Unconditionally stable element-by-element algorithms for dynamic problems

A collection of results is presented regarding the consistency, stability and accuracy of operator split methods and product formula algorithms for general nonlinear equations of evolution. These results are then applied to the structural dynamics problem. The basic idea is to exploit an element-by-element additive decomposition of a particular form of the discrete dynamic equations resulting from a finite element discretization. It is shown that such a particular form of the discrete dynamic equations is obtained when velocity and stress are taken as unknowns. By applying the general product formula technique to the element-by-element decomposition, unconditionally stable algorithms are obtained that involve only element coefficient matrices. The storage requirements and operation counts are comparable to those of explicit methods. The method places no restriction on the topology of the finite element mesh.     

Unconditionally stable implicit-explicit algorithms for coupled thermal stress waves

A new family of unconditionally stable implicit-explicit algorithms for transient coupled thermal stress waves is introduced. Its computer implementation aspects are presented. The mechanical displacement and temperature are used as independent variables and the resulting matrix equations are shown to be symmetric.     

Unconditionally stable algorithms for quasi-static elasto/visco-plastic finite element analysis

A new one-parameter family of implicit algorithms for quasi-static elasto/visco-plastic finite element analysis is proposed. For appropriate values of the parameter, the algorithms are shown to be unconditionally stable. Numerical tests in confirmation of the theory are presented.     

Unconditionally stable ADI scheme of higher-order for linear hyperbolic equations

An unconditionally stable alternating direction implicit (ADI) method of higher-order in space is proposed for solving two- and three-dimensional linear hyperbolic equations. The method is fourth-order in space and second-order in time. The solution procedure consists of a multiple use of one-dimensional matrix solver which produces a computational cost effective solver. Numerical experiments are conducted to compare the new scheme with the existing scheme based on second-order spatial discretization. The effectiveness of the new scheme is exhibited from the numerical results.     

Unconditionally stable C1-cubic spline collocation method for solving parabolic equations

This article aims to present a new approach based on C¹-cubic splines introduced by Sallam and Naim Anwar [Sallam, S. and Naim Anwar, M. (2000). Stabilized cubic C¹-spline collocation method for solving first-order ordinary initial value problems, Int. J. Comput. Math., 74, 87–96.], which is A-stable, for the time integration of parabolic equations (diffusion or heat equation). The introduced method is an example of the so-called method of lines (the solution is thought to consist of space discretization and time integration), which is an extension of the 1/3-Simpson's finite-difference scheme. Our main objective is to prove the unconditional stability of the proposed method as well as to show that the method is convergent and is of order O (h ²)?+?O (k ⁴) i.e. it is a fourth-order in time and second-order in space. Computational results also show that the method is relevant for long time interval problems.     

Unconditionally stable split-step finite difference time domain formulations for double-dispersive electromagnetic materials

Systematic split-step finite difference time domain (SS-FDTD) formulations, based on the general Lie–Trotter–Suzuki product formula, are presented for solving the time-dependent Maxwell equations in double-dispersive electromagnetic materials. The proposed formulations provide a unified tool for constructing a family of unconditionally stable algorithms such as the first order split-step FDTD (SS1-FDTD), the second order split-step FDTD (SS2-FDTD), and the second order alternating direction implicit FDTD (ADI-FDTD) schemes. The theoretical stability of the formulations is included and it has been demonstrated that the formulations are unconditionally stable by construction. Furthermore, the dispersion relation of the formulations is derived and it has been found that the proposed formulations are best suited for those applications where a high space resolution is needed. Two-dimensional (2-D) and 3-D numerical examples are included and it has been observed that the SS1-FDTD scheme is computationally more efficient than the ADI-FDTD counterpart, while maintaining approximately the same numerical accuracy. Moreover, the SS2-FDTD scheme allows using larger time step than the SS1-FDTD or ADI-FDTD and therefore necessitates less CPU time, while giving approximately the same numerical accuracy.     

Distributed estimation algorithms for nonlinear systems

In this paper, we consider the problem of combining the local conditional distributions of a random variable which have been generated by local observers having access to their private information. Sufficient statistics for the local distributions are communicated to a coordinator, who attempts to reconstruct the global centralized distribution using only the communicated statistics. We obtain a distributed processing algorithm which recovers exactly the centralized conditional distribution. The results can be applied in designing distributed hypothesis-testing algorithms for event-driven systems.     

Solution algorithms for nonlinear structural problems

The article discusses characteristic types of nonlinearities in structural systems. Various ways of illustrating such nonlinearities are suggested. Mathematical equations that form the basis for alternative numerical solution algorithms are stated. The use of the current stiffness parameter for characterizing nonlinear systems is discussed. Some new formulations and applications of this parameter are suggested. A new class of solution techniques utilizing simultaneous iteration on the loading parameter as well as the displacements is also proposed. By these techniques it is not necessary to formulate a nonlinear stiffness matrix and incrementation of displacement pattern can be used instead of load incrementation. No special load reversal criterion is normally required for passing limit points.     

Determination of the time-dependent reaction coefficient and the heat flux in a nonlinear inverse heat conduction problem

ABSTRACT

Diffusion processes with reaction generated by a nonlinear source are commonly encountered in practical applications related to ignition, pyrolysis and polymerization. In such processes, determining the intensity of reaction in time is of crucial importance for control and monitoring purposes. Therefore, this paper is devoted to such an identification problem of determining the time-dependent coefficient of a nonlinear heat source together with the unknown heat flux at an inaccessible boundary of a one-dimensional slab from temperature measurements at two sensor locations in the context of nonlinear transient heat conduction. Local existence and uniqueness results for the inverse coefficient problem are proved when the first three derivatives of the nonlinear source term are Lipschitz continuous functions. Furthermore, the conjugate gradient method (CGM) for separately reconstructing the reaction coefficient and the heat flux is developed. The ill-posedness is overcome by using the discrepancy principle to stop the iteration procedure of CGM when the input data is contaminated with noise. Numerical results show that the inverse solutions are accurate and stable.     

