Convergence Rate Research Articles

Lithospheric Deformation of Far‐Field Terranes in Response to the India–Asia Collision

ABSTRACTThe influence of the India–Asia collision is far‐reaching on the Cenozoic continental deformation in East Asia. Several cratonic lithospheres surrounding Tibet exhibit distinct lithospheric morphologies. However, the mechanisms driving these diverse responses of strong terranes remain incompletely understood. Here, we conduct thermo‐mechanical models to explore the effects of the width of the mobile belts, lithospheric properties and convergence rate on the deformation of the strong terranes. The model results reveal three different deformation modes. Far‐field strong terranes are vulnerable to delamination or underthrusting when the mobile belt is narrow and detached, whereas they remain largely undeformed when it is wide and strong. These deformation patterns align with the observed lithospheric structures around the Tibetan Plateau and are primarily influenced by the convergence rate and proximity to the collision front. Our findings provide new insights into the lithospheric deformation mechanisms of cratonic terranes in response to continental collision.

Lippmann-Schwinger equation representation of Green's function and its preconditioned generalized over-relaxation iterative solution in wavelet domain

The calculation of Green's function is the core of seismic forward and inverse methods based on integral operators. When the Lippmann-Schwinger (L-S) equation is used to calculate Green's function in strongly scattering media, both the Born scattering series and the numerical iterative method encounter issues of slow convergence or divergence. Although the renormalization method derived from quantum mechanics can effectively address the convergence problem of Born scattering series in strong scattering problems, it is acknowledgeed that the convergence conditions and rates of convergence of different reformulation series may vary, and no universal convergence reformulation scattering series exists. Numerical methods for solving integral equations tend to be more general and mathematically robust. In this work, we focus on the numerical solution method of L-S equations. By using a wavelet-domain preconditioner to a reformulated or equivalent Lippmann-Schwinger (L-S) equation, we present an iterative method for numerically solving the equivalent L-S equation aimed at improving the rate of convergence and iteration efficiency in strongly inhomogeneous media. Following Jakobsen et al. (2020), we first introduce a small imaginary component into the background wave number，then rewrite the L-S equation to derive the equivalent complex wave number L-S equation. This reformulation ensures that the coefficient matrix exhibits a banded structure after numerical discretization, allowing the wavelet coefficient matrix to maintain good sparsity. We employ a multi-level fill-in incomplete LU (ILU) factorization method along with a block ILU-based algebraic recursive multilevel solve (ARMS) method in the wavelet domain to generate sparse approximate inverses as preconditioning operators, thereby accelerating the convergence of the generalized successive over-relaxation (GSOR) iterative method. This method is applied to compute numerical Green's functions in strongly inhomogeneous media. Numerical results demonstrate that our method yields simulation outcomes consistent with those obtained from the direct method for solving the original real wave number L-S equation. By testing various preconditioners, we find that the ARMS preconditioner offers significant advantages in operator generation efficiency and non-zero element filling ratio, effectively accelerating the convergence of the GSOR iterative method while achieving higher computational efficiency.

Penalized spline estimation of principal components for sparse functional data: Rates of convergence

Penalized spline estimation of principal components for sparse functional data: Rates of convergence

General-Kindred physics-informed neural network to the solutions of singularly perturbed differential equations

Physics-informed neural networks (PINNs) have become a promising research direction in the field of solving partial differential equations (PDEs). Dealing with singular perturbation problems continues to be a difficult challenge in the field of PINN. The solution of singular perturbation problems often exhibits sharp boundary layers and steep gradients, and traditional PINN cannot achieve approximation of boundary layers. In this manuscript, we propose the General-Kindred physics-informed neural network (GKPINN) for solving singular perturbation differential equations (SPDEs). This approach utilizes asymptotic analysis to acquire prior knowledge of the boundary layer from the equation and establishes a novel network to assist PINN in approximating the boundary layer. It is compared with traditional PINN by solving examples of one-dimensional, two-dimensional, and time-varying SPDE equations. The research findings underscore the exceptional performance of our novel approach, GKPINN, which delivers a remarkable enhancement in reducing the L2 error by two to four orders of magnitude compared to the established PINN methodology. This significant improvement is accompanied by a substantial acceleration in convergence rates, without compromising the high precision that is critical for our applications. Furthermore, GKPINN still performs well in extreme cases with perturbation parameters of 1×10−38, demonstrating its excellent generalization ability.

