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Extension and convergence of sixth order Jarratt-type method
Suma Panathale Bheemaiah
Manipal Institute of Technology, Manipal Academy of Higher Education, India
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.
Optimal design for aeroacoustics with correlation data
Christian Aarset, Thorsten Hohage
University of Göttingen, Germany
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.
Source separation for Electron Paramagnetic Resonance Imaging
Mehdi Boussâa, Rémy Abergel, Sylvain Durand, Yves Frapart
Université Paris Cité, France
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.