Expected Convergence Rates Research Articles

Explicit stochastic advection algorithms for the regional-scale particle-resolved atmospheric aerosol model WRF-PartMC (v1.0)

Abstract. This paper presents the development of a stochastic particle method to simulate advection in regional-scale models with a particle-resolving aerosol representation. The new method is based on finite-volume discretizations with the flux terms interpreted as probabilities of particle transport between grid cells. We analyze the method in 1D and show that the stochastic particle sampling during transport injects energy at high spatial frequencies, which can be partially compensated for with the choice of a dissipative odd-order finite-volume scheme. We then apply the stochastic third- and fifth-order advection algorithms with monotonic limiters in WRF-PartMC, using idealized and realistic wind fields in 2D and 3D. In all cases we observe the expected convergence rates of the stochastic particle method to the finite-volume solution as the number of computational particles is increased. This work enables the use of particle-based aerosol models on the regional scale.

Simultaneous solution of implicitly defined curved, linear Timoshenko beams in two‐dimensional bulk domains

AbstractA novel formulation of curved Timoshenko beams which are defined by all level sets of a scalar function in a two‐dimensional bulk domain is presented, including a corresponding finite element method for the simultaneous solution of these beams. The required geometrical setup and differential operators are introduced and the model is defined in strong form. The weak form for the simultaneous solution involves the co‐area formula. A crucial aspect is how the difference vector, describing the rotational deformation in this model, is discretized wherefore two different approaches are shown in this paper. Higher‐order convergence studies in the residual errors confirm that the proposed method is valid and achieves expected convergence rates.

Mixed finite elements for Kirchhoff–Love plate bending

We present a mixed finite element method with triangular and parallelogram meshes for the Kirchhoff–Love plate bending model. Critical ingredient is the construction of low-dimensional local spaces and appropriate degrees of freedom that provide conformity in terms of a sufficiently large tensor space and that allow for any kind of physically relevant Dirichlet and Neumann boundary conditions. For Dirichlet boundary conditions and polygonal plates, we prove quasi-optimal convergence of the mixed scheme. An a posteriori error estimator is derived for the special case of the biharmonic problem. Numerical results for regular and singular examples illustrate our findings. They confirm expected convergence rates and exemplify the performance of an adaptive algorithm steered by our error estimator.

A Three‐Field Formulation for Two‐Phase Flow in Geodynamic Modeling: Toward the Zero‐Porosity Limit

AbstractTwo‐phase flow, a system where Stokes flow and Darcy flow are coupled, is of great importance in the Earth's interior, such as in subduction zones, mid‐ocean ridges, and hotspots. However, it remains challenging to solve the two‐phase equations accurately in the zero‐porosity limit, for example, when melt is fully frozen below solidus temperature. Here we propose a new three‐field formulation of the two‐phase system, with solid velocity (vs), total pressure (Pt), and fluid pressure (Pf) as unknowns, and present a robust finite‐element implementation, which can be used to solve problems in which domains of both zero porosity and non‐zero porosity are present. The reformulated equations include regularization to avoid singularities and exactly recover to the standard single‐phase incompressible Stokes problem at zero porosity. We verify the correctness of our implementation using the method of manufactured solutions and analytic solutions and demonstrate that we can obtain the expected convergence rates in both space and time. Example experiments, such as self‐compaction, falling block, and mid‐ocean ridge spreading show that this formulation can robustly resolve zero‐ and non‐zero‐porosity domains simultaneously, and can be used for a large range of applications in various geodynamic settings.

Consensus of Discrete-Time Leader-Following Linear Multi-Agent Systems Under Lyapunov-Function-Based Event-Triggered Mechanism

A novel Lyapunov-function-based event-triggered mechanism (ETM) for discrete-time linear leader-following multi-agent systems is proposed in this brief. Based on the constructed ETM, sufficient conditions to reach leader-following consensus asymptotically or exponentially are derived. Compared with the current results, the Lyapunov function no longer needs to be strictly monotonic decreasing under the designed ETM, which increases the triggering interval. The expected convergence rate can be achieved by the tunable parameters in the ETM. Particularly, the non-triviality of the introduced ETM is proved for some of the degraded systems. Finally, the obtained results are verified by a numerical example.

