositions (LOD) and allows to capture the time invariants with an accuracy of order $\mathcal{O}(H^6)$ with respect to the chosen mesh size $H$. This accuracy is preserved due to the conservation properties of the time stepping method. Furthermore, we prove that the resulting scheme approximates the exact solution in the $L^{\infty}(L^2)$-norm with order $\mathcal{O}(\tau^2 + H^4)$, where $\tau$ denotes the step size. The computational efficiency of the method is demonstrated in numerical experiments for a benchmark problem with known exact solution.

Numerical models of weather and climate critically depend on long-term stability of integrators for systems of hyperbolic conservation laws. While such stability is often obtained from (physical or numerical) dissipation terms, physical fidelity of such simulations also depends on properly preserving conserved quantities, such as energy, of the system. To address this apparent paradox, we develop a variational integrator for the shallow water equations that conserves energy, but dissipates potential enstrophy. Our approach follows the continuous selective decay framework [F. Gay-Balmaz and D. Holm. Selective decay by Casimir dissipation in inviscid fluids. Nonlinearity, 26(2):495, 2013], which enables dissipating an otherwise conserved quantity while conserving the total energy. We use this in combination with the variational discretization method [D. Pavlov, P. Mullen, Y. Tong, E. Kanso, J. Marsden and M. Desbrun. Structure-preserving discretization of incompressible fluids. Physica D: Nonlinear Phenomena, 240(6):443-458, 2011] to obtain a discrete selective decay framework. This is applied to the shallow water equations, both in the plane and on the sphere, to dissipate the potential enstrophy. The resulting scheme significantly improves the quality of the approximate solutions, enabling long-term integrations to be carried out.

To ensure preservation of local or global bounds for numerical solutions of conservation laws, we constrain a baseline finite element discretization using optimization-based (OB) flux correction. The main novelty of the proposed methodology lies in the use of flux potentials as control variables and targets of inequality-constrained optimization problems for numerical fluxes. In contrast to optimal control via general source terms, the discrete conservation property of flux-corrected finite element approximations is guaranteed without the need to impose additional equality constraints. Since the number of flux potentials is less than the number of fluxes in the multidimensional case, the potential-based version of optimal flux control involves fewer unknowns than direct calculation of optimal fluxes. We show that the feasible set of a potential-state potential-target (PP) optimization problem is nonempty and choose a primal-dual Newton method for calculating the optimal flux potentials. The results of numerical studies for linear advection and anisotropic diffusion problems in 2D demonstrate the superiority of the new OB-PP algorithms to closed-form flux limiting under worst-case assumptions.

We introduce a physically relevant stochastic representation of the rotating shallow water equations. The derivation relies mainly on a stochastic transport principle and on a decomposition of the fluid flow into a large-scale component and a noise term that models the unresolved flow components. As for the classical (deterministic) system, this scheme, referred to as modelling under location uncertainty (LU), conserves the global energy of any realization and provides the possibility to generate an ensemble of physically relevant random simulations with a good trade-off between the model error representation and the ensemble's spread. To maintain numerically the energy conservation feature, we combine an energy (in space) preserving discretization of the underlying deterministic model with approximations of the stochastic terms that are based on standard finite volume/difference operators. The LU derivation, built from the very same conservation principles as the usual geophysical models, together with the numerical scheme proposed can be directly used in existing dynamical cores of global numerical weather prediction models. The capabilities of the proposed framework is demonstrated for an inviscid test case on the f-plane and for a barotropically unstable jet on the sphere.

We introduce a new algorithm for expected log-likelihood maximization in situations where the objective function is multi-modal and/or has saddle points, that we term G-PFSO. The key idea underpinning G-PFSO is to define a sequence of probability distributions which (a) is shown to concentrate on the target parameter value and (b) can be efficiently estimated by means of a standard particle filter algorithm. These distributions depends on a learning rate, where the faster the learning rate is the faster is the rate at which they concentrate on the desired parameter value but the lesser is the ability of G-PFSO to escape from a local optimum of the objective function. To conciliate ability to escape from a local optimum and fast convergence rate, the proposed estimator exploits the acceleration property of averaging, well-known in the stochastic gradient literature. Based on challenging estimation problems, our numerical experiments suggest that the estimator introduced in this paper converges at the optimal rate, and illustrate the practical usefulness of G-PFSO for parameter inference in large datasets. If the focus of this work is expected log-likelihood maximization the proposed approach and its theory apply more generally for optimizing a function defined through an expectation.

We introduce a new preconditioner for a recently developed pressure-robust hybridized discontinuous Galerkin (HDG) finite element discretization of the Stokes equations. A feature of HDG methods is the straightforward elimination of degrees-of-freedom defined on the interior of an element. In our previous work (J. Sci. Comput., 77(3):1936--1952, 2018) we introduced a preconditioner for the case in which only the degrees-of-freedom associated with the element velocity were eliminated via static condensation. In this work we introduce a preconditioner for the statically condensed system in which the element pressure degrees-of-freedom are also eliminated. In doing so the number of globally coupled degrees-of-freedom are reduced, but at the expense of a more difficult problem to analyse. We will show, however, that the Schur complement of the statically condensed system is spectrally equivalent to a simple trace pressure mass matrix. This result is used to formulate a new, provably optimal preconditioner. Through numerical examples in two- and three-dimensions we show that the new preconditioned iterative method converges in fewer iterations, has superior conservation properties for inexact solves, and is faster in CPU time when compared to our previous preconditioner.

This paper presents and analyzes a discontinuous Galerkin method for the compressible three-phase flow problem in porous media. We use a first order time extrapolation which allows us to solve the equations implicitly and sequentially. We show that the discrete problem is well-posed, and obtain a priori error estimates. Our numerical results validate the theoretical results, i.e. the algorithm converges with first order, under different setups that involve variable density and effects of gravity.

We propose finite-volume schemes for general continuity equations which preserve positivity and global bounds that arise from saturation effects in the mobility function. In the particular case of gradient flows, the schemes dissipate the free energy at the fully discrete level. Moreover, these schemes are generalised to coupled systems of non-linear continuity equations, such as multispecies models in mathematical physics or biology, preserving the bounds and the dissipation of the energy whenever applicable. These results are illustrated through extensive numerical simulations which explore known behaviours in biology and showcase new phenomena not yet described by the literature.

We study the random reshuffling (RR) method for smooth nonconvex optimization problems with a finite-sum structure. Though this method is widely utilized in practice such as the training of neural networks, its convergence behavior is only understood in several limited settings. In this paper, under the well-known Kurdyka-Lojasiewicz (KL) inequality, we establish strong limit-point convergence results for RR with appropriate diminishing step sizes, namely, the whole sequence of iterates generated by RR is convergent and converges to a single stationary point in an almost sure sense. In addition, we derive the corresponding rate of convergence, depending on the KL exponent and the suitably selected diminishing step sizes. When the KL exponent lies in $[0,\frac12]$, the convergence is at a rate of $\mathcal{O}(t^{-1})$ with $t$ counting the iteration number. When the KL exponent belongs to $(\frac12,1)$, our derived convergence rate is of the form $\mathcal{O}(t^{-q})$ with $q\in (0,1)$ depending on the KL exponent. The standard KL inequality-based convergence analysis framework only applies to algorithms with a certain descent property. Remarkably, we conduct convergence analysis for the non-descent RR with diminishing step sizes based on the KL inequality, which generalizes the standard KL analysis framework. We summarize our main steps and core ideas in an analysis framework, which is of independent interest. As a direct application of this framework, we also establish similar strong limit-point convergence results for the shuffled proximal point method.