Within the framework of $ p $-adaptive flux reconstruction, we aim to construct efficient polynomial multigrid ($p$MG) preconditioners for implicit time integration of the Navier--Stokes equations using Jacobian-free Newton--Krylov (JFNK) methods. We hypothesise that in pseudo transient continuation (PTC), as the residual drops, the frequency of error modes that dictates the convergence rate gets higher and higher. We apply nonlinear $p$MG solvers to stiff steady problems at low Mach number ($\mathrm{Ma}=10^{-3}$) to verify our hypothesis. It is demonstrated that once the residual drops by a few orders of magnitude, improved smoothing on intermediate $ p $-sublevels will not only maintain the stability of $ p $MG at large time steps but also improve the convergence rate. For the unsteady Navier--Stokes equations, we elaborate how to construct nonlinear preconditioners using pseudo transient continuation for the matrix-free generalized minimal residual (GMRES) method used in explicit first stage, singly diagonally implicit Runge--Kutta (ESDIRK) methods, and linearly implicit Rosenbrock--Wanner (ROW) methods. Given that at each time step the initial guess in the nonlinear solver is not distant from the converged solution, we recommend a two-level $p\{p_0\text{-}p_0/2\} $ or even $ p\{p_0\text{-}(p_0-1)\} $ $p$-hierarchy for optimal efficiency with a matrix-based smoother on the coarser level based on our hypothesis. It is demonstrated that insufficient smoothing on intermediate $p$-sublevels will deteriorate the performance of $p$MG preconditioner greatly. (See full abstract in the paper.)
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Integration：Integration, the VLSI Journal。
Explanation：集成，VLSI雜志。
Publisher：Elsevier。
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Two novel parallel Newton-Krylov Balancing Domain Decomposition by Constraints (BDDC) and Dual-Primal Finite Element Tearing and Interconnecting (FETI-DP) solvers are here constructed, analyzed and tested numerically for implicit time discretizations of the three-dimensional Bidomain system of equations. This model represents the most advanced mathematical description of the cardiac bioelectrical activity and it consists of a degenerate system of two non-linear reaction-diffusion partial differential equations (PDEs), coupled with a stiff system of ordinary differential equations (ODEs). A finite element discretization in space and a segregated implicit discretization in time, based on decoupling the PDEs from the ODEs, yields at each time step the solution of a non-linear algebraic system. The Jacobian linear system at each Newton iteration is solved by a Krylov method, accelerated by BDDC or FETI-DP preconditioners, both augmented with the recently introduced {\em deluxe} scaling of the dual variables. A polylogarithmic convergence rate bound is proven for the resulting parallel Bidomain solvers. Extensive numerical experiments on linux clusters up to two thousands processors confirm the theoretical estimates, showing that the proposed parallel solvers are scalable and quasi-optimal.
Escaping from saddle points and finding local minimum is a central problem in nonconvex optimization. Perturbed gradient methods are perhaps the simplest approach for this problem. However, to find $(\epsilon, \sqrt{\epsilon})$-approximate local minima, the existing best stochastic gradient complexity for this type of algorithms is $\tilde O(\epsilon^{-3.5})$, which is not optimal. In this paper, we propose LENA (Last stEp shriNkAge), a faster perturbed stochastic gradient framework for finding local minima. We show that LENA with stochastic gradient estimators such as SARAH/SPIDER and STORM can find $(\epsilon, \epsilon_{H})$-approximate local minima within $\tilde O(\epsilon^{-3} + \epsilon_{H}^{-6})$ stochastic gradient evaluations (or $\tilde O(\epsilon^{-3})$ when $\epsilon_H = \sqrt{\epsilon}$). The core idea of our framework is a step-size shrinkage scheme to control the average movement of the iterates, which leads to faster convergence to the local minima.
The monotone variational inequality is a central problem in mathematical programming that unifies and generalizes many important settings such as smooth convex optimization, two-player zero-sum games, convex-concave saddle point problems, etc. The extragradient method by Korpelevich [1976] is one of the most popular methods for solving monotone variational inequalities. Despite its long history and intensive attention from the optimization and machine learning community, the following major problem remains open. What is the last-iterate convergence rate of the extragradient method for monotone and Lipschitz variational inequalities with constraints? We resolve this open problem by showing a tight $O\left(\frac{1}{\sqrt{T}}\right)$ last-iterate convergence rate for arbitrary convex feasible sets, which matches the lower bound by Golowich et al. [2020]. Our rate is measured in terms of the standard gap function. The technical core of our result is the monotonicity of a new performance measure -- the tangent residual, which can be viewed as an adaptation of the norm of the operator that takes the local constraints into account. To establish the monotonicity, we develop a new approach that combines the power of the sum-of-squares programming with the low dimensionality of the update rule of the extragradient method. We believe our approach has many additional applications in the analysis of iterative methods.