Adaptive stabilized finite elements via residual minimization onto bubble enrichments

The Adaptive Stabilized Finite Element method (AS-FEM) developed in [16] combines the idea of the residual minimization method with the inf-sup stability offered by the discontinuous Galerkin (DG) frameworks. As a result, the discretizations deliver stabilized conforming approximations and residual representatives in a DG space that can drive automatic adaptivity. In this work, we extend AS-FEM by considering continuous enriched test spaces. Thus, we propose a residual minimization method on a stable Continuous Interior Penalty (CIP) formulation, which considers a C0-conforming trial FEM space and a test space based on the bubble enrichment of that trial space. In our numerical experiments, the test space choice significantly reduces the total degrees of freedom compared to the DG test spaces of [16], while recovering the expected convergence rate for the error in the corresponding trial space norm.

Sinc Collocation Method to Simulate the Fractional Partial Integro-Differential Equation with a Weakly Singular Kernel

In this article, we develop an efficient numerical scheme for dealing with fractional partial integro-differential equations (FPIEs) with a weakly singular kernel. The weight and shift Grünwald difference (WSGD) operator is adopted to approximate a time fractional derivative and the Sinc collocation method is applied for discretizing the spatial derivative.The exponential convergence of our proposed method is demonstrated in detail. Finally, numerical evidence is employed to verify the theoretical results and confirm the expected convergence rate.

Convergence of an asynchronous block-coordinate forward-backward algorithm for convex composite optimization

AbstractIn this paper, we study the convergence properties of a randomized block-coordinate descent algorithm for the minimization of a composite convex objective function, where the block-coordinates are updated asynchronously and randomly according to an arbitrary probability distribution. We prove that the iterates generated by the algorithm form a stochastic quasi-Fejér sequence and thus converge almost surely to a minimizer of the objective function. Moreover, we prove a general sublinear rate of convergence in expectation for the function values and a linear rate of convergence in expectation under an error bound condition of Tseng type. Under the same condition strong convergence of the iterates is provided as well as their linear convergence rate.

Volume Integral Equations and Single-Trace Formulations for Acoustic Wave Scattering in an Inhomogeneous Medium

AbstractWe study frequency domain acoustic scattering at a bounded, penetrable, and inhomogeneous obstacleΩ−⊂Rd\Omega^{-}\subset\mathbb{R}^{d},d=2,3d=2,3. By defining constant reference coefficients, a representation formula for the pressure field is derived. It contains a volume integral operator, related to the one in the Lippmann–Schwinger equation. Besides, it features integral operators defined on∂Ω−\partial\Omega^{-}and closely related to boundary integral equations of single-trace formulations (STF) for transmission problems with piecewise constant coefficients. We show well-posedness of the continuous variational formulation and asymptotic convergence of Galerkin discretizations. Numerical experiments in 2D validate our expected convergence rates.

Adaptive Nonintrusive Reconstruction of Solutions to High-Dimensional Parametric PDEs

Numerical methods for random parametric PDEs can greatly benefit from adaptive refinement schemes, in particular when functional approximations are computed as in stochastic Galerkin and stochastic collocations methods. This work is concerned with a nonintrusive generalization of the adaptive Galerkin finite element method with residual-based error estimation. It combines the nonintrusive character of a randomized least squares method with the a posteriori error analysis of stochastic Galerkin methods. The proposed approach uses the variational Monte Carlo method to obtain a quasi-optimal low-rank approximation of the Galerkin projection in a highly efficient hierarchical tensor format. We derive an adaptive refinement algorithm which is steered by a reliable error estimator. Opposite to stochastic Galerkin methods, the approach is easily applicable to a wide range of problems, enabling a fully automated adjustment of all discretization parameters. Benchmark examples with affine and (unbounded) lognormal coefficient fields illustrate the performance of the nonintrusive adaptive algorithm, showing the expected convergence rates of single-level strategies.

