Convergence Rate Of Order Research Articles

Pinning Control of Dynamical Networks With Optimal Convergence Rate

Dynamical systems coupled in the form of a complex networks have been exploited for modeling numerous real-world distributed systems. Due to the decentralized nature of dynamical networks, and inaccessibility or high control cost, in some applications, it is desired to employ pinning control techniques. A natural question that comes to mind is how one can select the proper pinned nodes and their feedback gains, (i.e., the performance metrics) to optimize the convergence rate to the homogeneous stationary state (i.e., the performance index) while the total pinning cost is kept below a given limit. This optimization problem has been converted into a semidefinite programming problem, and the optimal feedback gains have been derived analytically. Based on these results, for a given set of pinned nodes, an algorithm for determining the optimal feedback gains has been developed. Furthermore, it is shown that the edges between the pinned nodes do not impact the optimal results, and adding an edge between free nodes or between free and pinned nodes modifies the optimal convergence rate in nondecreasing order. Based on the derived analytical results, several interesting properties have been discovered. For symmetric networks, the nodes within a vertex orbit have the same optimal feedback gains, which are independent of the edges within each vertex orbit. For a number of topologies, closed-form formulas are provided for the optimal results. For a network of Chua systems, the impact of cut number is illustrated by comparing the optimal results for different topologies.

Local discontinuous Galerkin method for a third‐order singularly perturbed problem of reaction–diffusion type

AbstractA third‐order singularly perturbed problem of reaction–diffusion type is solved numerically by the local discontinuous Galerkin (LDG) method. By using the local Gauss–Radau projection for the primal variable u, but local L2 projection for the two auxiliary variables p and q, we obtain an optimal convergence rate of order (or up to a logarithmic factor) in the energy norm, uniformly in the singular perturbation parameter. Here is the degree of the piecewise polynomials used in the discrete space and N is the number of mesh intervals. The layer‐adapted meshes are used including a standard Shishkin mesh, a Bakhvalov–Shishkin mesh, and a Bakhvalov‐type mesh. Numerical experiments are given to show the sharpness of our theoretical results.

Convergence of Uniformity Criteria and the Application in Numerical Integration

Quasi-Monte Carlo (QMC) methods have been successfully used for the estimation of numerical integrations arising in many applications. In most QMC methods, low-discrepancy sequences have been used, such as digital nets and lattice rules. In this paper, we derive the convergence rates of order of some improved discrepancies, such as centered L2-discrepancy, wrap-around L2-discrepancy, and mixture discrepancy, and propose a randomized QMC method based on a uniform design constructed by the mixture discrepancy and Baker’s transformation. Moreover, the numerical results show that the proposed method has better approximation than the Monte Carlo method and many other QMC methods, especially when the number of dimensions is less than 10.

Convergence rates for the Allen–Cahn equation with boundary contact energy: the non-perturbative regime

We extend the recent rigorous convergence result of Abels and Moser (SIAM J Math Anal 54(1):114–172, 2022. https://doi.org/10.1137/21M1424925) concerning convergence rates for solutions of the Allen–Cahn equation with a nonlinear Robin boundary condition towards evolution by mean curvature flow with constant contact angle. More precisely, in the present work we manage to remove the perturbative assumption on the contact angle being close to 90^circ . We establish under usual double-well type assumptions on the potential and for a certain class of boundary energy densities the sub-optimal convergence rate of order {varepsilon ^{frac{1}{2}}} for general contact angles alpha in (0,pi ). For a very specific form of the boundary energy density, we even obtain from our methods a sharp convergence rate of order varepsilon ; again for general contact angles alpha in (0,pi ). Our proof deviates from the popular strategy based on rigorous asymptotic expansions and stability estimates for the linearized Allen–Cahn operator. Instead, we follow the recent approach by Fischer et al. (SIAM J Math Anal 52(6):6222–6233, 2020. https://doi.org/10.1137/20M1322182), thus relying on a relative entropy technique. We develop a careful adaptation of their approach in order to encode the constant contact angle condition. In fact, we perform this task at the level of the notion of gradient flow calibrations. This concept was recently introduced in the context of weak-strong uniqueness for multiphase mean curvature flow by Fischer et al. (arXiv:2003.05478v2).