Enhancing improved advection upstream splitting method on triangular grids: A hybrid approach for improved stability and accuracy in compressible flow simulations

This paper introduces NAUSM+M+AUFS (New Improved Advection Upstream Splitting Method Plus Artificially Upstream Flux Vector Splitting), a novel hybrid computational scheme for simulating compressible flows on triangular grids. The AUSM+M (Improved Advection Upstream Splitting Method) method is enhanced through two key modifications to boost numerical stability and robustness in high Mach number and hypersonic flows. The first modification redefines the interfacial numerical sound velocity, reducing shock anomalies and improving shock-capturing by integrating velocity and characteristic sound speed parameters. The second modification addresses the insufficiency of the pressure flux dissipation term at supersonic speeds by introducing a formulation that increases dissipation proportionally to the Mach number, thereby enhancing performance in high-speed flows. These enhancements constitute the NAUSM+M method. The NAUSM+M+AUFS scheme combines the strengths of NAUSM+M and AUFS (Artificially Upstream Flux Vector Splitting) methods, particularly in overcoming the limitations of NAUSM+M in handling shock instabilities and the carbuncle phenomenon on structure triangular grids. A dynamic switching function adjusts the weighting between NAUSM+M and AUFS, optimizing accuracy and stability based on local flow conditions. Numerical tests demonstrate that NAUSM+M+AUFS significantly outperforms AUSM+M, NAUSM+M, and AUFS, effectively eliminating the carbuncle phenomenon and providing smooth shock wave contours. In steady flow analysis, the new hybrid method achieves convergence speeds comparable to AUFS and shows 15% to 45% superior convergence accelerating than AUSM+M, depending on the convergence rate. In addition, in steady flow analysis, the accuracy of NAUSM+M+AUFS is 46% better than that of AUFS. This approach represents a significant advancement, offering a robust, accurate, and efficient solution for high-speed aerodynamic simulations, with broad applicability across various compressible flow challenges.

Simple yet effective adaptive activation functions for physics-informed neural networks

Physics-informed neural networks (PINNs) gained widespread advancements in solving differential equations, where the performance tightly hinges on the choice of activation functions that are inefficient when selected manually. To tackle this issue, we propose two straightforward yet powerful adaptive activation functions: a weighted average function that adjusts activation functions by directly manipulating their weights, and a L2-normalization function that compresses learnable parameters. These methods ensure a consistent sum of weights for each activation function, thereby enhancing optimization efficiency. We assess the performance of these approaches across a range of differential equation problems, encompassing Poisson equation, Wave equation, Burgers equation, Navier-Stokes equation, and linear/nonlinear solid mechanics problems. Through comparisons with exact solutions, we demonstrate significant improvements in convergence rate and solution accuracy. Our results underscore the efficacy of these techniques, providing a simple yet promising pathway for augmenting PINN performance across diverse applications. The source codes and software implementation are available at https://github.com/jzhange/AAF-for-PINNs.

Acoustic and soliton propagation using fully-discrete energy preserving partially implicit scheme in homogeneous and heterogeneous mediums

This study presents an energy preserving partially implicit scheme for the simulation of wave propagation in homogeneous and heterogeneous mediums. Despite its implicit nature, the developed scheme does not require any explicit numerical or analytical inversion of the coefficient matrix. Theoretical analysis and numerical experiments are performed to validate the energy preserving properties of the fully-discrete scheme. Convergence analysis is also performed to assess the rate of convergence of the developed scheme. The efficiency and accuracy of the developed scheme are validated by numerical solutions of wave propagation in layered heterogeneous mediums. Furthermore, simulations of soliton propagation following nonlinear sine-Gordon and Klein-Gordon equations in homogeneous and heterogeneous mediums are discussed. Numerical solutions are also compared with the results available in the literature. The present method accurately resolves the physical characteristics of the chosen problems, competing well with existing multi-stage time-integration methods. Moreover, it has significantly lower computational complexity than the four-stage, fourth-order Runge-Kutta-Nyström method.

