Convergence Of Numerical Methods Research Articles

A comparative study of two globally convergent numerical methods for acoustic tomography

The comparative study of two globally convergent numerical methods for acoustic tomography is carried out in two dimensions. These are the boundary control method and the quasi-reversibility method. The novelty is that in the latter a nonlinear inverse problem is reduced to a family of the linear integral equation of the first kind via the Lavrentiev approach and this reduction is used within the quasi-reversibility method. The analysis of its stability is carried out via Carleman estimates. The computational effectiveness of these methods is tested in the numerical experiments with the smooth and discontinuous coefficients to be recovered from the tomographic data.

Small Collaboration: Advanced Numerical Methods for Nonlinear Hyperbolic Balance Laws and Their Applications

This small collaborative workshop brought together experts from the Sino-German project working in the field of advanced numerical methods for hyperbolic balance laws. These are particularly important for compressible fluid flows and related systems of equations. The investigated numerical methods were finite volume/finite difference, discontinuous Galerkin methods, and kinetic-type schemes. We have discussed challenging open mathematical research problems in this field, such as multidimensional shock waves, interfaces with different phases or efficient and problem suited adaptive algorithms. Consequently, our main objective was to discuss novel high-order accurate schemes that reliably approximate underlying physical models and preserve important physically relevant properties. Theoretical questions concerning the convergence of numerical methods and proper solution concepts were addressed as well.

The case of SU(3) criticality in spin-2 chains

It was proposed in [(https://doi.org/10.1103/PhysRevLett.114.145301){Chen et al., Phys. Rev. Lett. $\mathbf{114}$, 145301 (2015)}] that spin-2 chains display an extended critical phase with enhanced SU$(3)$ symmetry. This hypothesis is highly unexpected for a spin-2 system and, as we argue, would imply an unconventional mechanism for symmetry emergence. Yet, the absence of convenient critical points for renormalization group perturbative expansions, allied with the usual difficulty in the convergence of numerical methods in critical or small-gapped phases, renders the verification of this hypothetical SU$(3)$-symmetric phase a non-trivial matter. By tracing parallels with the well-understood phase diagram of spin-1 chains and searching for signatures robust against finite-size effects, we draw criticism on the existence of this phase. We perform non-Abelian density matrix renormalization group studies of multipolar static correlation function, energy spectrum scaling, single-mode approximation, and entanglement spectrum to shed light on the problem. We determine that the hypothetical SU$(3)$ spin-2 phase is, in fact, dominated by ferro-octupolar correlations and also observe a lack of Luttinger-liquid-like behavior in correlation functions that suggests that is perhaps not critical. We further construct an infinite family of spin-$S$ systems with similar ferro-octupolar-dominated quasi-SU$(3)$-like phenomenology; curiously, we note that the spin-3 version of the problem is located in a subspace of exact G$_2$ symmetry, making this a point of interest for search of Fibonacci topological properties in magnetic systems.

Solving an inverse elliptic coefficient problem by convex non-linear semidefinite programming

Several applications in medical imaging and non-destructive material testing lead to inverse elliptic coefficient problems, where an unknown coefficient function in an elliptic PDE is to be determined from partial knowledge of its solutions. This is usually a highly non-linear ill-posed inverse problem, for which unique reconstructability results, stability estimates and global convergence of numerical methods are very hard to achieve. The aim of this note is to point out a new connection between inverse coefficient problems and semidefinite programming that may help addressing these challenges. We show that an inverse elliptic Robin transmission problem with finitely many measurements can be equivalently rewritten as a uniquely solvable convex non-linear semidefinite optimization problem. This allows to explicitly estimate the number of measurements that is required to achieve a desired resolution, to derive an error estimate for noisy data, and to overcome the problem of local minima that usually appears in optimization-based approaches for inverse coefficient problems.

Multi-phase algorithm design for accurate and efficient model fitting

Recent research applies soft computing techniques to fit software reliability growth models. However, runtime performance and the distribution of the distance from an optimal solution over multiple runs must be explicitly considered to justify the practical utility of these approaches, promote comparison, and support reproducible research. This paper presents a meta-optimization framework to design stable and efficient multi-phase algorithms for fitting software reliability growth models. The approach combines initial parameter estimation techniques from statistical algorithms, the global search properties of soft computing, and the rapid convergence of numerical methods. Designs that exhibit the best balance between runtime performance and accuracy are identified. The approach is illustrated through nonhomogeneous Poisson process and covariate software reliability growth models, including a cross-validation step on data sets not used to identify designs. The results indicate the nonhomogeneous Poisson process model considered is too simple to benefit from soft computing because it incurs additional runtime with no increase in accuracy attained. However, a multi-phase design for the covariate software reliability growth model consisting of the bat algorithm followed by a numerical method achieves better performance and converges consistently, compared to a numerical method only. The proposed approach supports higher-dimensional covariate software reliability growth model fitting suitable for implementation in a tool.

