Stefan Jakobsson

Rational radial basis function interpolation with applications to antenna design

Functions with poles occur in many branches of applied mathematics which involve resonance phenomena. Such functions are challenging to interpolate, in particular in higher dimensions. In this paper we develop a technique for interpolation with quotients of two radial basis function (RBF) expansions to approximate such functions as an alternative to rational approximation. Since the quotient is not uniquely determined we introduce an additional constraint, the sum of the RBF-norms of the numerator and denominator squared should be minimal subjected to a norm condition on the function values. The method was designed for antenna design applications and we show by examples that the scattering matrix for a patch antenna as a function of some design parameters can be approximated accurately with the new method. In many cases, e.g. in antenna optimization, the function evaluations are time consuming, and therefore it is important to reduce the number of evaluations but still obtain a good approximation. A sensitivity analysis of the new interpolation technique is carried out and it gives indications how efficient adaptation methods could be devised. A family of such methods are evaluated on antenna data and the results show that much performance can be gained by choosing the right method.

Particle Swarm Optimization Applied to EEG Source Localization of Somatosensory Evoked Potentials

One of the most important steps in presurgical diagnosis of medically intractable epilepsy is to find the precise location of the epileptogenic foci. Electroencephalography (EEG) is a noninvasive tool commonly used at epilepsy surgery centers for presurgical diagnosis. In this paper, a modified particle swarm optimization (MPSO) method is used to solve the EEG source localization problem. The method is applied to noninvasive EEG recording of somatosensory evoked potentials (SEPs) for a healthy subject. A 1 mm hexahedra finite element volume conductor model of the subjects head was generated using T1-weighted magnetic resonance imaging data. Special consideration was made to accurately model the skull and cerebrospinal fluid. An exhaustive search pattern and the MPSO method were then applied to the peak of the averaged SEP data and both identified the same region of the somatosensory cortex as the location of the SEP source. A clinical expert independently identified the expected source location, further corroborating the source analysis methods. The MPSO converged to the global minima with significantly lower computational complexity compared to the exhaustive search method that required almost 3700 times more evaluations.

Application of particle swarm optimization in epileptic spike EEG source localization

Surgical therapy has become an important therapeutic alternative for patients with medically intractable epilepsy. Correct and anatomically precise localization of an epileptic focus is essential to decide if resection of brain tissue is possible. The inverse problem in EEG-based source localization is to determine the location of the brain sources that are responsible for the measured potentials at the scalp electrodes. We propose a new global optimization method based on particle swarm optimization (PSO) to solve the epileptic spike EEG source localization inverse problem. In a forward problem a modified subtraction method is proposed to reduce the computational time. The good accuracy and fast convergence are demonstrated for 2D and 3D cases with realistic head models. The results from the new method are promising for use in the pre-surgical clinic in the future.

Non-invasive EEG source localization using particle swarm optimization: A clinical experiment

One of the most important steps of pre-surgical diagnosis in patients with medically intractable epilepsy is to find the precise location of the epileptogenic foci. An Electroencephalography (EEG) is a non-invasive standard tool used at epilepsy surgery center for pre-surgical diagnosis. In this paper a modified particle swarm optimization (MPSO) method is applied to a real EEG data, i.e., a somatosensory evoked potentials (SEPs) measured from a healthy subject, to solve the EEG source localization problem. A high resolution 1 mm hexahedra finite element volume conductor model of the subjects head was generated using T1-weighted magnetic resonance imaging data. An exhaustive search pattern and the MPSO method were then applied to the peak of the averaged SEPs data. The non-invasive EEG source analysis methods localized the somatosensory cortex area where our clinical expert expected the received SEPs. The proposed inverse problem solver found the global minima with acceptable accuracy and reasonable number of iterations.

Multiobjective Optimization Applied to Design of PIFA Antennas

In this paper multiobjective optimization is applied to antenna design. The optimization algorithm is a novel response surface method based on approximation with radial basis functions. It is combined with CAD and mesh generation software, and electromagnetic solvers. To demonstrate the procedure we optimize the geometric design and feed position of a PIFA antenna located on a ground plane.

A time domain beam propagation method for chirped signals

Chirped pulses are used in time stretch analog-to-digital converters to stretch high speed electric signals so that they can be measured with conventional detectors. Here a time domain beam propagation method for strongly chirped signals propagating along a specified direction is presented. By examining the propagation of chirped pulses in waveguides we derive a phase factor which captures the rapidly oscillating part of the chirped pulse. We then solve for the slowly varying envelope with respect to this phase factor in a time window moving with the pulse. The new method is applied to simulation of electro-optical modulators for chirped pulses with promising results.

Approximation of Antenna Data with Rational Radial Basis Function Interpolation

We present a recently developed method to interpolate functions with poles by quotients of two radial basis functions (RBF) expansions. This method combines rational function’s ability to approximate function with poles with RBF interpolation’s flexibility for scattered data interpolation in any space dimension. The method is tested on an approximation problem for antenna data.