Stochastic partial differential equations (SPDEs) are the mathematical tool of choice for modelling spatiotemporal PDE-dynamics under the influence of randomness. Based on the notion of mild solution of an SPDE, we introduce a novel neural architecture to learn solution operators of PDEs with (possibly stochastic) forcing from partially observed data. The proposed Neural SPDE model provides an extension to two popular classes of physics-inspired architectures. On the one hand, it extends Neural CDEs and variants -- continuous-time analogues of RNNs -- in that it is capable of processing incoming sequential information arriving irregularly in time and observed at arbitrary spatial resolutions. On the other hand, it extends Neural Operators -- generalizations of neural networks to model mappings between spaces of functions -- in that it can parameterize solution operators of SPDEs depending simultaneously on the initial condition and a realization of the driving noise. By performing operations in the spectral domain, we show how a Neural SPDE can be evaluated in two ways, either by calling an ODE solver (emulating a spectral Galerkin scheme), or by solving a fixed point problem. Experiments on various semilinear SPDEs, including the stochastic Navier-Stokes equations, demonstrate how the Neural SPDE model is capable of learning complex spatiotemporal dynamics in a resolution-invariant way, with better accuracy and lighter training data requirements compared to alternative models, and up to 3 orders of magnitude faster than traditional solvers.
Stochastic evolution equations with compensated Poisson noise are considered in the variational approach with monotone and coercive coefficients. Here the Poisson noise is assumed to be time-homogeneous with $\sigma$-finite intensity measure on a metric space. By using finite element methods and Galerkin approximations, some explicit and implicit discretizations for this equation are presented and their convergence is proved. Polynomial growth condition and linear growth condition are assumed on the drift operator, respectively for the implicit and explicit schemes.
This paper introduces a new simulation-based inference procedure to model and sample from multi-dimensional probability distributions given access to i.i.d. samples, circumventing the usual approaches of explicitly modeling the density function or designing Markov chain Monte Carlo. Motivated by the seminal work on distance and isomorphism between metric measure spaces, we propose a new notion called the Reversible Gromov-Monge (RGM) distance and study how RGM can be used to design new transform samplers to perform simulation-based inference. Our RGM sampler can also estimate optimal alignments between two heterogeneous metric measure spaces $(\mathcal{X}, \mu, c_{\mathcal{X}})$ and $(\mathcal{Y}, \nu, c_{\mathcal{Y}})$ from empirical data sets, with estimated maps that approximately push forward one measure $\mu$ to the other $\nu$, and vice versa. Analytic properties of the RGM distance are derived; statistical rate of convergence, representation, and optimization questions regarding the induced sampler are studied. Synthetic and real-world examples showcasing the effectiveness of the RGM sampler are also demonstrated.
In this work, we introduce a novel approach to formulating an artificial viscosity for shock capturing in nonlinear hyperbolic systems by utilizing the property that the solutions of hyperbolic conservation laws are not reversible in time in the vicinity of shocks. The proposed approach does not require any additional governing equations or a priori knowledge of the hyperbolic system in question, is independent of the mesh and approximation order, and requires the use of only one tunable parameter. The primary novelty is that the resulting artificial viscosity is unique for each component of the conservation law which is advantageous for systems in which some components exhibit discontinuities while others do not. The efficacy of the method is shown in numerical experiments of multi-dimensional hyperbolic conservation laws such as nonlinear transport, Euler equations, and ideal magnetohydrodynamics using a high-order discontinuous spectral element method on unstructured grids.
This paper introduces a novel approach to compute the numerical fluxes at the cell boundaries for a cell-centered conservative numerical scheme. Explicit gradients used in deriving the reconstruction polynomials are replaced by high-order gradients computed by compact finite differences, referred to as implicit gradients in this paper. The new approach has superior dispersion and dissipation properties in comparison to the compact reconstruction approach. A problem-independent shock capturing approach via Boundary Variation Diminishing (BVD) algorithm is used to suppress oscillations for the simulation of flows with shocks and material interfaces. Several numerical test cases are carried out to verify the proposed method's capability using the implicit gradient method for compressible flows.
We study an implicit finite-volume scheme for non-linear, non-local aggregation-diffusion equations which exhibit a gradient-flow structure, recently introduced by Bailo, Carrillo, and Hu (2020). Crucially, this scheme keeps the dissipation property of an associated fully discrete energy, and does so unconditionally with respect to the time step. Our main contribution in this work is to show the convergence of the method under suitable assumptions on the diffusion functions and potentials involved.
In this paper, we provide a general framework for studying multi-agent online learning problems in the presence of delays and asynchronicities. Specifically, we propose and analyze a class of adaptive dual averaging schemes in which agents only need to accumulate gradient feedback received from the whole system, without requiring any between-agent coordination. In the single-agent case, the adaptivity of the proposed method allows us to extend a range of existing results to problems with potentially unbounded delays between playing an action and receiving the corresponding feedback. In the multi-agent case, the situation is significantly more complicated because agents may not have access to a global clock to use as a reference point; to overcome this, we focus on the information that is available for producing each prediction rather than the actual delay associated with each feedback. This allows us to derive adaptive learning strategies with optimal regret bounds, even in a fully decentralized, asynchronous environment. Finally, we also analyze an "optimistic" variant of the proposed algorithm which is capable of exploiting the predictability of problems with a slower variation and leads to improved regret bounds.
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