Generalized Nesterov’s accelerated proximal gradient algorithms with convergence rate of order o(1/k2)

The accelerated gradient method initiated by Nesterov is now recognized to be one of the most powerful tools for solving smooth convex optimization problems. This method improves significantly the convergence rate of function values from O(1/k) of the standard gradient method down to \(O(1/k^2)\). In this paper, we present two generalized variants of Nesterov’s accelerated proximal gradient method for solving composition convex optimization problems in which the objective function is represented by the sum of a smooth convex function and a nonsmooth convex part. We show that with suitable ways to pick the sequences of parameters, the convergence rate for the function values of this proposed method is actually of order \(o(1/k^2)\). Especially, when the objective function is p-uniformly convex for \(p>2\), the convergence rate is of order \(O\left( \ln k/k^{2p/(p-2)}\right)\), and the convergence is linear if the objective function is strongly convex. By-product, we derive a forward–backward algorithm generalizing the one by Attouch–Peypouquet (SIAM J Optim 26(3):1824–1834, 2016), which produces a convergence sequence with a convergence rate of the function values of order \(o(1/k^2)\). Initial computational experiments for solving linear inverse problems with the \(l_1\)-regularization demonstrate the capabilities of the proposed algorithms.

Fast Augmented Lagrangian Method in the convex regime with convergence guarantees for the iterates

This work aims to minimize a continuously differentiable convex function with Lipschitz continuous gradient under linear equality constraints. The proposed inertial algorithm results from the discretization of the second-order primal-dual dynamical system with asymptotically vanishing damping term addressed by Boţ and Nguyen (J. Differential Equations 303:369–406, 2021), and it is formulated in terms of the Augmented Lagrangian associated with the minimization problem. The general setting we consider for the inertial parameters covers the three classical rules by Nesterov, Chambolle–Dossal and Attouch–Cabot used in the literature to formulate fast gradient methods. For these rules, we obtain in the convex regime convergence rates of order {mathcal {O}}left( 1/k^{2} right) for the primal-dual gap, the feasibility measure, and the objective function value. In addition, we prove that the generated sequence of primal-dual iterates converges to a primal-dual solution in a general setting that covers the two latter rules. This is the first result which provides the convergence of the sequence of iterates generated by a fast algorithm for linearly constrained convex optimization problems without additional assumptions such as strong convexity. We also emphasize that all convergence results of this paper are compatible with the ones obtained in Boţ and Nguyen (J. Differential Equations 303:369–406, 2021) in the continuous setting.

A Hybrid High-Order Method for Quasilinear Elliptic Problems of Nonmonotone Type

In this paper, we design and analyze a Hybrid-High Order (HHO) approximation for a class of quasilinear elliptic problems of nonmonotone type. The proposed method has several advantages, for instance, it supports arbitrary order of approximation and general polytopal meshes. The key ingredients involve local reconstruction and high-order stabilization terms. Existence and uniqueness of the discrete solution are shown by Brouwer's fixed point theorem and contraction result. A priori error estimate is shown in discrete energy norm that shows optimal order convergence rate. Numerical experiments are performed to substantiate the theoretical results.

