Motivated by Fredholm theory, we develop a framework to establish the convergence of spectral methods for operator equations $\mathcal L u = f$. The framework posits the existence of a left-Fredholm regulator for $\mathcal L$ and the existence of a sufficiently good approximation of this regulator. Importantly, the numerical method itself need not make use of this extra approximant. We apply the framework to finite-section and collocation-based numerical methods for solving differential equations with periodic boundary conditions and to solving Riemann--Hilbert problems on the unit circle. We also obtain improved results concerning the approximation of eigenvalues of differential operators with periodic coefficients.
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This study investigates a class of initial-boundary value problems pertaining to the time-fractional mixed sub-diffusion and diffusion-wave equation (SDDWE). To facilitate the development of a numerical method and analysis, the original problem is transformed into a new integro-differential model which includes the Caputo derivatives and the Riemann-Liouville fractional integrals with orders belonging to (0,1). By providing an a priori estimate of the solution, we have established the existence and uniqueness of a numerical solution for the problem. We propose a second-order method to approximate the fractional Riemann-Liouville integral and employ an L2 type formula to approximate the Caputo derivative. This results in a method with a temporal accuracy of second-order for approximating the considered model. The proof of the unconditional stability of the proposed difference scheme is established. Moreover, we demonstrate the proposed method's potential to construct and analyze a second-order L2-type numerical scheme for a broader class of the time-fractional mixed SDDWEs with multi-term time-fractional derivatives. Numerical results are presented to assess the accuracy of the method and validate the theoretical findings.
The strong convergence of numerical methods for stochastic differential equations (SDEs) for $t\in[0,\infty)$ is proved. The result is applicable to any one-step numerical methods with Markov property that have the finite time strong convergence and the uniformly bounded moment. In addition, the convergence of the numerical stationary distribution to the underlying one can be derived from this result. To demonstrate the application of this result, the strong convergence in the infinite horizon of the backward Euler-Maruyama method in the $L^p$ sense for some small $p\in (0,1)$ is proved for SDEs with super-linear coefficients, which is also a a standalone new result. Numerical simulations are provided to illustrate the theoretical results.
This paper presents a decentralized algorithm for solving distributed convex optimization problems in dynamic networks with time-varying objectives. The unique feature of the algorithm lies in its ability to accommodate a wide range of communication systems, including previously unsupported ones, by abstractly modeling the information exchange in the network. Specifically, it supports a novel communication protocol based on the "over-the-air" function computation (OTA-C) technology, that is designed for an efficient and truly decentralized implementation of the consensus step of the algorithm. Unlike existing OTA-C protocols, the proposed protocol does not require the knowledge of network graph structure or channel state information, making it particularly suitable for decentralized implementation over ultra-dense wireless networks with time-varying topologies and fading channels. Furthermore, the proposed algorithm synergizes with the "superiorization" methodology, allowing the development of new distributed algorithms with enhanced performance for the intended applications. The theoretical analysis establishes sufficient conditions for almost sure convergence of the algorithm to a common time-invariant solution for all agents, assuming such a solution exists. Our algorithm is applied to a real-world distributed random field estimation problem, showcasing its efficacy in terms of convergence speed, scalability, and spectral efficiency. Furthermore, we present a superiorized version of our algorithm that achieves faster convergence with significantly reduced energy consumption compared to the unsuperiorized algorithm.
It is well known that the Euler method for approximating the solutions of a random ordinary differential equation $\mathrm{d}X_t/\mathrm{d}t = f(t, X_t, Y_t)$ driven by a stochastic process $\{Y_t\}_t$ with $\theta$-H\"older sample paths is estimated to be of strong order $\theta$ with respect to the time step, provided $f=f(t, x, y)$ is sufficiently regular and with suitable bounds. Here, it is proved that, in many typical cases, further conditions on the noise can be exploited so that the strong convergence is actually of order 1, regardless of the H\"older regularity of the sample paths. This applies for instance to additive or multiplicative It\^o process noises (such as Wiener, Ornstein-Uhlenbeck, and geometric Brownian motion processes); to point-process noises (such as Poisson point processes and Hawkes self-exciting processes, which even have jump-type discontinuities); and to transport-type processes with sample paths of bounded variation. The result is based on a novel approach, estimating the global error as an iterated integral over both large and small mesh scales, and switching the order of integration to move the critical regularity to the large scale. The work is complemented with numerical simulations illustrating the strong order 1 convergence in those cases, and with an example with fractional Brownian motion noise with Hurst parameter $0 < H < 1/2$ for which the order of convergence is $H + 1/2$, hence lower than the attained order 1 in the examples above, but still higher than the order $H$ of convergence expected from previous works.
The spectral decomposition of a symmetric, second-order tensor is widely adopted in many fields of Computational Mechanics. As an example, in elasto-plasticity under large strain and rotations, given the Cauchy deformation tensor, it is a fundamental step to compute the logarithmic strain tensor. Recently, this approach has been also adopted in small-strain isotropic plasticity to reconstruct the stress tensor as a function of its eigenvalues, allowing the formulation of predictor-corrector return algorithms in the invariants space. These algorithms not only reduce the number of unknowns at the constitutive level, but also allow the correct handling of stress states in which the plastic normals are undefined, thus ensuring a better convergence with respect to the standard approach. While the eigenvalues of a symmetric, second-order tensor can be simply computed as a function of the tensor invariants, the computation of its eigenbasis can be more difficult, especially when two or more eigenvalues are coincident. Moreover, when a Newton-Rhapson algorithm is adopted to solve nonlinear problems in Computational Mechanics, also the tensorial derivatives of the eigenbasis, whose computation is still more complicate, are required to assemble the tangent matrix. A simple and comprehensive method is presented, which can be adopted to compute a closed form representation of a second-order tensor, as well as their derivatives with respect to the tensor itself, allowing a simpler implementation of spectral decomposition of a tensor in Computational Mechanics applications.