Finite Element Calculation of Photonic Band Structures for Frequency Dependent Materials

Band structure calculation of frequency dependent photonic crystals has important applications. The associated eigenvalue problem is nonlinear and the development of convergent numerical methods is challenging. In this paper, we formulate the band structure problem as the eigenvalue problem of a holomorphic Fredholm operator function of index zero. Lagrange finite elements are used to discretize the operators. The convergence of the eigenvalues is proved using the abstract approximation theory for holomorphic operator functions. Then a spectral indicator method is developed to practically compute the eigenvalues. Numerical examples are presented to validate the theory and show the effectiveness of the proposed method.

Efficient graph‐based tensile strength simulations of random fiber structures

AbstractIn this paper, we propose a model‐simulation framework for virtual tensile strength testing of random fiber structures, such as those in nonwoven materials. The focus is on efficient handling of the problem‐inherent multi‐scales and randomness. In particular, the interplay between random microstructure and deterministic structural production‐related features on the macro‐scale makes classical homogenization‐based approaches computationally complex and costly. In our approach we model the fiber structure as graph‐based and of truss‐type, equipped with a nonlinear elastic material law. The tensile strength test is described by a sequence of force equilibria with varied boundary conditions. Its embedding into a singularly perturbed dynamical system is advantageous with respect to statements on solution theory and convergence of numerical methods. A problem‐tailored data reduction provides additional speed‐up, while Monte‐Carlo simulations account for randomness. This work serves as a proof of concept and opens the field for optimization.

Investigation of a Hybrid Kinematic Calibration Method for the “Sina” Surgical Robot

Calibrating the inverse kinematics of complex robots is often a challenging task. Finding analytical solutions is not always possible and the convergence of numerical methods is not guaranteed. The model-free approaches, based on machine learning and artificial intelligence, are fast and easy to work, however, they need a huge amount of experimental training data to provide acceptable results. In this article, we proposed a hybrid method to benefit the advantage of both model-based and model-free approaches. The forward kinematics of the robot is calibrated using a model-based approach, and its inverse kinematics using a neural network. Hence, while there is no need to solve the nonlinear inverse kinematic equations, the training data of the neural network is generated artificially by the calibrated forward kinematic model. The implementation of the proposed methodology on slave robot of the “Sina” surgical system revealed reasonably good results. The accuracy improved by ${\bf 53\%}$ and ${\bf 43\%}$ for position and orientation, respectively, after calibration.

Mean-square convergence of numerical methods for random ordinary differential equations

The aim of this work is to analyze the mean-square convergence rates of numerical schemes for random ordinary differential equations (RODEs). First, a relation between the global and local mean-square convergence order of one-step explicit approximations is established. Then, the global mean-square convergence rates are investigated for RODE-Taylor schemes for general RODEs, Affine-RODE-Taylor schemes for RODEs with affine noise, and Ito-Taylor schemes for RODEs with Ito noise, respectively. The theoretical convergence results are demonstrated through numerical experiments.

Visco-elastic dampers in structural buildings and numerical solution with spline collocation methods

The dynamic system of a structure utilized with visco-elastic dampers can be modeled by fractional differential equations. All the resulted systems of fractional differential equations can be represented in a state space and can be transformed into a system of multi-term fractional differential equations of order 1. Considering the presence of indeterministic exogenous force like earthquake we need powerful, convergent and reliable numerical methods to simulate the response of this dynamical systems. Therefore, spline collocations method has been proposed and studied for solving system of multi-term fractional differential equations of order 1. A rigorous mathematical analysis is provided to show the efficiency and effectiveness of the method. To this end, we apply a functional analysis framework to obtain convergence and superconvergence properties of the proposed methods on the graded mesh. Some numerical experiments are provided to confirm the theoretical results. Finally, this method is used for simulating the response of a 4-story building under El Centro earthquake excitation.