Learning transport processes with machine intelligence

Transport processes ruled by complex micro-physics and impractical to theoretical investigation may exhibit emergent behavior describable by mathematical expressions. Such information, while implicitly contained in the results of microscopic-scale numerical simulations close to first principles or experiments is not in a form suitable for macroscopic modelling. Here we present a machine learning approach that leverages such information to deploy micro-physics informed transport flux representations applicable to a continuum mechanics description. One issue with deep neural networks, arguably providing the most generic of such representations, is their noisiness which is shown to break the performance of numerical schemes. The matter is addressed and a methodology suitable for schemes characterised by second order convergence rate is presented. The capability of the methodology is demonstrated through an idealized study of the long standing problem of heat flux suppression relevant to fusion and cosmic plasmas. Symbolic representations, although potentially less generic, are straightforward to use in numerical schemes and theoretical analysis, and can be even more accurate as shown by the application to the same problem of an advanced symbolic regression tool. These results are a promising initial step to filling the gap between micro and macro in this important area of modeling.

The convergence rate for difference approximations to fractional boundary value problems

We study fractional differential equations defined in a bounded domain and with Dirichlet boundary conditions. To discretize these equations and in particular the fractional derivative of order α we consider the Grünwald–Letnikov approximation which is first order accurate when we have an open domain but it can be of lower order in bounded domains and sometimes not consistent. Because the fractional derivative is nonlocal, the accuracy of its approximation near the boundary is strongly affected by the cut of the domain. However, as we will prove, the numerical methods are convergent in the discrete L∞ and L2 norms and the first order rate of convergence can be recovered. In some cases, regarding the order of accuracy of the fractional derivative approximation, we can go down an order α, from one, without destroying the convergence of the scheme.

Split-step theta Milstein methods for SDEs with non-globally Lipschitz diffusion coefficients

The present work aims to analyze mean-square convergence rates of split-step theta Milstein methods with method parameters θ∈[12,1] for stochastic differential equations with non-globally Lipschitz diffusion coefficients. The expected convergence rate of order one is successfully proved under more relaxed conditions, compared to existing relevant results. Numerical examples are also offered to confirm the theoretical findings.

Error Estimates of the Godunov Method for the Multidimensional Compressible Euler System

We derive a priori error estimates of the Godunov method for the multidimensional compressible Euler system of gas dynamics. To this end we apply the relative energy principle and estimate the distance between the numerical solution and the strong solution. This yields also the estimates of the L^2-norms of the errors in density, momentum and entropy. Under the assumption, that the numerical density is uniformly bounded from below by a positive constant and that the energy is uniformly bounded from above and stays positive, we obtain a convergence rate of 1/2 for the relative energy in the L^1-norm, that is to say, a convergence rate of 1/4 for the L^2-error of the numerical solution. Further, under the assumption—the total variation of the numerical solution is uniformly bounded, we obtain the first order convergence rate for the relative energy in the L^1-norm, consequently, the numerical solution converges in the L^2-norm with the convergence rate of 1/2. The numerical results presented are consistent with our theoretical analysis.

Time-dependent Hamiltonian Simulation of Highly Oscillatory Dynamics and Superconvergence for Schrödinger Equation

We propose a simple quantum algorithm for simulating highly oscillatory quantum dynamics, which does not require complicated quantum control logic for handling time-ordering operators. To our knowledge, this is the first quantum algorithm that is both insensitive to the rapid changes of the time-dependent Hamiltonian and exhibits commutator scaling. Our method can be used for efficient Hamiltonian simulation in the interaction picture. In particular, we demonstrate that for the simulation of the Schrödinger equation, our method exhibits superconvergence and achieves a surprising second order convergence rate, of which the proof rests on a careful application of pseudo-differential calculus. Numerical results verify the effectiveness and the superconvergence property of our method.

A systematic approach for potentials on closely packed cells using the null-field boundary integral equation in conjunction with the degenerate kernel and eigenfunction expansion

A systematic approach based on the null-field integral formula is used to determine the electric potential of a tissue with many cells stimulated by remote electric fields. When the cells are very close to each other, the problem becomes nearly singular and the accuracy of the solution deteriorates. However, in the proposed approach, the highly accurate results are obtained because the separable kernel (degenerate kernel) and eigenfunction expansion are introduced to capture the geometry property in the integral formulation. Only boundary nodes are required instead of boundary elements to satisfy the boundary conditions or interface conditions. The proposed approach could be seen as one kind of meshless and semi-analytical methods. In addition, the error just stems from the number of truncation terms of the eigenfuntion expansion and the convergence rate of exponential order is better than the linear order of the conventional boundary element method. For the problem of closely packed cells, the boundary density of sharp variation could be accurately simulated or captured by increasing the number of terms of eigenfunctions. Finally, the acceptable results are shown to see the efficiency and accuracy of the proposed approach by the given numerical examples including one, three and twenty cells.

