On the Quotients of Regular Operators

Approximation of multipliers by regular operators

Indagationes Mathematicae ◽

10.1016/0019-3577(91)90003-p ◽

1991 ◽

Vol 2 (2) ◽

pp. 159-170 ◽

Cited By ~ 3

Author(s):

Wolfgang Arendt ◽

Jürgen Voigt

Keyword(s):

Regular Operators

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Regular operators

Functional Analysis and Its Applications ◽

10.1007/bf01075953 ◽

1968 ◽

Vol 2 (2) ◽

pp. 175-176 ◽

Cited By ~ 2

Author(s):

G. M. Vainikko ◽

Yu. B. Umanskii

Keyword(s):

Regular Operators

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Hierarchical and Multi-resolution Neuron Network Computing for Complex Inverse Problems-An Approach Based on Brain-inspired Computing and Learning Method

Current Chinese Computer Science ◽

10.2174/2665997201666210329104604 ◽

2021 ◽

Vol 01 ◽

Author(s):

Mingyong Zhou

Keyword(s):

Deep Learning ◽

Inverse Problems ◽

Radar Imaging ◽

Network Computing ◽

Neuron Network ◽

Impedance Tomography ◽

General Complex ◽

Mathematical Algorithms ◽

Ill Posed ◽

Regular Operators

Background: Complex inverse problems such as Radar Imaging and CT/EIT imaging are well investigated in mathematical algorithms with various regularization methods. However it is difficult to obtain stable inverse solutions with fast convergence and high accuracy at the same time due to the ill-posed property and non-linear property. Objective: In this paper, we propose a hierarchical and multi-resolution scalable method from both algorithm perspective and hardware perspective to achieve fast and accurate solu-tions for inverse problems by taking radar and EIT imaging as examples. Method: We present an extension of discussion on neuromorphic computing as brain-inspired computing method and the learning/training algorithm to design a series of problem specific AI “brains” (with different memristive values) to solve a general complex ill-posed inverse problems that are traditionally solved by mathematical regular operators. We design a hierarchical and multi-resolution scalable method and an algorithm framework to train AI deep learning neuron network and map into the memristive circuit so that the memristive val-ues are optimally obtained. We propose FPGA as an emulation implementation for neuro-morphic circuit as well. Result: We compared the methodology between our approach and traditional regulariza-tion method. In particular we use Electrical Impedance Tomography (EIT) and Radar imaging as typical examples to compare how to design an AI deep learning neuron network architec-tures to solve inverse problems. Conclusion: With EIT imaging as a typical example, we show that any moderate complex inverse problem, as long as it can be described as combinational problem, AI deep learning neuron network is a practical alternative approach to try to solve the inverse problems with any given expected resolution accuracy, as long as the neuron network width is large enough and computational power is strong enough for all combination samples training purpose.

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