Sieve estimation of the accelerated mean model based on panel count data

Panel count data are gathered when subjects are examined at discrete times during a study, and only the number of recurrent events occurring before each examination time is recorded. We consider a semiparametric accelerated mean model for panel count data in which the effect of the covariates is to transform the time scale of the baseline mean function. Semiparametric inference for the model is inherently challenging because the finite-dimensional regression parameters appear in the argument of the (infinite-dimensional) functional parameter, i.e., the baseline mean function, leading to the phenomenon of bundled parameters. We propose sieve pseudolikelihood and likelihood methods to construct the random criterion function for estimating the model parameters. An inexact block coordinate ascent algorithm is used to obtain these estimators. We establish the consistency and rate of convergence of the proposed estimators, as well as the asymptotic normality of the estimators of the regression parameters. Novel consistent estimators of the asymptotic covariances of the estimated regression parameters are derived by leveraging the counting process associated with the examination times. Comprehensive simulation studies demonstrate that the optimization algorithm is much less sensitive to the initial values than the Newton–Raphson method. The proposed estimators perform well for practical sample sizes, and are more efficient than existing methods. An example based on real data shows that due to this efficiency gain, the proposed method is better able to detect the significance of practically meaningful covariates than an existing method.

Unified algorithms for distributed regularized linear regression model

In recent years, distributed statistical models have received increasing attention for large-scale data analysis. On the one hand, data sets come from multiple data sources, and are stored in different locations due to limited bandwidth and storage, or privacy protocols, directly centralizing all data together is impossible. On the other hand, the size of data is so large that it is difficult or inefficient to analyze data together. There are two main research aspects to using distributed statistical models to analyze large-scale data. The first one is to study the statistical convergence rate under some mild assumptions. The second one is to establish fast and efficient optimization algorithms considering the property of the loss function. There is a lot of research on the first aspect, but relatively little research on the second one. Motivated by this, we consider the construction of unified algorithms for distributed linear regression with different losses and regularizers. As a result, we designed two type methods, proximal alternating direction method of multipliers (pADMM) and distributed accelerated proximal gradient method with line-search (DAPGL). In order to demonstrate the efficiency of the proposed algorithms, we perform numerical experiments on the distributed Huber-Lasso model and Huber-Group-Lasso model. In view of the numerical results, we can observe that these two algorithms are more competitive than some of state-of-art algorithms. In particular, DAPGL algorithm performs better than pADMM in most cases.

A block α-circulant based preconditioned MINRES method for wave equations

In this work, we propose an absolute value block α-circulant preconditioner for the minimal residual (MINRES) method to solve an all-at-once system arising from the discretization of wave equations. Motivated by the absolute value block circulant preconditioner proposed in [E. McDonald, J. Pestana, and A. Wathen. SIAM J. Sci. Comput., 40(2):A1012-A1033, 2018], we propose an absolute value version of the block α-circulant preconditioner. Since the original block α-circulant preconditioner is non-Hermitian in general, it cannot be directly used as a preconditioner for MINRES. Our proposed preconditioner is the first Hermitian positive definite variant of the block α-circulant preconditioner for the concerned wave equations, which fills the gap between block α-circulant preconditioning and the field of preconditioned MINRES solver. The matrix-vector multiplication of the preconditioner can be fast implemented via fast Fourier transforms. Theoretically, we show that for a properly chosen α the MINRES solver with the proposed preconditioner achieves a linear convergence rate independent of the matrix size. To the best of our knowledge, this is the first attempt to generalize the original absolute value block circulant preconditioner in the aspects of both theory and performance the concerned problem. Numerical experiments are given to support the effectiveness of our preconditioner, showing that the expected optimal convergence can be achieved.

Convergence Rate for Randomly Weighted Sums of Random Variables and Its Application

Convergence Rate for Randomly Weighted Sums of Random Variables and Its Application

Application of an Improved Ant Colony Optimization Algorithm of Hybrid Strategies Using Scheduling for Patient Management in Hospitals

To balance the convergence speed and solution diversity and enhance optimization performance when addressing large-scale optimization problems, this research study presents an improved ant colony optimization (ICMPACO) technique. Its foundations include the co-evolution mechanism, the multi-population strategy, the pheromone diffusion mechanism, and the pheromone updating method. The suggested ICMPACO approach separates the ant population into elite and common categories and breaks the optimization problem into several sub-problems to boost the convergence rate and prevent slipping into the local optimum value. To increase optimization capacity, the pheromone update approach is applied. Ants emit pheromone at a certain spot, and that pheromone progressively spreads to a variety of nearby regions thanks to the pheromone diffusion process. Here, the real gate assignment issue and the travelling salesman problem (TSP) are chosen for the validation of the performance for the optimization of the ICMPACO algorithm. The experiment's findings demonstrate that the suggested ICMPACO method can successfully solve the gate assignment issue, find the optimal optimization value in resolving TSP, provide a better assignment outcome, and exhibit improved optimization ability and stability. The assigned efficiency is comparatively higher than earlier ones. With an assigned efficiency of 83.5%, it can swiftly arrive at the ideal gate assignment outcome by assigning 132 patients to 20 gates of hospital testing rooms. To minimize the patient's overall hospital processing time, this algorithm was specifically employed with a better level of efficiency to create appropriate scheduling in the hospital.