Convergent numerical methods for parabolic equations with reversed time via a new Carleman estimate

The key tool of this paper is a new Carleman estimate for an arbitrary parabolic operator of the second order for the case of reversed time data. This estimate works on an arbitrary time interval. On the other hand, the previously known Carleman estimate for the reversed time case works only on a sufficiently small time interval. First, a stability estimate is proven. Next, the quasi-reversibility numerical method is proposed for an arbitrary time interval for the linear case. This is unlike a sufficiently small time interval in the previous work. The convergence rate for the quasi-reversibility method is established. Finally, the quasilinear parabolic equation with reversed time is considered. A weighted globally strictly convex Tikhonov-like functional is constructed. The weight is the Carleman weight function which is involved in that Carleman estimate. The global convergence of the gradient projection method to the exact solution is proved for this functional.

Dynamic Co-Simulation Methods for Combined Transmission-Distribution System With Integration Time Step Impact on Convergence

Combined transmission and distribution systems (CoTDS) simulation for power systems requires development of algorithms and software that are numerically stable and at the same time accurately simulate dynamic events that can occur in practical systems. The time scales of simulation can be orders of magnitude apart making the combined simulation extremely challenging. This has led to increased research in applying co-simulation techniques for integrated simulation of the two systems. In this paper, two methods for dynamic co-simulation of CoTDS are proposed using parallel and series computation of the transmission system and distribution systems. The convergence of numerical methods in co-simulation is addressed from a fundamental mathematical perspective and the impact of the integration time-step is studied on proposed CoTDS dynamic co-simulation methods. Both these co-simulation methods are validated against total system simulation in a single time-domain simulation environment. The proposed CoTDS co-simulation method is also validated against commercial EMTP dynamic simulation. CoTDS dynamic co-simulation is further demonstrated on a New England 39-bus transmission system with ten load buses replaced by distribution systems including a 5780-node distribution system on one load bus for a detailed fault-induced delayed voltage recovery simulation study.

Reduction and Minimality of Coexhausters

V.F. Demyanov introduced exhausters for the study of nonsmooth functions. These are families of convex compact sets that enable one to represent the main part of the increment of a considered function in a neighborhood of the studied point as MaxMin or MinMax of linear functions. Optimality conditions were described in terms of these objects. This provided a way for constructing new algorithms for solving nondifferentiable optimization problems. Exhausters are defined not uniquely. It is obvious that the smaller an exhauster, the less are the computational expenses when working with it. Thus, the problem of reduction of an available family arises. For the first time, this problem was considered by V.A. Roshchina. She proposed conditions for minimality and described some methods of reduction in the case when these conditions are not satisfied. However, it turned out that the exhauster mapping is not continuous in the Hausdorff metrics, which leads to the problems with convergence of numerical methods. To overcome this difficulty, Demyanov proposed the notion of coexhausters. These objects enable one to represent the main part of the increment of the considered function in a neighborhood of the studied point in the form of MaxMin or MinMax of affine functions. One can define a class of functions with the continuous coexhauster mapping. Optimality conditions can be stated in terms of these objects too. But coexhausters are also defined not uniquely. The problem of reduction of coexhausters is considered in this paper for the first time. Definitions of minimality proposed by Roshchina are used. In contrast to ideas proposed in the works of Roshchina, the minimality conditions and the technique of reduction developed in this paper have a clear and transparent geometric interpretation.

Convergence of the Tamed Euler Method for Stochastic Differential Equations with Piecewise Continuous Arguments Under Non-global Lipschitz Continuous Coefficients

ABSTRACTIn this paper, we deal with the strong convergence of numerical methods for stochastic differential equations with piecewise continuous arguments (SEPCAs) with at most polynomially growing drift coefficients and global Lipschitz continuous diffusion coefficients. An explicit and time-saving tamed Euler method is used to solve this type of SEPCAs. We show that the tamed Euler method is bounded in pth moment. And then the convergence of the tamed Euler method is proved. Moreover, the convergence order is one-half. Several numerical simulations are shown to verify the convergence of this method.

Convexification of restricted Dirichlet-to-Neumann map

Abstract By our definition, “restricted Dirichlet-to-Neumann (DN) map” means that the Dirichlet and Neumann boundary data for a coefficient inverse problem (CIP) are generated by a point source running along an interval of a straight line. On the other hand, the conventional DN data can be generated, at least sometimes, by a point source running along a hypersurface. CIPs with restricted DN data are non-overdetermined in the n-dimensional case, with n ≥ 2 {n\geq 2} . We develop, in a unified way, a general and radically new numerical concept for CIPs with restricted DN data for a broad class of PDEs of second order, such as, e.g., elliptic, parabolic and hyperbolic ones. Namely, using Carleman weight functions, we construct globally convergent numerical methods. Hölder stability and uniqueness are also proved. The price we pay for these features is a well-acceptable one in the numerical analysis, that is, we truncate a certain Fourier-like series with respect to some functions depending only on the position of the point source. At least three applications are imaging of land mines, crosswell imaging and electrical impedance tomography.