EM algorithms for nonlinear mixed effects models

Implementing the Monte Carlo EM algorithm (MCEM) algorithm for finding maximum likelihood estimates (MLEs) in the nonlinear mixed effects model (NLMM) has encountered a great deal of difficulty in obtaining samples used for estimating the E step due to the intractability of the target distribution. Sampling methods such as Markov chain techniques and importance sampling have been used to alleviate such difficulty. The advantage of Markov chains is that they are applicable to a wider range of distributions than the approaches based on independent samples. However, in many cases the computational cost of Markov chains is significantly greater than that of independent samplers. The MCEM algorithms based on independent samples allow for straightforward assessment of Monte Carlo error and can be considerably more efficient than those based on Markov chains when an efficient candidate distribution is chosen, which forms the motivation of this paper. The proposed MCEM algorithm in this paper uses samples obtained from an easy-to-simulate and efficient importance distribution so that the computational intensity and complexity is much reduced. Moreover, the proposed MCEM algorithm preserves the flexibility introduced by independent samples in gauging Monte Carlo error and thus allows the Monte Carlo sample size to increase with the number of EM iterations. We also introduce an EM algorithm using Gaussian quadrature approximations (GQEM) for the E step. In low-dimensional cases, the GQEM algorithm is more efficient than the proposed MCEM algorithm and thus can be used as an alternative. The performances of the proposed EM methods are compared to the existing ML estimators using real data examples and simulations.     

Nonparametric control algorithms for nonlinear fading memorysystems

We develop an algorithm to control an unknown nonlinear fading memory discrete-time system. Our approach is based on nonparametric regression techniques rather than traditional feedback control. We discuss a procedure that, given a desired periodic output and a tolerance, produces an acceptable output from any system in a wide class. The algorithm uses as data past inputs (which are selected by the algorithm) and corresponding (possibly noisy) output observations and needs no extra parametric knowledge or restrictions on the system. We also present an algorithm that produces an output that converges to the desired output when stricter conditions are imposed on the system. Our approach, in its current form, however, cannot be used to control open-loop unstable systems     

Efficient Steffensen-type algorithms for solving nonlinear equations

We introduce a Steffensen-type method (STTM) for solving nonlinear equations in a Banach space setting. Then, we present a local convergence analysis for (STTM) using recurrence relations. Numerical examples validating our theoretical results are also provided in this study to show that (STTM) is faster than other methods [I.K. Argyros, J. Ezquerro, J.M. Gutiérrez, M. Hernández, and S. Hilout, On the semilocal convergence of efficient Chebyshev-Secant-type methods, J. Comput. Appl. Math. 235 (2011), pp. 3195–3206; J.A. Ezquerro and M.A. Hernández, An optimization of Chebyshev's method, J. Complexity 25 (2009), pp. 343–361] using similar convergence conditions.     

A test case for nonlinear difference algorithms

A family of test problems with explicit analytical solutions is given which is sufficiently nontrivial for the realistic examination of nonlinear algorithms. The following applications are shown: the test and comparison of some first and second order accurate schemes including nonlinear restglied analysis, the synthetic computation of the error distribution and a treatment of the boundary-neighbour problem.     

Distributed nonlinear control algorithms for network consensus

In this paper, we develop a thermodynamic framework for addressing consensus problems for nonlinear multiagent dynamical systems with fixed and switching topologies. Specifically, we present distributed nonlinear static and dynamic controller architectures for multiagent coordination. The proposed controller architectures are predicated on system thermodynamic notions resulting in controller architectures involving the exchange of information between agents that guarantee that the closed-loop dynamical network is consistent with basic thermodynamic principles.     

Smoothing algorithms for nonlinear distributed parameter systems

Fixed interval and fixed-lag smoothing algorithms are developed for a class of noisy nonlinear distributed parameter systems with unknown volume disturbances. Using a least-squares estimation criterion a two point boundary value problem is obtained. The solution of this boundary value problem is obtained by converting it into an initial value problem using a Riccati transformation. Using a Gaussian assumption for the disturbances, approximate error covariance expressions are developed. The possible instability of the fixed-lag smoothing algorithm is noted. An alternate fixed-interval algorithm is developed to obviate storage requirements. The applicability of these algorithms is illustrated with an example from nuclear reactor kinetics.     

GPU-based parallel algorithms for sparse nonlinear systems

In this work we describe some parallel algorithms for solving nonlinear systems using CUDA (Compute Unified Device Architecture) over a GPU (Graphics Processing Unit). The proposed algorithms are based on both the Fletcher–Reeves version of the nonlinear conjugate gradient method and a polynomial preconditioner type based on block two-stage methods. Several strategies of parallelization and different storage formats for sparse matrices are discussed. The reported numerical experiments analyze the behavior of these algorithms working in a fine grain parallel environment compared with a thread-based environment.     

A system of algorithms for stable human recognition

The principles for designing multimodal biometric systems for improving the reliability of human recognition systems are discussed. The problem is solved by the example of a combination of two biometric human features that can be obtained without any contact with the registering sensor, namely, face images and records of the human voice. The identification technique developed involves face, speech, and voice recognition. Note that the data is processed in real time. The following identification scheme is employed: detection of human presence-face detection-enquiry of speech data-joint recognition of the facial and vocal features. The methods and algorithms employed for solving all posed problems are described, and the rules for making joint decisions are addressed. The results of testing the described algorithms are presented.