Spectral shift function for a discretized continuum

A spectral shift function (SSF) is an important object in the scattering theory which is related both to the spectral density and to the scattering matrix. In the paper, it is shown how to employ the SSF formalism to solve scattering problems when the continuum is discretized, e.g. when solving a scattering problem in a finite volume or in the representation of some finite square-integrable basis. A new algorithm is proposed for reconstructing integrated densities of states and the SSF using a union of discretized spectra corresponding to a set of Gaussian bases with the shifted scale parameters. The examples given show that knowledge of the discretized spectra of the total and asymptotic Hamiltonians is sufficient to find the scattering partial phase shifts at any required energy, as well as the resonances parameters.

Convergence analysis of two finite element methods for the modified Maxwell's Steklov eigenvalue problem

The modified Maxwell's Steklov eigenvalue problem is a new problem arising from the study of inverse electromagnetic scattering problems. In this paper, we investigate two finite element methods for this problem and perform the convergence analysis. Moreover, the monotonic convergence of the discrete eigenvalues computed by one of the methods is analyzed.

A Real-time Breast Hyperthermia Monitoring Scheme Based on Processing of Microwave Scattering Parameters with Deep Learning

This study proposes a new method based on deep learning to determine whether the temperature values ​​are at an appropriate level during the use of microwave hyperthermia method in the treatment of breast cancer. To implement our method, we utilize the temperature dependent dielectric properties of biological tissues to generate the heating scenarios that simulates the thermal behavior of biological tissue during the breast cancer hyperthermia treatment. Using the temperature-dependent dielectric properties we designated corresponding temperature thresholds, next, we labeled the malignant tumor region and the healthy tissue region in accordance with the pre-determined thresholds. In addition, scattering problems are solved based on treatment (hot or heated) and pre-treatment (cool) scenarios. Using the difference between hot and cool states, we train, test, and validate the CNN. Our main purpose in the project is to determine whether the tissue is heated in the desired temperature region using only the single frequency differential scattered electric field data.

A Real-time Breast Hyperthermia Monitoring Scheme Based on Processing of Microwave Scattering Parameters with Deep Learning

This study proposes a new method based on deep learning to determine whether the temperature values ​​are at an appropriate level during the use of microwave hyperthermia method in the treatment of breast cancer. To implement our method, we utilize the temperature dependent dielectric properties of biological tissues to generate the heating scenarios that simulates the thermal behavior of biological tissue during the breast cancer hyperthermia treatment. Using the temperature-dependent dielectric properties we designated corresponding temperature thresholds, next, we labeled the malignant tumor region and the healthy tissue region in accordance with the pre-determined thresholds. In addition, scattering problems are solved based on treatment (hot or heated) and pre-treatment (cool) scenarios. Using the difference between hot and cool states, we train, test, and validate the CNN. Our main purpose in the project is to determine whether the tissue is heated in the desired temperature region using only the single frequency differential scattered electric field data.

Boundary element methods for the wave equation based on hierarchical matrices and adaptive cross approximation

Time-domain Boundary Element Methods (BEM) have been successfully used in acoustics, optics and elastodynamics to solve transient problems numerically. However, the storage requirements are immense, since the fully populated system matrices have to be computed for a large number of time steps or frequencies. In this article, we propose a new approximation scheme for the Convolution Quadrature Method powered BEM, which we apply to scattering problems governed by the wave equation. We use $${\mathscr {H}}^2$$ H 2 -matrix compression in the spatial domain and employ an adaptive cross approximation algorithm in the frequency domain. In this way, the storage and computational costs are reduced significantly, while the accuracy of the method is preserved.

Resolution of Born Scattering in Curve Geometries: Aspect-Limited Observations and Excitations

In inverse scattering problems, the most accurate possible imaging results require plane waves impinging from all directions and scattered fields observed in all observation directions around the object. Since this full information is infrequently available in actual applications, this paper is concerned with the mathematical analysis and numerical simulations to estimate the achievable resolution in object reconstruction from the knowledge of the scattered far-field when limited data are available at a single frequency. The investigation focuses on evaluating the Number of Degrees of Freedom (NDF) and the Point Spread Function (PSF), which accounts for reconstructing a point-like unknown and depends on the NDF. The discussion concerns objects belonging to curve geometries, in this case, circumference and square scatterers. In addition, since the exact evaluation of the PSF can only be accomplished numerically, an approximated closed-form evaluation is introduced and compared with the exact one. The approximation accuracy of the PSF is verified by numerical results, at least within its main lobe region, which is the most critical as far as the resolution discussion is concerned. The main result of the analysis is the space variance of the PSF for the considered geometries, showing that the resolution is different over the investigation domain. Finally, two numerical applications of the PSF concept are shown, and their relevance in the presence of noisy data is outlined.