This paper proposes a general optimization framework to improve the spectral and energy efficiency (EE) of ultra-reliable low-latency communication (URLLC) simultaneous-transfer-and-receive (STAR) reconfigurable intelligent surface (RIS)-assisted interference-limited systems with finite block length (FBL). This framework can solve a large variety of optimization problems in which the objective and/or constraints are linear functions of the rates and/or EE of users. Additionally, the framework can be applied to any interference-limited system with treating interference as noise as the decoding strategy at receivers. We consider a multi-cell broadcast channel as an example and show how this framework can be specialized to solve the minimum-weighted rate, weighted sum rate, global EE and weighted EE of the system. We make realistic assumptions regarding the (STAR-)RIS by considering three different feasibility sets for the components of either regular RIS or STAR-RIS. Our results show that RIS can substantially increase the spectral and EE of URLLC systems if the reflecting coefficients are properly optimized. Moreover, we consider three different transmission strategies for STAR-RIS as energy splitting (ES), mode switching (MS), and time switching (TS). We show that STAR-RIS can outperform a regular RIS when the regular RIS cannot cover all the users. Furthermore, it is shown that the ES scheme outperforms the MS and TS schemes.
We consider additive Schwarz methods for boundary value problems involving the $p$-Laplacian. Although the existing theoretical estimates indicate a sublinear convergence rate for these methods, empirical evidence from numerical experiments demonstrates a linear convergence rate. In this paper, we narrow the gap between these theoretical and empirical results by presenting a novel convergence analysis. Firstly, we present an abstract convergence theory of additive Schwarz methods written in terms of a quasi-norm. This quasi-norm exhibits behavior similar to the Bregman distance of the convex energy functional associated to the problem. Secondly, we provide a quasi-norm version of the Poincar'{e}--Friedrichs inequality, which is essential for deriving a quasi-norm stable decomposition for a two-level domain decomposition setting. By utilizing these two key elements, we establish a new bound for the linear convergence rate of the methods.
Nested sampling (NS) computes parameter posterior distributions and makes Bayesian model comparison computationally feasible. Its strengths are the unsupervised navigation of complex, potentially multi-modal posteriors until a well-defined termination point. A systematic literature review of nested sampling algorithms and variants is presented. We focus on complete algorithms, including solutions to likelihood-restricted prior sampling, parallelisation, termination and diagnostics. The relation between number of live points, dimensionality and computational cost is studied for two complete algorithms. A new formulation of NS is presented, which casts the parameter space exploration as a search on a tree data structure. Previously published ways of obtaining robust error estimates and dynamic variations of the number of live points are presented as special cases of this formulation. A new online diagnostic test is presented based on previous insertion rank order work. The survey of nested sampling methods concludes with outlooks for future research.
In this paper, we consider a new approach for semi-discretization in time and spatial discretization of a class of semi-linear stochastic partial differential equations (SPDEs) with multiplicative noise. The drift term of the SPDEs is only assumed to satisfy a one-sided Lipschitz condition and the diffusion term is assumed to be globally Lipschitz continuous. Our new strategy for time discretization is based on the Milstein method from stochastic differential equations. We use the energy method for its error analysis and show a strong convergence order of nearly $1$ for the approximate solution. The proof is based on new H\"older continuity estimates of the SPDE solution and the nonlinear term. For the general polynomial-type drift term, there are difficulties in deriving even the stability of the numerical solutions. We propose an interpolation-based finite element method for spatial discretization to overcome the difficulties. Then we obtain $H^1$ stability, higher moment $H^1$ stability, $L^2$ stability, and higher moment $L^2$ stability results using numerical and stochastic techniques. The nearly optimal convergence orders in time and space are hence obtained by coupling all previous results. Numerical experiments are presented to implement the proposed numerical scheme and to validate the theoretical results.
In this work, we investigate the interval generalized Sylvester matrix equation ${\bf{A}}X{\bf{B}}+{\bf{C}}X{\bf{D}}={\bf{F}}$ and develop some techniques for obtaining outer estimations for the so-called united solution set of this interval system. First, we propose a modified variant of the Krawczyk operator which causes reducing computational complexity to cubic, compared to Kronecker product form. We then propose an iterative technique for enclosing the solution set. These approaches are based on spectral decompositions of the midpoints of ${\bf{A}}$, ${\bf{B}}$, ${\bf{C}}$ and ${\bf{D}}$ and in both of them we suppose that the midpoints of ${\bf{A}}$ and ${\bf{C}}$ are simultaneously diagonalizable as well as for the midpoints of the matrices ${\bf{B}}$ and ${\bf{D}}$. Some numerical experiments are given to illustrate the performance of the proposed methods.
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