Isogeometric mixed collocation of nearly-incompressible electromechanics in finite deformations for cardiac muscle simulations

We present an extension of isogeometric collocation to coupled cardiac electromechanical problems. We develop a staggered solution scheme that takes advantage of isogeometric collocation to reduce the computational effort in the simulation of the mechanical step, guaranteeing high accuracy for all field variables.We mainly focus on (i) the strategy adopted to couple the electrical and mechanical sub-problems, (ii) the possibility of handling different meshes to better represent the spatial scales, (iii) and the mitigation of volumetric locking. To this end, we propose a suitable mixed formulation for finite elasticity.Several numerical tests demonstrate that the mixed formulation retrieves the expected convergence rates under h-refinement and the effectiveness of the proposed solution scheme for electromechanics.

Stochastic linearized generalized alternating direction method of multipliers: Expected convergence rates and large deviation properties

AbstractAlternating direction method of multipliers (ADMM) receives much attention in the field of optimization and computer science, etc. The generalized ADMM (G-ADMM) proposed by Eckstein and Bertsekas incorporates an acceleration factor and is more efficient than the original ADMM. However, G-ADMM is not applicable in some models where the objective function value (or its gradient) is computationally costly or even impossible to compute. In this paper, we consider the two-block separable convex optimization problem with linear constraints, where only noisy estimations of the gradient of the objective function are accessible. Under this setting, we propose a stochastic linearized generalized ADMM (called SLG-ADMM) where two subproblems are approximated by some linearization strategies. And in theory, we analyze the expected convergence rates and large deviation properties of SLG-ADMM. In particular, we show that the worst-case expected convergence rates of SLG-ADMM are $\mathcal{O}\left( {{N}^{-1/2}}\right)$ and $\mathcal{O}\left({\ln N} \cdot {N}^{-1}\right)$ for solving general convex and strongly convex problems, respectively, where N is the iteration number, similarly hereinafter, and with high probability, SLG-ADMM has $\mathcal{O}\left ( \ln N \cdot N^{-1/2} \right ) $ and $\mathcal{O}\left ( \left ( \ln N \right )^{2} \cdot N^{-1} \right ) $ constraint violation bounds and objective error bounds for general convex and strongly convex problems, respectively.

1-Bit Compressive Sensing for Efficient Federated Learning Over the Air

For distributed learning among collaborative users, this paper develops and analyzes a communication-efficient scheme for federated learning (FL) over the air, which incorporates 1-bit compressive sensing (CS) into analog-aggregation transmissions. To facilitate design parameter optimization, we analyze the efficacy of the proposed scheme by deriving a closed-form expression for the expected convergence rate. Our theoretical results unveil the tradeoff between convergence performance and communication efficiency as a result of the aggregation errors caused by sparsification, dimension reduction, quantization, signal reconstruction and noise. Then, we formulate a joint optimization problem to mitigate the impact of these aggregation errors through joint optimal design of worker scheduling and power scaling policy. An enumeration-based method is proposed to solve this non-convex problem, which is optimal but becomes computationally infeasible as the number of devices increases. For scalable computing, we resort to the alternating direction method of multipliers (ADMM) technique to develop an efficient implementation that is suitable for large-scale networks. Simulation results show that our proposed 1-bit CS based FL over the air achieves comparable performance to the ideal case where conventional FL without compression and quantification is applied over error-free aggregation, at much reduced communication overhead and transmission latency.

