A sixth order convergence of Jarratt-type method for solving nonlinear equations is considered. Weaker assumptions on the derivative of the involved operator is made, contrary to the earlier studies. The convergence analysis does not depend on the Taylor series expansion and this increases the applicability of the proposed method. Numerical examples and Basins of attractions of the method are provided in this study.

[1] I.K. Argyros , S. Hilout. On the local convergence of fast two-step Newton-like methods for solving nonlinear equations: Journal of Computational and Applied Mathematics 245:1-9, 2013.

[2] A. Cordero , M.A. Hernández-Verón , N. Romero , J.R. Torregrosa. Semilocal convergence by using recurrence relations for a fifth-order method in Banach spaces: Journal of computational and applied mathematics,volume(273):205-213, 2015.

[3] S. George , I.K. Argyros , P. Jidesh , M. Mahapatra, M. Saeed. Convergence Analysis of a Fifth-Order Iterative Method Using Recurrence Relations and Conditions on the First Derivative: Mediterranean Journal of Mathematics,volume(18):1-12, 2021.

[4] P. Jarratt. Some fourth order multipoint iterative methods for solving equations: Mathematics of computation, Vol(20):434-437, 1966.

[5] H. Ren. On the local convergence of a deformed Newton’s method under Argyros-type condition, Journal of Mathematical Analysis and Applications, 321(1):396-404. 2006.

[6] S. Singh, D.K. Gupta, E. Martínez , J.L. Hueso. Semilocal convergence analysis of an iteration of order five using recurrence relations in Banach spaces: Mediterranean Journal of Mathematics. volume(13):4219-4235, 2016.

A key problem in aeroacoustics is the inverse problem of estimating an unknown random source from correlation data sampled from surrounding sensors. We study optimal design for this and related problems, that is, we identify the sensor placement minimising covariance of the solution to the inverse random source problem, while remaining sparse. To achieve this, we discuss the assumption of gaussianity and how to adapt this to our setting of correlation data, and demonstrate how this model can lead to sparse designs for aeroacoustic experiments.

Electron Paramagnetic Resonance Imaging (EPRI) is a versatile imaging modality that enables the study of free radical molecules or atoms from materials $\textit{in-vitro}$ to$ \textit{in-vivo}$ appplication in biomedical research. Clinical applications are currently under investigation. While recent advancements in EPRI techniques have made it possible to study a single free radical, or source, inside the imaging device [1], the reconstruction of multiple sources, or source separation, remains a challenging task. The state-of-the-art technique heavily relies on time-consuming acquisition and voxel-wise direct inverse methods, which are prone to artifacts and do not leverage the spatial consistency of the source images to reconstruct. To address this issue, we propose a variational formulation of the source separation problem with a Total Variation $\textit{a-priori}$, which emphasizes the spatial consistency of the source. This approach drastically reduces the needed number of acquisitions without sacrificing the quality of the source separation. An EPRI experimental study has been conducted, and we will present some of the results obtained.

[1] S. Durand, Y.-M. Frapart, M. Kerebel. Electron paramagnetic resonance image reconstruction with total variation and curvelets regularization. Inverse Problems, 33(11):114002, 2017.