Order Selection Scheme Research Articles

Perturbation Approach for Order Selections of Two-Sided Oblique Projection-Based Interconnect Reductions

An order selection scheme for two-sided oblique projection-based interconnect reduction will be investigated. It will provide a guideline for terminating the conventional nonsymmetric Pade via Lanczos (PVL) iteration process. By exploring the relationship of the system Grammians of the original network and those of the reduced network, it can be shown that the system matrix of the reduced-order system generated by the two-sided oblique projection can also be expressed as those of the original interconnect model with some additive perturbations. The perturbation matrix only involves bi-orthogonal vectors at the previous step of the nonsymmetric Lanczos algorithm. This perturbation matrix will provide the stopping criteria in the order selection scheme and achieve the desired accuracy of the approximate transfer function.

Error Estimations of Arnoldi-Based Interconnect Model-Order Reductions

Projection-based model reductions become a necessity for efficient interconnect modeling and simulations. In order to choose the order of the reduced system that can really reflect the essential dynamics of the original interconnect, the residual error of the transfer function can be considered as a stopping criteria to terminate the Arnoldi iteration process. Analytical expressions of this residual error are derived in detail. Furthermore, it can be found that the approximate transfer function can also be expressed as the original interconnect model with some additive perturbations. The perturbation matrix only involves resultant vectors at the previous step of the Arnoldi algorithm. These error information will provide a guideline for the order selection scheme used in the Krylov subspace model-order algorithm.

Overlap bias in the case-crossover design, with application to air pollution exposures

The case-crossover design uses cases only, and compares exposures just prior to the event times to exposures at comparable control, or 'referent' times, in order to assess the effect of short-term exposure on the risk of a rare event. It has commonly been used to study the effect of air pollution on the risk of various adverse health events. Proper selection of referents is crucial, especially with air pollution exposures, which are shared, highly seasonal, and often have a long-term time trend. Hence, careful referent selection is important to control for time-varying confounders, and in order to ensure that the distribution of exposure is constant across referent times, a key assumption of this method. Yet the referent strategy is important for a more basic reason: the conditional logistic regression estimating equations commonly used are biased when referents are not chosen a priori and are functions of the observed event times. We call this bias in the estimating equations overlap bias. In this paper, we propose a new taxonomy of referent selection strategies in order to emphasize their statistical properties. We give a derivation of overlap bias, explore its magnitude, and consider how the bias depends on properties of the exposure series. We conclude that the bias is usually small, though highly unpredictable, and easily avoided.

Applications of Lobatto polynomials to an adaptive finite element method: A posteriori error estimates for HP-Adaptivity and Grid-to-Grid Interpolation

Hp-adaptive finite element codes require methods for estimating the error at several spatial orders and for interpolating solutions between grids. Lobatto polynomial-based techniques are presented for both. An interpolation error-based error estimation strategy for a posteriori error estimates is generalized to yield asymptotically exact error estimates one order higher than the computed solution. The estimates involve high-order derivatives of the solution that must be approximated from the computed solution. Differentiating a ``Taylor-like'' series for error in the Lobatto interpolant and using the weak form of the equations yields the correct derivative approximations. This leads to a more robust order selection strategy. Interpolation between grids is done over each element using the Lobatto interpolating polynomial. Explicit formulas for the inverse of the resulting Lobatto interpolation matrices are given. Computational results illustrate the theory.

Continuous numerical methods for ODEs with defect control

Over the last decade several general-purpose numerical methods for ordinary differential equations (ODEs) have been developed which generate a continuous piecewise polynomial approximation that is defined for all values of the independent variable in the range of interest. For such methods it is possible to introduce measures of the quality of the approximate solution based on how well the piecewise polynomial satisfies the ODE. This leads naturally to the notion of “defect-control”. Numerical methods that adopt error estimation and stepsize selection strategies in order to control the magnitude of the associated defect can be very effective and such methods are now being widely used. In this paper we will review the advantages of this class of numerical methods and present examples of how they can be effectively applied. We will focus on numerical methods for initial value problems (IVPs) and boundary value problems (BVPs) where most of the developments have been introduced but we will also discuss the implications and related developments for other classes of ODEs such as delay differential equations (DDEs) and differential algebraic equations (DAEs).

Maximum a posteriori maximum entropy order determination

An instance crucial to most problems in signal processing is the selection of the order of a presupposed model. Examples are the determination of the putative number of signals present in white Gaussian noise or the number of noise-contaminated sources impinging on a passive sensor array. It is shown that maximum a posteriori Bayesian arguments, coupled with maximum entropy considerations, offer an operational and consistent model order selection scheme, competitive with the minimum description length criterion.

A Runge-Kutta starter for a multistep method for differential-algebraic systems with discontinuous effects

In simulation problems, e.g. in multibody dynamics, realistic models usually incorporate discontinuous effects necessitating frequent restarts of multistep methods. This makes the order and stepsize selection schemes and in particular the starting phase of such methods crucial for their efficiency. In this paper we present a Runge-Kutta starter for an Adams-Bashforth-Moulton predictor-corrector method by giving a detailed description of its construction as well as the ideas behind it with a special emphasis on an appropriate error estimator. Finally we demonstrate its usefulness on some numerical examples.

A New Order Selection Strategy for Ordinary Differential Equation Solvers

The normal order selection strategy used by a variable order, variable stepsize ordinary differential equation solver is based on selecting the formula that can use the largest stepsize while meeting the accuracy constraint. This strategy does not take into account the stiffness or nonstiffness of the differential system. By considering how the stiffness of the differential system influences order selection, a new selection strategy is proposed that is biased in the presence of stiffness toward the lower-order formulas. This new order selection strategy has been incorporated into a version of a widely used ordinary differential equation solver LSODE. The modified solver is shown to be greatly more efficient for problems that the original solver cannot solve efficiently, namely, problems where the Jacobian of the differential system has some eigenvalues of large modulus close to the imaginary axis. On other problems that the unmodified solver solves efficiently, the modified solver still performs with comparable efficiency.

An efficient algorithm reduces computing time in solving a system of stiff ordinary differential equations

The problem of stiff differential equations arises in many computer-aided design techniques, particularly in the transient analysis of network simulation. Special multistep methods are used to solve the first-order stiff nonlinear differential equations. Instability and a large number of steps are encountered during simulation. Different techniques such as step and order selection schemes, procedures for changing step and order, may reduce the number of steps while preserving stability. An improved algorithm is presented using BDF formulas given by Brayton et at. and leads to reducing computer time by controlling the number of integration steps, but also the number of Newton iterations, the number of Jacobian matrix evaluations and other parameters, without producing additional errors or instability phenomena. Experimental results are shown.