Low dispersion finite volume/element discretization of the enhanced Green–Naghdi equations for wave propagation, breaking and runup on unstructured meshes

We study a hybrid approach combining a finite volume (FV) and a finite element (FE) method to solve a fully-nonlinear and weakly-dispersive depth averaged wave propagation model. The FV method is used to solve the underlying hyperbolic shallow water system, while a standard P1 finite element method is used to solve the elliptic system associated to the dispersive correction. We study the impact of several numerical aspects: the impact of the reconstruction used in the hyperbolic phase; the representation of the FV data in the FE method used in the elliptic phase and their impact on the theoretical accuracy of the method; the well-posedness of the overall method. For the first element we proposed a systematic implementation of an iterative reconstruction providing on arbitrary meshes up to third order solutions, full second order first derivatives, as well as a consistent approximation of the second derivatives. These properties are exploited to improve the assembly of the elliptic solver, showing dramatic improvement of the finale accuracy, if the FV representation is correctly accounted for. Concerning the elliptic step, the original problem is usually better suited for an approximation in H(div) spaces. However, it has been shown that perturbed problems involving similar operators with a small Laplace perturbation are well behaved in H1. We show, based on both heuristic and strong numerical evidence, that numerical dissipation plays a major role in stabilizing the coupled method, and not only providing convergent results, but also providing the expected convergence rates. Finally, the full mode, coupling a wave breaking closure previously developed by the authors, is thoroughly tested on standard benchmarks using unstructured grids with sizes comparable or coarser than those usually proposed in literature.

Stable and decoupled schemes for an electrohydrodynamics model

In this paper, we study numerical solutions of an electrohydrodynamics model. The considered model appears in the description of electric convection dynamics arising from unipolar charge injection on the boundary of insulating liquid, which is a coupling of the Navier–Stokes equations, charge transfer equation, and potential energy equation. A class of stable numerical schemes is proposed and analysed for this coupled equation system. The advantage of the proposed schemes is twofold: (1) they are unconditionally stable, consequently the choice of time step size only concerns the accuracy requirement; (2) they decouple the charge density and potential energy from the Navier–Stokes equations, and therefore can be implemented efficiently. The numerical examples provided in the paper show that the proposed schemes achieve the expected convergence rate, and can be used to accurately simulate the changes of flow field and electric field induced by the electrical convection. We first consider the case of constant density, then extend the construction, analysis, and validation of the schemes to the case of variable density.

ENO-ET: a reconstruction scheme based on extended ENO stencil and truncated highest-order term

High-order numerical methods for hyperbolic balance laws must circumvent Godunov’s theorem, for which non-linear spatial reconstruction is a successful procedure; this allows the design of non-linear semi-discrete and fully discrete high-order finite volume methods. The Essentially-Non-Oscillatory (ENO) method pioneered by Harten and collaborators in 1987 was a fundamental breakthrough in providing the bases for constructing schemes of both high-order of accuracy in smooth regions and essentially non-oscillatory at discontinuities. Several variations of ENO have since appeared in the literature, notably the Weighted Essentially-Non-Oscillatory (WENO) method of Jiang and Shu. In the present article, after revisiting the main elements of reconstruction methods, including their analysis, we propose a new, conservative ENO-type reconstruction procedure, called ENO-ET, that preserves the main advantages of the classical ENO and successfully overcomes its well-known weaknesses. The newly proposed reconstruction scheme is systematically assessed and compared with the best existing reconstruction schemes. We do so in two ways: (i) the methods are purely regarded as local, non-linear interpolation objects and (ii) the reconstruction methods are implemented in the context of high-order fully-discrete ADER numerical schemes for solving hyperbolic balance laws. The resulting schemes are of arbitrary order of accuracy in both space and time. In this paper the methods are implemented up to seventh order of accuracy and tested on carefully chosen numerical examples. Tests solved include the linear advection equation, the Euler equations of gas dynamics and the blood flow equations. Particular features of interest here are: (a) attaining theoretically expected convergence rates for smooth solutions arising from demanding initial conditions, (b) compute essentially non-oscillatory profiles for discontinuous solutions and (c) attain successfully steady-steady state solutions by time marching. As compared to existing methods, the newly proposed ENO-ET scheme performs very well for all these problems.