Efficient second-order accurate exponential time differencing for time-fractional advection-diffusion-reaction equations with variable coefficients

Time-fractional advection-diffusion-reaction type equations are useful for characterizing anomalous transport processes. In this paper, linearly implicit as well as explicit generalized exponential time differencing (GETD) schemes are proposed for solving a class of such equations having time-space dependent coefficients. The implicit scheme, being unconditionally stable, is robust in handling the numerical instabilities in problems where the advection term is dominant. Regarding the error analysis, uniformly optimal second-order convergence rates are derived using time-graded meshes to counter the effect of the inherent singularity of the continuous solution. Implementation of generalized exponential integrators requires computing the action of Mittag-Leffler function of matrices on a vector, or on a matrix in the case of the implicit scheme. For cost-effective implementation, using global Padé approximants these computation tasks get reduced to solving linear systems. A new approach based on Sylvester equation formulation of the resulting linear systems is developed in this paper. This technique leads to significantly faster algorithms for implementing the GETD schemes. Numerical experiments are provided to illustrate the theoretical findings and to assert the efficiency of the Sylvester equation based approach. Application of this approach to an existing GETD scheme for solving a nonlinear subdiffusion problem is also discussed.

On applying deflation and flexible preconditioning to the adaptive Simpler GMRES method for Sylvester tensor equations

Due to their multidimensional structure, the use of tensors when solving linear matrix equations has been one of the most explored topics over the last decade. In particular, the development of Krylov subspace methods for solving tensor equations is currently attracting a great deal of interest. In this work, we develop the adaptive simpler GMRES method based on tensor format (Ad-SGMRES−BTF) for solving Sylvester tensor equations, in order to address the numerical instability of the simpler version of GMRES based on tensor format (SGMRES−BTF). In addition, an efficient Krylov subspace method, formed by the integration of adaptive deflation and preconditioning and called FAd-SGMRES-D−BTF(m,k) -in which m is the restart parameter and k is the number of harmonic Ritz pairs-, is proposed. Note that the idea behind the FAd-SGMRES-D−BTF(m,k) method is to reuse certain key information, which can be obtained from the current tensor Krylov subspace in the next cycle of the FAd-SGMRES-D−BTF(m) method. It is also important to note that FAd-SGMRES-D−BTF(m,k) represents the first attempt to synchronously combine flexible preconditioning and deflation techniques, with the aim of improving the convergence rate and reducing the computational cost of Ad-SGMRES−BTF(m). Finally, the computational cost of each cycle of the proposed methods when applied to a three-dimensional Sylvester tensor equation is estimated. Moreover, numerical experiments on synthetic and real-world applications demonstrate the potential of Ad-SGMRES−BTF and FAd-SGMRES-D−BTF(m) to solve Sylvester tensor equations.

Reducing polynomial degree by one for inner-stage operators affects neither stability type nor accuracy order of the Runge–Kutta discontinuous Galerkin method

The Runge–Kutta (RK) discontinuous Galerkin (DG) method is a mainstream numerical algorithm for solving hyperbolic equations. In this paper, we use the linear advection equation in one and two dimensions as a model problem to prove the following results: For an arbitrarily high-order RKDG scheme in Butcher form, as long as we use the P k P^k approximation in the final stage, even if we drop the k k th-order polynomial modes and use the P k − 1 P^{k-1} approximation for the DG operators at all inner RK stages, the resulting numerical method still maintains the same type of stability and convergence rate as those of the original RKDG method. Numerical examples are provided to validate the analysis. The numerical method analyzed in this paper is a special case of the Class A RKDG method with stage-dependent polynomial spaces proposed by Chen, Sun, and Xing [arXiv preprint, arXiv:2402.15150, 2024]. Our analysis provides theoretical justifications for employing cost-effective and low-order spatial discretization at specific RK stages for developing more efficient DG schemes without affecting stability type and accuracy order of the original method.