Distributed Learning With Sparsified Gradient Differences

A very large number of communications are typically required to solve distributed learning tasks, and this critically limits scalability and convergence speed in wireless communications applications. In this paper, we devise a <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</b> radient <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</b> escent method with <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</b> parsification and <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</b> rror <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</b> orrection ( <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">GD-SEC</b> ) to improve the communications efficiency in a general worker-server architecture. Motivated by a variety of wireless communications learning scenarios, GD-SEC reduces the number of bits per communication from worker to server with no degradation in the order of the convergence rate. This enables larger scale model learning without sacrificing convergence or accuracy. At each iteration of GD-SEC, instead of directly transmitting the entire gradient vector, each worker computes the difference between its current gradient and a linear combination of its previously transmitted gradients, and then transmits the sparsified gradient difference to the server. A key feature of GD-SEC is that any given component of the gradient difference vector will not be transmitted if its magnitude is not sufficiently large. An error correction technique is used at each worker to compensate for the error resulting from sparsification. We prove that GD-SEC is guaranteed to converge for strongly convex, convex, and nonconvex optimization problems with the same order of convergence rate as GD. Furthermore, if the objective function is strongly convex, GD-SEC has a fast linear convergence rate. Numerical results not only validate the convergence rate of GD-SEC but also explore the communication bit savings it provides. Given a target accuracy, GD-SEC can significantly reduce the communications load compared to the best existing algorithms without slowing down the optimization process.

On the efficiency of 5(4) RK-embedded pairs with high order compact scheme and Robin boundary condition for options valuation

When solving the American options with or without dividends, numerical methods often obtain lower convergence rates if further treatment is not implemented even using high-order schemes. In this article, we present a fast and explicit fourth-order compact scheme for solving the free boundary options. In particular, the early exercise features with the asset option and option sensitivity are computed based on a coupled of nonlinear PDEs with fixed boundaries for which a high order analytical approximation is obtained. Furthermore, we implement a new treatment at the left boundary by introducing a third-order Robin boundary condition. Rather than computing the optimal exercise boundary from the analytical approximation, we simply obtain it from the asset option based on the linear relationship at the left boundary. As such, a high order convergence rate can be achieved. We validate by examples that the improvement at the left boundary yields a fourth-order convergence rate without further implementation of mesh refinement, Rannacher time-stepping, and/or smoothing of the initial condition. Furthermore, we extensively compare the performance of our present method with several 5(4) Runge–Kutta pairs and observe that Dormand and Prince and Bogacki and Shampine 5(4) pairs are faster and provide more accurate numerical solutions. Based on numerical results and comparison with other existing methods, we can validate that the present method is very fast and provides more accurate solutions with very coarse grids.

Convergence Analysis and Numerical Studies for Linearly Elastic Peridynamics with Dirichlet-Type Boundary Conditions