BEV-SGD: Best Effort Voting SGD Against Byzantine Attacks for Analog-Aggregation-Based Federated Learning Over the Air

As a promising distributed learning technology, analog aggregation based federated learning over the air (FLOA) provides high communication efficiency and privacy provisioning under the edge computing paradigm. When all edge devices (workers) simultaneously upload their local updates to the parameter server (PS) through commonly shared time-frequency resources, the PS obtains the averaged update only rather than the individual local ones. While such a concurrent transmission and aggregation scheme reduces the latency and communication costs, it unfortunately renders FLOA vulnerable to Byzantine attacks. Aiming at Byzantine-resilient FLOA, this paper starts from analyzing the channel inversion (CI) mechanism that is widely used for power control in FLOA. Our theoretical analysis indicates that although CI can achieve good learning performance in the benign scenarios, it fails to work well with limited defensive capability against Byzantine attacks. Then, we propose a novel scheme called the best effort voting (BEV) power control policy that is integrated with stochastic gradient descent (SGD). Our BEV-SGD enhances the robustness of FLOA to Byzantine attacks, by allowing all the workers to send their local updates at their maximum transmit power. Under worst-case attacks, we derive the expected convergence rates of FLOA with CI and BEV power control policies, respectively. The rate comparison reveals that our BEV-SGD outperforms its counterpart with CI in terms of better convergence behavior, which is verified by experimental simulations.

Efficient multi-level hp-finite elements in arbitrary dimensions

We present an efficient algorithmic framework for constructing multi-level hp-bases that uses a data-oriented approach that easily extends to any number of dimensions and provides a natural framework for performance-optimized implementations. We only operate on the bounding faces of finite elements without considering their lower-dimensional topological features and demonstrate the potential of the presented methods using a newly written open-source library. First, we analyze a Fichera corner and show that the framework does not increase runtime and memory consumption when compared against the classical p-version of the finite element method. Then, we compute a transient example with dynamic refinement and derefinement, where we also obtain the expected convergence rates and excellent performance in computing time and memory usage.

High-order pressure estimates for Navier-Stokes Runge-Kutta solvers using stage pseudo-pressures

This note addresses the question of whether stage pseudo-pressures can be used to build high-accuracy instantaneous pressure estimates in the context of projection-based Navier-Stokes solvers. The construction of such estimates using inter-stage pseudo-pressures is not straightforward, especially when time-dependent boundary conditions are present. New formulas are introduced that circumvent the limitation imposed by this type of boundary condition. While additional numerical costs might be involved in the construction of such estimates, the formulas are robust and show how one is able to recover high-order estimates of the pressure from within a Runge-Kutta pressure-projection scheme. The proposed formulas are then numerically tested on a two-dimensional channel flow simulation with unsteady inflow condition and show expected convergence rates.

Solving Stochastic Optimization with Expectation Constraints Efficiently by a Stochastic Augmented Lagrangian-Type Algorithm

This paper considers the problem of minimizing a convex expectation function with a set of inequality convex expectation constraints. We propose a stochastic augmented Lagrangian-type algorithm—namely, the stochastic linearized proximal method of multipliers—to solve this convex stochastic optimization problem. This algorithm can be roughly viewed as a hybrid of stochastic approximation and the traditional proximal method of multipliers. Under mild conditions, we show that this algorithm exhibits [Formula: see text] expected convergence rates for both objective reduction and constraint violation if parameters in the algorithm are properly chosen, where K denotes the number of iterations. Moreover, we show that, with high probability, the algorithm has a [Formula: see text] constraint violation bound and [Formula: see text] objective bound. Numerical results demonstrate that the proposed algorithm is efficient. History: Accepted byAntonio Frangioni, Area Editor for Design & Analysis ofAlgorithms—Continuous. Funding: This work was supported by the National Natural Science Foundation of China [Grants 11971089, 11731013, 12071055, and 11871135]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplementary Information [ https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.1228 ] or is available from the IJOC GitHub software repository ( https://github.com/INFORMSJoC ) at http://dx.doi.org/10.5281/zenodo.6818229 .