Age-dependent production and replacement strategies in failure-prone manufacturing systems

A failure-prone manufacturing system that consists of one machine producing one type of product is studied. The random phenomena examined are machine breakdowns and repairs. We assume that the machine undergoes a progressive deterioration while in operation and that the machine failure rate is a function of its age. The aging of the machine (the dynamics of the machine age) is assumed to be an increasing function of its production rate. Corrective maintenance activities are imperfect and restore the age of the machine to as-bad-as-old conditions. When a failure occurs, the machine can be repaired, and during production, the machine can be replaced, depending on its age. When the replacement action is selected, the machine is replaced by a new and identical one. The decision variables are the production rate and the replacement policy. The objective of this article is to address the simultaneous production and replacement policy optimization problem in the context of manufacturing with deterioration and imperfect repairs satisfying the customer demand and minimizing the total cost, which includes costs associated with inventory, backlog, production, repair and replacement, over an infinite planning horizon. We thoroughly explore the impact of the machine aging on the production and replacement policies. Particular attention is paid to the verification of underlying mathematical results that guarantee the existence of optimal solutions and the convergence of numerical methods. Due to imperfect repairs, the dynamics of the system is affected by the system history, and semi-Markov processes have to be used for modeling. Optimality conditions in the form of the Hamilton–Jacobi–Bellman equations are developed, and numerical methods are used to obtain the optimal control policies (production (rate) and replacement policies). A numerical example is given to illustrate the proposed approach, and an extensive sensitivity analysis is presented to confirm the structure of the obtained control policies.

Approximate solutions to the initial value problem for some compressible flows

For the natural initial conditions L 1 in the density field (more generally a positive bounded Radon measure) and L ∞ in the velocity field, we obtain global approximate solutions to the Cauchy problem for the 3-D systems of isothermal and isentropic gases and the 2-D shallow water equations. We obtain a sequence of functions which are differentiable in time and continuous in space and tend to satisfy the equations in the sense of distributions in the space variables and in the strong sense in the time variable. The method of construction relies on the study of a specific family of two ODEs in a classical Banach space (one for the continuity equation and one for the Euler equation). Standard convergent numerical methods for the solution of these ODEs can be used to provide concrete approximate solutions. It has been checked in numerous cases in which the solutions of systems of fluid dynamics are known that our construction always gives back the known solutions. It is also proved that it gives the classical analytic solutions in the domain of application of the Cauchy–Kovalevskaya theorem.

Error analysis of waveform relaxation method for semi-linear partial differential equations

The waveform relaxation (WR) method is investigated for semi-linear partial differential equations. Explicit error estimation is given for the iteration error. A way to combine WR with convergent numerical methods is proposed, the error of the combined method is analyzed and its convergence is proven. The effect of the application of time windows is discussed. Numerical tests are presented to confirm the theoretical results.

A Seminumerical Transient Analysis of Switched Capacitor Converters

Abstract —The paper proposes an algebraic approach to tran-sient analysis of switched capacitor converters (SCCs). To thisend, the SCC is considered as an analog computer that solvesthe system of linear equations by an iterative method. The resultof each iteration represents a point in the step response, whereasthe function between the points is reconstructed. To simplifythe derivation of the iterative method, each SCC topology isconsidered as a first-order circuit. The developed method iswritten as a recurrence relation and under certain conditions canbe reduced to the well-known Kaczmarz method. The theoreticalpredictions were verified by simulations and experimentally andagree well with the results obtained by the equivalent resistanceconcept. Index Terms —Analog computers, charge pumps, convergenceof numerical methods, discrete-time systems, iterative methods,switched capacitor circuits. I. I NTRODUCTION S WITCHED capacitor converters (SCCs), also known ascharge pumps, have been found to be useful in low-power applications due to their compatibility with integratedcircuit technology. It is well known that the SCC exhibit highefficiency only when their output voltage,

Investigating Convergence Patterns for Numerical Methods Using Data Analysis

The article investigates the patterns that arise in the convergence of numerical methods, particularly those in the errors involved in successive iterations, using data analysis and curve fitting methods. In particular, the results obtained are used to convey a deeper level of understanding of the concepts of linear, quadratic, and cubic convergence. In addition, the issue estimating the number of additional correct digits on each successive iteration is also considered.