The nonlocal models of peridynamics have successfully predicted fractures and deformations for a variety of materials. In contrast to local mechanics, peridynamic boundary conditions must be defined on a finite volume region outside the body. Therefore, theoretical and numerical challenges arise in order to properly formulate Dirichlet-type nonlocal boundary conditions, while connecting them to the local counterparts. While a careless imposition of local boundary conditions leads to a smaller effective material stiffness close to the boundary and an artificial softening of the material, several strategies were proposed to avoid this unphysical surface effect. In this work, we study convergence of solutions to nonlocal state-based linear elastic model to their local counterparts as the interaction horizon vanishes, under different formulations and smoothness assumptions for nonlocal Dirichlet-type boundary conditions. Our results provide explicit rates of convergence that are sensitive to the compatibility of the nonlocal boundary data and the extension of the solution for the local model. In particular, under appropriate assumptions, constant extensions yield \(\frac{1}{2}\) order convergence rates and linear extensions yield \(\frac{3}{2}\) order convergence rates. With smooth extensions, these rates are improved to quadratic convergence. We illustrate the theory for any dimension \(d\ge 2\) and numerically verify the convergence rates with a number of two-dimensional benchmarks, including linear patch tests, manufactured solutions, and domains with curvilinear surfaces. Numerical results show a first-order convergence for constant extensions and second-order convergence for linear extensions, which suggests a possible room of improvement in the future convergence analysis.

A Two-Level Nonconforming Rotated Quadrilateral Finite Element Method for the Stationary Navier–Stokes Equations

In this paper, we propose a two-level nonconforming rotated finite element (TNRFE) method for solving the Navier–Stokes equations. A new nonconforming rotated finite element (NRFE) method was proposed by Douglas added by conforming bubbles to velocity and discontinuous piecewise constant to the pressure on quadrilateral elements possessing favorable stability properties. The TNRFE method involves solving a small Navier–Stokes problem on a coarse mesh with mesh size H and a large linearized Navier–Stokes problem on a fine mesh with mesh size h by the NRFE method. If we choose h = O H 2 , the TNRFE method gives the convergence rate of the same order as that of the NRFE method. Compared with the NRFE method, the TNRFE method can save a large amount of CPU time. In this paper, the stability of the approximate solutions and the error estimates are proved. Finally, the numerical experiments are given, and results indicate that the method is practicable and effective.

An efficient algorithm for the numerical evaluation of pseudo differential operator with error estimation

&lt;abstract&gt;&lt;p&gt;In this paper we introduce an efficient and new numerical algorithm for evaluating a pseudo differential operator. The proposed algorithm is time saving and fruitful. The theoretical as well as numerical error estimation of the algorithm is established, together with its stability analysis. We have provided numerical illustrations and established that the numerical findings echo the analytical findings. The proposed technique has a convergence rate of order three. CPU time of computation is also listed. Trueness of numerical findings are validated using figures.&lt;/p&gt;&lt;/abstract&gt;

High order one-step methods for backward stochastic differential equations via Itô-Taylor expansion

&lt;p style='text-indent:20px;'&gt;In this work, by combining the Feynman-Kac formula with an Itô-Taylor expansion, we propose a class of high order one-step schemes for backward stochastic differential equations, which can achieve at most six order rate of convergence and only need the terminal conditions on the last one step. Numerical experiments are carried out to show the efficiency and high order accuracy of the proposed schemes.&lt;/p&gt;

Nonlinear Evolutionary PDE-Based Refinement of Optical Flow

The goal of this paper is to propose two nonlinear variational models for obtaining a refined motion estimation from an image sequence. Both the proposed models can be considered as a part of a generalized framework for an accurate estimation of physics-based flow fields such as rotational and fluid flow. The first model is novel in the sense that it is divided into two phases: the first phase obtains a crude estimate of the optical flow and then the second phase refines this estimate using additional constraints. The correctness of this model is proved using an evolutionary PDE approach. The second model achieves the same refinement as the first model, but in a standard manner, using a single functional. A special feature of our models is that they permit us to provide efficient numerical implementations through the first-order primal-dual Chambolle-Pock scheme. Both the models are compared in the context of accurate estimation of angle by performing an anisotropic regularization of the divergence and curl of the flow respectively. We observe that, although both the models obtain the same level of accuracy, the two-phase model is more efficient. In fact, we empirically demonstrate that the single-phase and the two-phase models have convergence rates of order O(1/N2) and O(1/N) respectively.