inverse problem
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J. Stepan ◽  
T. del Pino Aleman ◽  
J. Trujillo Bueno

2022 ◽  
Roland Lombard ◽  
Rabia Yekken

Abstract We want to thank our colleague F. Fernandez for his interest and his careful reading of our paper "The inverse problem from discrete spectrum in the D = 2 dimensional space". We are confused to have left a number of mistakes in the manuscript.

2022 ◽  
Francisco Marcelo Fernandez

Abstract We analyse a method for the construction of the potential-energy function from the moments of the ground-state density. The sum rule on which some expressions are based appear to be wrong, as well as the moments and potential-energy functions derived for some illustrative examples.

Zeng Hui ◽  
Li Ying ◽  
Wang Lingyue ◽  
Yin Ning ◽  
Yang Shuo

Electroencephalography (EEG) inverse problem is a typical inverse problem, in which the electrical activity within the brain is reconstructed based on EEG data collected from the scalp electrodes. In this paper, the four-layer concentric head model is used for simulation firstly, four deep neural network models including a multilayer perceptron (MLP) model and three convolutional neural networks (CNNs) are adopted to solve EEG inverse problem based on equal current dipole (ECD) model. In the simulations, 100,000 samples are generated randomly, of which 60% are used for network training and 20% are used for cross-validation. Eventually, the generalization performance of the model using the optimal function is measured by the errors in the rest 20% testing set. The experimental results show that the absolute error, relative error, mean positioning error and standard deviation of the four models are extremely low. The CNN with 6 convolutional layers and 3 pooling layers (CNN-3) is the best model. Its absolute error is about 0.015, its relative error is about 0.005, and its dipole position error is 0.040±0.029 cm. Furthermore, we use CNN-3 for source localization of the real EEG data in Working Memory. The results are in accord with physiological experience. The deep neural network method in our study needs fewer calculation parameters, takes less time, and has better positioning results.

Oliver Bodart ◽  
Valérie Cayol ◽  
Farshid Dabaghi ◽  
Jonas Koko

2022 ◽  
Abdelhak Hadj

Abstract This study This work deals with an inverse problem for the harmonic equation to recover a Robin coefficient on a non-accessible part of a circle from Cauchy data measured on an accessible part of that circle. By assuming that the available data has a Fourier expansion, we adopt the Modified Collocation Trefftz Method (MCTM) to solve this problem. We use the truncation regularization method in combination with the collocation technique to approximate the solution, and the conjugate gradient method to obtain the coefficients, thus completing the missing Cauchy data. We recommend the least squares method to achieve a better stability. Finally, we illustrate the feasibility of this method with numerical examples.

2022 ◽  
Hanne Kekkonen

Abstract We consider the statistical non-linear inverse problem of recovering the absorption term f>0 in the heat equation with given sufficiently smooth functions describing boundary and initial values respectively. The data consists of N discrete noisy point evaluations of the solution u_f. We study the statistical performance of Bayesian nonparametric procedures based on a large class of Gaussian process priors. We show that, as the number of measurements increases, the resulting posterior distributions concentrate around the true parameter generating the data, and derive a convergence rate for the reconstruction error of the associated posterior means. We also consider the optimality of the contraction rates and prove a lower bound for the minimax convergence rate for inferring f from the data, and show that optimal rates can be achieved with truncated Gaussian priors.

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