Right-hand Side Function Research Articles

Optimal adaptive solution of initial-value problems with unknown singularities

The optimal solution of initial-value problems in ODEs is well studied for smooth right-hand side functions. Much less is known about the optimality of algorithms for singular problems. In this paper, we study the (worst case) solution of scalar problems with a right-hand side function having r continuous bounded derivatives in R , except for an unknown singular point. We establish the minimal worst case error for such problems (which depends on r similarly as in the smooth case), and define optimal adaptive algorithms. The crucial point is locating an unknown singularity of the solution by properly adapting the grid. We also study lower bounds on the error of an algorithm for classes of singular problems. In the case of a single singularity with nonadaptive information, or in the case of two or more singularities, the error of any algorithm is shown to be independent of r.

Initialization of a Nonlinear Identification Algorithm Applied to Laboratory Plant Data

New techniques for recursive identification of systems described by nonlinear ordinary differential equation models are discussed. The model is of black-box state space type, where the right-hand side function is estimated as a multi-variate polynomial in the states and inputs, with the parameters selected to be the polynomial coefficients. An algorithm based on Kalman filtering techniques is derived, where a numerical differentiation scheme, used for generation of approximate state variables is a key ingredient. The Kalman-filter-based algorithm is, for example, suitable for initialization of a previously published recursive prediction error method (RPEM) based on the same model. In this brief, the algorithm performance of the Kalman-filter-based method is compared to that of the RPEM using a numerical example. Another example shows that the success rate of the RPEM is increased from 70% to 100%, when the proposed algorithm is used for generation of initial estimates for the RPEM. The Kalman-filter-based algorithm is also used for finding initial parameters for the RPEM when applied to live data from a laboratory process - a system of cascaded tanks. Based on the experimental results, this brief discusses advantages and disadvantages of different algorithms and differentiation schemes.

A New Methodology for the General Multiparametric Mixed-Integer Linear Programming (MILP) Problems

In this paper, a general algorithm is developed to address the multiparametric mixed-integer linear programming (mpMILP) problem with uncertain parameters in the left-hand side (LHS), right-hand side (RHS), and objective function coefficients simultaneously. The algorithm is based on a general multiparametric linear programming (mpLP) algorithm, which is derived using the optimality conditions of standard linear programming (LP) problem. The intent of the proposed framework is to propose a general framework to address different uncertainties in the process engineering problems. In addition, the paper also discusses the solution of the special problem where LHS uncertainty is included in the optimization model as a coefficient of continuous variable, and it notes the high computational complexity needed to retrieve the rigorous solution of large-scale problems, because of the nonlinearities of the objective function and nonconvex critical regions. Several numerical examples are presented to illustrate the effectiveness and applicability of the proposed method.

Asymptotic expansions of singular solutions for [formula omitted]-D Protter problems

Four-dimensional boundary value problems for the nonhomogeneous wave equation are studied, which are analogues of Darboux problems in the plane. The smoothness of the right-hand side function of the wave equation is decisive for the behavior of the solution of the boundary value problem. It is shown that for each n ∈ N there exists such a right-hand side function from C n , for which the uniquely determined generalized solution has a strong power-type singularity at one boundary point. This singularity is isolated at the vertex of the characteristic cone and does not propagate along the cone. The present article describes asymptotic expansions of the generalized solutions in negative powers of the distance to this singularity point. Some necessary and sufficient conditions for existence of regular solutions, or solutions with fixed order of singularity, are derived and additionally some a priori estimates for the singular solutions are given.

Recursive prediction error identification and scaling of non-linear state space models using a restricted black box parameterization

A recursive prediction error algorithm for identification of systems described by non-linear ordinary differential equation (ODE) models is presented. The model is an ODE model, parameterized with coefficients of a multi-variable polynomial that describes one component of the right-hand side function of the ODE. This avoids over-parameterization problems. The selected model can also handle systems with more complicated right-hand side structure, by identification of a local input–output equivalent system in the coordinate system of the selected state variables. A novel technique based on scaling of the sampling period is proposed. The technique can improve the conditioning of the identification problem, thereby enhancing the chances of convergence to the correct minimum of the criterion. The algorithm is applied to live data from a system consisting of two cascaded tanks, with promising results. A MATLAB software package, which implements the proposed algorithm and a set of support scripts, can be freely downloaded from http://www.it.uu.se/research/reports/.

