# Registered Data

Contents

- 1 [CT032]
- 1.1 [01233] A Study of the Spectra-Cutoff Imaging Method of Multiple Scattering in Isotropic Point-Like Discrete Random Media
- 1.2 [01754] Spectral-Cutoff for Imaging of Multiple Scattering in Isotropic Point-Like Discrete Random Media
- 1.3 [01413] Inverted finite elements method: principles and recent advances.
- 1.4 [02278] On the fourth order semipositone problem in $\mathbb{R}^N$
- 1.5 [00958] Existence results and numerical approximation for a quasilinear elliptic system

# [CT032]

## [01233] A Study of the Spectra-Cutoff Imaging Method of Multiple Scattering in Isotropic Point-Like Discrete Random Media

**Session Date & Time**: 2C (Aug.22, 13:20-15:00)**Type**: Contributed Talk**Abstract**: Imaging in random media is an important and interesting subject of inverse problems, relevant to a wide range of physical and engineering contexts, such as seismic imaging, remote sensing, medical imaging, wireless communications, and nondestructive testing. In this talk, we show that imaging becomes difficult to perform in random media when multiple scattering is too strong to cause image distortion arising from the underlying interactions of multiply scattered waves at resonance frequencies. The Foldy-Lax-Lippmann-Schwinger, (FLLS), formalism, which is employed for the multiply scattered waves, in the frequency domain, in the case of an ensemble of randomly distributed point-like scatterers. The scattering matrix representing the (FLLS) formalism is a non-Hermitian Euclidean random matrix. According to the eigenvalue distribution of the scattering matrix, we present our approach to restore the distorted images by cutting off the sharp frequency responses in the resonance regime due to strong multiple scattering. Finally, we show the use of this approach for imaging in discrete random media with numerical simulations and also discuss the limitations and future research direction.**Classification**:__35J05__,__35P15__,__35P25__,__47B06__,__78A46__**Author(s)**:**Ray-Hon Sun**(Stanford University (while working on this research))

## [01754] Spectral-Cutoff for Imaging of Multiple Scattering in Isotropic Point-Like Discrete Random Media

**Session Date & Time**: 2C (Aug.22, 13:20-15:00)**Type**: Contributed Talk**Abstract**: To image objects in the discrete random medium composed of isotropic point-like scatterers, the resonances with multiple scattering in the medium can interfere with imaging and result in poor quality of images. To solve this problem, we present a spectral-cutoff method, which is derived based on the random matrix theory, to filter out the undesired responses in the resonance regime to recover the damaged images. Finally, we demonstrate this method for boosting imaging with numerical simulations.**Classification**:__35J05__,__35P15__,__47B06__,__78A46__,__94A12__**Author(s)**:**Ray-Hon Sun**(Stanford University)- Ray-Hon Sun (Stanford University)

## [01413] Inverted finite elements method: principles and recent advances.

**Session Date & Time**: 2C (Aug.22, 13:20-15:00)**Type**: Contributed Talk**Abstract**: The purpose of this talk is to expose the general principles of the inverted finite element method. The method, introduced by the speaker, aims to solve partial differential equations, especially elliptic ones, in unbounded domains without truncation. After a rigorous analysis of its convergence, we present some numerical results obtained with this method in the case of several multidimensional problems arising in physics (micromagnetism, quantum physics, etc.).**Classification**:__35J15__,__65N30__,__65N99__,__65Z05__**Author(s)**:**Tahar Zamene BOULMEZAOUD**(University of Versailles SQY, University of Paris-Saclay and University of Victoria )

## [02278] On the fourth order semipositone problem in $\mathbb{R}^N$

**Session Date & Time**: 2C (Aug.22, 13:20-15:00)**Type**: Contributed Talk**Abstract**: For $N \geq 5$ and $a>0$, we consider the following semipositone problem \begin{align*} \Delta^2 u= g(x)f_a(u) \text { in } \mathbb{R}^N, \, \text{ and } \, u \in \mathcal{D}^{2,2}(\mathbb{R}^N),\ \ \ \qquad \quad \mathrm{(SP)} \end{align*} where $g \in L^1_{loc}(\mathbb{R}^N)$ is an indefinite weight function, $f_a:\mathbb{R} \to \mathbb{R}$ is a continuous function that satisfies $f_a(t)=-a$ for $t \in \mathbb{R}^-$, and $\mathcal{D}^{2,2}(\mathbb{R}^N)$ is the completion of $\mathcal{C}_c^{\infty}(\mathbb{R}^N)$ with respect to $(\int_{\mathbb{R}^N} (\Delta u )^2)^{1/2}$. For $f_a$ satisfying subcritical nonlinearity and a weaker Ambrosetti-Rabinowitz type growth condition, we find the existence of $a_1>0$ such that for each $a \in (0,a_1)$, $\mathrm{(SP)}$ admits a mountain pass solution. Further, we show that the mountain pass solution is positive if $a$ is near zero. For the positivity, we derive uniform regularity estimates of the solutions of $\mathrm{(SP)}$ for certain ranges in $(0,a_1)$, relying on the Riesz potential of the biharmonic operator.**Classification**:__35J35__,__35J91__,__35J08__,__35B09__,__Variational methods for higher-order elliptic equations, Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian, Green’s functions for elliptic equations, Positive solutions to PDEs__**Author(s)**:**Nirjan Biswas**(Tata Institute of Fundamental Research CAM, Bengaluru)- Ujjal Das (Technion - Israel Institute of Technology)
- Abhishek Sarkar (Indian Institute of Technology Jodhpur)

## [00958] Existence results and numerical approximation for a quasilinear elliptic system

**Session Date & Time**: 2C (Aug.22, 13:20-15:00)**Type**: Contributed Talk**Abstract**: We analyse, in the context of anisotropic Sobolev spaces, the existence and the numerical simulation of a capacity solution to a coupled nonlinear elliptic system. We consider the case of a non-uniformly elliptic problem with a quadratic growth in the gradient. The system may be regarded as a generalization of the so-called thermistor problem.**Classification**:__35J47__,__35J70__,__47H05__,__46E35__,__65N12__**Author(s)**:**Hajar Talbi**(Moulay Ismail University)