Survey Paper on Quantum Deep Reinforcement Learning for Robot Navigation Tasks

Abstract: Quantum deep reinforcement learning is an integration of the principles of quantum computing with deep reinforcement learning in an effort to advance robot navigation tasks. Due to the explosion of the state and action spaces, traditionally DRL methods are limited in scalability and efficiency. QDRL, however, uses quantum superposition and entanglement for more efficient processing of big amounts of information-it enables robots to learn optimal policies for navigating complex environments. We introduce here a new QDRL framework in which quantum neural networks encode policy functions and value estimates. We demonstrate how quantum circuits can take advantage of multi-dimensional state representations in order to strengthen the power of a robot in changing its environment over time. Benchmark experiments on navigation tasks reflect phenomenal acceleration in learning performance along with better decision-making accuracy compared to classical DRL methods. We then consider integrating such quantum algorithms, as Grover's search, to improve exploration strategies in sparse reward settings. The experiments show that QDRL speeds up convergence rates and provides robots with the possibility to react more rapidly to unexpected obstacles. It discusses new applications for quantum computing of autonomous systems, potentially applied to robotics, autonomous vehicles, and other environments which require complex decision making. Some lines of work in the future are optimization of the resources needed for quantum computers and hybrid architectures for the resolution of classical problems in navigation

Investigation of the approximate solution of one class of curvilinear integral equations by the projection method

UDC 517.9 We prove the existence theorem for the normal derivative of the double-layer potential and establish the formula for its evaluation. A new method for the construction of quadrature formulas for the normal derivatives of simple- and double-layer potentials is developed, and the error estimates are obtained for the constructed quadrature formulas. By using these quadrature formulas, the integral equation of the exterior Dirichlet boundary-value problem for the Helmholtz equation in two-dimensional space is replaced by a system of algebraic equations, and the existence and uniqueness of the solution to this system is proved. The convergence of the solution of the system of algebraic equations to the exact solution of the integral equation at the control points is proved and the convergence rate of the method is determined.

Mesh optimization for the virtual element method: How small can an agglomerated mesh become?

We present an optimization procedure for generic polygonal or polyhedral meshes, tailored for the Virtual Element Method (VEM). Once the local quality of the mesh elements is analyzed through a quality indicator specific to the VEM, groups of elements are agglomerated to optimize the global mesh quality. A user-set parameter regulates the percentage of mesh elements, and consequently of faces, edges, and vertices, to be removed. This significantly reduces the total number of degrees of freedom associated with a discrete problem defined over the mesh with the VEM, particularly for high-order formulations. We show how the VEM convergence rate is preserved in the optimized meshes, and the approximation errors are comparable with those obtained with the original ones. We observe that the optimization has a regularization effect over low-quality meshes, removing the most pathological elements. In such cases, these “badly-shaped” elements yield a system matrix with very large condition number, which may cause the VEM to diverge, while the optimized meshes lead to convergence. We conclude by showing how the optimization of a real CAD model can be used effectively in the simulation of a time-dependent problem.

Refining uniform approximation algorithm for low-rank Chebyshev embeddings

Abstract Nowadays, low-rank approximations are a critical component of many numerical procedures. Traditionally the problem of low-rank approximation of matrices is solved in unitary invariant norms such as Frobenius or spectral norm due to the existence of efficient methods for constructing approximations. However, recent results discover the potential of low-rank approximations in the Chebyshev norm, which naturally arises in many applications. In this paper, we investigate the problem of uniform approximation of vectors, which is the main component in the low-rank approximations of matrices in the Chebyshev norm. The principal novelty of this paper is the accelerated algorithm for solving uniform approximation problems. We also analyze the iterative procedure of the proposed algorithm and demonstrate that it has a geometric convergence rate. Finally, we provide an extensive numerical evaluation, which demonstrates the effectiveness of the proposed procedures.

A Global Method for Approximating Caputo Fractional Derivatives—An Application to the Bagley–Torvik Equation

In this paper, we propose a global numerical method for approximating Caputo fractional derivatives of order α(Dαf)(y)=1Γ(m−α)∫0y(y−x)m−α−1f(m)(x)dx,y>0, with m−1<α≤m,m∈N. The numerical procedure is based on approximating f(m) by the m-th derivative of a Lagrange polynomial, interpolating f at Jacobi zeros and some additional nodes suitably chosen to have corresponding logarithmically diverging Lebsegue constants. Error estimates in a uniform norm are provided, showing that the rate of convergence is related to the smoothness of the function f according to the best polynomial approximation error and depending on order α. As an application, we approximate the solution of a Volterra integral equation, which is equivalent in some sense to the Bagley–Torvik initial value problem, using a Nyström-type method. Finally, some numerical tests are presented to assess the performance of the proposed procedure.