Least-squares periodic signal modeling using orbits of nonlinear ODEs and fully automated spectral analysis

Periodic signals can be modeled by means of second-order nonlinear ordinary differential equations (ODEs). The right-hand side function of the ODE is parameterized in terms of known basis functions. The least-squares algorithm developed for estimating the coefficients of these basis functions gives biased estimates, especially at low signal-to-noise ratios. This is due to noise contributions to the periodic signal and its derivatives evaluated using finite difference approximations. In this paper a fully automated spectral analysis (ASA) technique is used to eliminate these noise contributions. A simulation study shows that using the ASA technique significantly improves the performance of the least-squares estimator.

Periodic boundary value problems on time scales

We extend the results concerning periodic boundary value problems from the continuous calculus to time scales. First we use the Schauder fixed point theorem and the concept of lower and upper solutions to prove the existence of the solutions and then we investigate a monotone iterative method which could generate some of them. Since this method does not work on each time scale, a condition containing a Lipschitz constant of right-hand side function and the supremum of the graininess function is introduced.

Linearly implicit Runge–Kutta methods and approximate matrix factorization

Linearly implicit Runge–Kutta methods are a class of suitable time integrators for initial value problems of ordinary differential systems whose right-hand side function can be written as the sum of a stiff linear part and a nonlinear term. Such systems arise for instance after spatial discretization of taxis–diffusion–reaction systems from mathematical biology. When approximate matrix factorization is used for efficiently solving the stage equations appearing in these methods, then the order of the methods is reduced to one. In this paper we analyse this fact and propose an appropriate and efficient correction to achieve order two while preserving the main stability properties of the underlying method. Numerical experiments with LIRK3 [Appl. Numer. Math. 37 (2001) 535] illustrating the theory are provided. In the case of taxis–diffusion–reaction systems, the corrected method compares well with other suitable schemes.

An adaptive N-body algorithm of optimal order

Picard iteration is normally considered a theoretical tool whose primary utility is to establish the existence and uniqueness of solutions to first-order systems of ordinary differential equations (ODEs). However, in 1996, Parker and Sochacki [Neural, Parallel, Sci. Comput. 4 (1996)] published a practical numerical method for a certain class of ODEs, based upon modified Picard iteration, that generates the Maclaurin series of the solution to arbitrarily high order. The applicable class of ODEs consists of first-order, autonomous systems whose right-hand side functions (generators) are projectively polynomial; that is, they can be written as polynomials in the unknowns. The class is wider than might be expected. The method is ideally suited to the classical N-body problem, which is projectively polynomial. Here, we recast the N-body problem in polynomial form and develop a Picard-based algorithm for its solution. The algorithm is highly accurate, parameter-free, and simultaneously adaptive in time and order. Test cases for both benign and chaotic N-body systems reveal that optimal order is dynamic. That is, in addition to dependency upon N and the desired accuracy, optimal order depends upon the configuration of the bodies at any instant.

Stability analysis of nonlinear retarded difference equations in Banach spaces

In this paper, we give new sufficient conditions for asymptotic stability and instability of nonlinear difference equations with delays in infinite-dimensional spaces x( k + 1) = f( k, x( k), x( k−1), … x( k− r))), x( k) ϵ X, k = 0, 1,2,…, where X is a Banach space. Our results are obtained using a general comparison condition on the right-hand side function f( k,.), which generalizes the stability and instability results obtained by Yang, Miminis, Naulin, and Vanegas. An application to the stabilizability problem of retarded control systems and illustrative examples of the obtained results are given.

The positivity of low-order explicit Runge-Kutta schemes applied in splitting methods

Splitting methods are frequently used for the solution of large stiff initial value problems of ordinary differential equations with an additively split right-hand side function. Such systems arise, for instance, as method of lines discretizations of evolutionary partial differential equations in many applications. We consider the choice of explicit Runge-Kutta (RK) schemes in implicit-explicit splitting methods. Our main objective is the preservation of positivity in the numerical solution of linear and nonlinear positive problems while maintaining a sufficient degree of accuracy and computational efficiency. A three-stage second-order explicit RK method is proposed which has optimized positivity properties. This method compares well with standard s-stage explicit RK schemes of order s, s = 2, 3. It has advantages in the low accuracy range, and this range is interesting for an application in splitting methods. Numerical results are presented.

