Shape parameter selection for multi-quadrics function method in solving electromagnetic boundary value problems

Abstract

Purpose – The multi-quadrics (MQ) function is a kind of radial basis function. And the MQ method has been successfully adopted as a type of meshless method in solving electromagnetic boundary value problems. However, the accuracy of MQ interpolation or solving equations is severely influenced by shape parameter. Thus the purpose of this paper is to propose a case-independent shape parameter selection strategy from the aspect of coefficient matrix condition number analysis. Design/methodology/approach – The condition number of coefficient matrix is investigated. It is shown that the condition number is only a function of shape parameter and MQ node number, and is irrelevant to the interpolated function which means case-independent. The effective condition number which takes into account the interpolated function is introduced. Then, the relation between the relative root mean square error and condition number is analyzed. Three numerical experiments as transmission line, cable channel and grounding metal box model were carried out. Findings – In the numerical experiments, there is an approximate linear relationship between the logarithm of the condition number and shape parameter, an approximate quadratic relationship with node number. And the optimal shape parameter is corresponding to the early stage of condition number oscillation. Originality/value – This paper proposed a case-independent shape parameter selection strategy. For a finite precision computation, the upper limit of the condition number is predetermined. Therefore, the shape parameter can be chosen where condition number oscillates in early stage.

A hybrid radial basis functions collocation technique to numerically solve fractional advection–diffusion models

“…Note that, effective condition number is used here as a check to observe early stage oscillation in the condition number of system matrix. Anyhow, if κ (Φ ϵ ) exceeds this bound but κ eff (Φ ϵ ) is comparatively small, the computed results can be considered still stable and accurate (see e.g., [5, 38]).…”

“…Using these facts, researchers have used condition number of the system matrix to optimally locate shape parameter. The notable works are Cheng et al [5], Sarra [10, 37], Zhang et al [38], and Haq and Hussain [39]. It is to mention that the Schaback uncertainty principle holds in case of standard bases [40], however, this issue can be alleviated by modifying or changing the bases [40–43].…”

Certain Relations Between the Main Matrix Condition Number and Multiquadric Shape Parameter in the Non-Symmetric Kansa Method

“…Haq and Hussain [20] solved the time-fractional Black-Scholes equations using the Kansa method. They formulated an algorithm for determining the quasi-optimal shape parameter value based on Zhang et al's paper [21]. This paper proposed a caseindependent shape parameter selection strategy.…”