Dual Reciprocity Boundary Element Method for Magnetohydrodynamic Flow Using Radial Basis Functions

A dual reciprocity boundary element method is given to obtain the solution in terms of velocity and induced magnetic field for the study of MHD (magnetohydrodynamic) flow through a rectangular duct having insulating walls. The equations are transformed to two types of nonlinear Poisson equations and the right-hand sides in these equations are approximated using combinations of two classes of radial basis functions (the value of the function and its normal derivatives are utilized for approximation). Computations are carried out for several values of the Hartman number (0 h M h 10) by using constant boundary elements. Comparisons are made for two types of formulations and for traditional and osculatory type approximations of the right-hand side functions. It is found that osculatory interpolation gives better results than traditional interpolation and the type of the Poisson equation, which contains derivative of the unknown function, is better than the other type, which contains unknown function only. The results for velocity and induced magnetic field are illustrated by some selected graphs.

An h–p adaptive spectral element method for Stokes flow

An h–p adaptive spectral element method for three-dimensional axisymmetric domains featuring an element residual a posteriori error indicator, equidistribution of the error amongst the elements in the initial mesh and simultaneous h and p refinements is presented. The mortar element method is used to efficiently handle the nonconforming decompositions that arise in the adaptive process. Results obtained by applying the adaptive strategy to Stokes flow for different orders of regularity of the righthand side function are compared with the results obtained using a uniform p refinement. The results obtained show the advantages of using the proposed strategy.

DIAGONAL-IMPLICITLY ITERATED RUNGE–KUTTA METHODS ON DISTRIBUTED MEMORY MACHINES

We consider diagonal-implicitly iterated Runge–Kutta methods which are one-step methods for stiff ordinary differential equations providing embedded solutions for stepsize control. In these methods, algorithmic parallelism is introduced at the expense of additional computations. In this paper, we concentrate on the algorithmic structure of these Runge–Kutta methods and consider several parallel variants of the method exploiting algorithmic and data parallelism in different ways. Our aim is to investigate whether these variants lead to good performance on current distributed memory machines such as the Intel Paragon and the IBM SP2. As test application we use ordinary differential equations with dense and sparse right-hand side functions.

Analysis of approximate factorization in iteration methods

We consider the systems of ordinary differential equations (ODEs) obtained by spatial discretization of multi-dimensional partial differential equations. In order to solve the initial value problem (IVP) for such ODE systems numerically, we need a stiff IVP solver, because the Lipschitz constant associated with the right-hand side function f becomes increasingly large as the spatial resolution is refined. Stiff IVP solvers are necessarily implicit, so that we are faced with the problem of solving large systems of implicit relations. In the solution process of the implicit relations one may exploit the fact that the right-hand side function f can often be split into functions f i which contain only the discretizations of derivatives with respect to one spatial dimension. In this paper, we analyze iterative solution methods based on approximate factorization which are suitable for implementation on parallel computer systems. In particular, we derive convergence and stability regions.

Euler-Chebyshev methods for integro-differential equations

We construct and analyse explicit methods for solving initial value problems for systems of differential equations with expensive right-hand side functions whose Jacobian has its stiff eigenvalues along the negative axis. Such equations arise after spatial discretization of parabolic integro-differential equations of Volterra or Fredholm type with nonstiff integral parts. The methods to be developed in this paper may be interpreted as stabilized forward Euler methods. They require only one right-hand side evaluation per step and the construction of a stabilizing matrix. This matrix should be tuned to the class of problems to be integrated. In the case of parabolic integro-differential equations, the stabilizing matrix will be based on Chebyshev polynomials and will be constructed by means of recursions satisfied by these polynomials. This construction is related to the construction of the intermediate stages in the so-called Runge-Kutta-Chebyshev methods for ordinary differential equations. In analogy with these methods, we shall call the stabilized Euler methods, Euler-Chebyshev methods. They are second-order accurate, and although they are explicit, their stepsize restriction is not prescribed by the stiff eigenvalues. For integro-differential equations in which the parabolic part consists of a one-dimensional diffusion term, we describe an efficient implementation of the stabilizing matrix, which is based on factorization properties of Chebyshev polynomials.

Reaction diffusion equations and quadratic convergence

In this paper, the method of generalized quasilinearization has been extended to reaction diffusion equations. The extension includes earlier known results as special cases. The earlier results developed are when (i) the right‐hand side function is the sum of a convex and concave function, and (ii) the right‐hand function can be made convex by adding a convex function. In our present result, if the monotone iterates are mildly nonlinear, we establish the quadratic convergence as in the quasilinearization method. If the iterates are totally linear then the iterates converge semi‐quadratically.

Parallel Linear System Solvers for Runge-Kutta Methods

If the nonlinear systems arising in implicit Runge-Kutta methods like the Radau IIA methods are iterated by (modified) Newton, then we have to solve linear systems whose matrix of coefficients is of the form I-A ⊗ hJ with A the Runge-Kutta matrix and J an approximation to the Jacobian of the righthand side function of the system of differential equations. For larger systems of differential equations, the solution of these linear systems by a direct linear solver is very costly, mainly because of the LU-decompositions. We try to reduce these costs by solving the linear systems by a second (inner) iteration process. This inner iteration process is such that each inner iteration again requires the solution of a linear system. However, the matrix of coefficients in these new linear systems is of the form I - B ⊗ hJ where B is similar to a diagonal matrix with positive diagonal entries. Hence, after performing a similarity transformation, the linear systems are decoupled into s subsystems, so that the costs of the LU-decomposition are reduced to the costs of s LU-decompositions of dimension d. Since these LU-decompositions can be computed in parallel, the effective LU-costs on a parallel computer system are reduced by a factor s 3 . It will be shown that matrices B can be constructed such that the inner iterations converge whenever A and J have their eigenvalues in the positive and nonpositive halfplane, respectively. The theoretical results will be illustrated by a few numerical examples. A parallel implementation on the four-processor Cray-C98/4256 shows a speed-up ranging from at least 2.4 until at least 3.1 with respect to RADAU5 applied in one-processor mode.

Qualitative simulation in the limit

Limit properties of the QSIM algorithm with respect to the spurious behaviours have been analysed in this short paper, in the case of approximating landmark sets and non-qualitative right-hand side functions. It has been shown that at least two types of spurious behaviour do not disappear in the limit. The first type corresponds to the identically constant solution and appears when the solution passes the zero location of the right-hand side function, i.e. at a critical point. The second type is a consequence of the rigid sign-based description of qualitative directions and it may appear in every multivariable case (for sets of qualitative differential equations). The commonly applied global filters do not remove the source of the problems. A mathematically correct solution would be using the extensions of the usual numerical methods for integrating ordinary differential equations operating on intervals.

Fast solution methods for fredholm integral equations of the second kind

The main purpose of this paper is to describe a fast solution method for one-dimensional Fredholm integral equations of the second kind with a smooth kernel and a non-smooth right-hand side function. Let the integral equation be defined on the interval [?1, 1]. We discretize by a Nystrom method with nodes {cos(?j/N)} j =0/N . This yields a linear system of algebraic equations with an (N+1)×(N+1) matrixA. GenerallyN has to be chosen fairly large in order to obtain an accurate approximate solution of the integral equation. We show by Fourier analysis thatA can be approximated well by $$\mathop A\limits^ \sim $$ , a low-rank modification of the identity matrix. ReplacingA by $$\mathop A\limits^ \sim $$ in the linear system of algebraic equations yields a new linear system of equations, whose elements, and whose solution $$\tilde x$$ , can be computed inO (N logN) arithmetic operations. If the kernel has two more derivatives than the right-hand side function, then $$\tilde x$$ is shown to converge optimally to the solution of the integral equation asN increases. We also consider iterative solution of the linear system of algebraic equations. The iterative schemes use bothA andA. They yield the solution inO (N 2) arithmetic operations under mild restrictions on the kernel and the right-hand side function. Finally, we discuss discretization by the Chebyshev-Galerkin method. The techniques developed for the Nystrom method carry over to this discretization method, and we develop solution schemes that are faster than those previously presented in the literature. The schemes presented carry over in a straightforward manner to Fredholm integral equations of the second kind defined on a hypercube.