About Data-driven Integration of Ill-posed Geophysical Tomography and Geotechnical Logging Data

2016 ◽  
Author(s):  
H. Paasche
Keyword(s):  
Data Driven ◽  
Ill Posed ◽  
Geophysical Tomography

scholarly journals Conditional Invertible Neural Networks for Medical Imaging

2021 ◽  
Vol 7 (11) ◽  
pp. 243
Author(s):  
Alexander Denker ◽  
Maximilian Schmidt ◽  
Johannes Leuschner ◽  
Peter Maass
Keyword(s):  
Neural Networks ◽  
Medical Imaging ◽  
Inverse Problems ◽  
Low Dose ◽  
Data Driven ◽  
Resonance Imaging ◽  
Point Estimates ◽  
Ill Posed ◽  
Low Dose Computed Tomography

Over recent years, deep learning methods have become an increasingly popular choice for solving tasks from the field of inverse problems. Many of these new data-driven methods have produced impressive results, although most only give point estimates for the reconstruction. However, especially in the analysis of ill-posed inverse problems, the study of uncertainties is essential. In our work, we apply generative flow-based models based on invertible neural networks to two challenging medical imaging tasks, i.e., low-dose computed tomography and accelerated medical resonance imaging. We test different architectures of invertible neural networks and provide extensive ablation studies. In most applications, a standard Gaussian is used as the base distribution for a flow-based model. Our results show that the choice of a radial distribution can improve the quality of reconstructions.

On a characteristic property of conditionally well-posed problems

2015 ◽  
Vol 23 (3) ◽  
Author(s):  
Mikhail Y. Kokurin
Keyword(s):  
Inverse Problems ◽  
Input Data ◽  
Characteristic Property ◽  
Optimization Problems ◽  
Data Driven ◽  
Error Models ◽  
Regularizing Algorithms ◽  
Stochastic Error ◽  
Ill Posed ◽  
Well Posed

AbstractThe aim of this paper is to discuss and illustrate the fact that conditionally well-posed problems stand out among all ill-posed problems as being regularizable via an operator independent of the level of errors in input data. We give examples of corresponding purely data driven regularizing algorithms for various classes of conditionally well-posed inverse problems and optimization problems in the context of deterministic and stochastic error models.

scholarly journals Extract the information from big data with randomly distributed noise

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jin Cheng ◽  
Jiantang Zhang ◽  
Min Zhong
Keyword(s):  
Big Data ◽  
Confidence Interval ◽  
Unique Solvability ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Data Driven ◽  
Numerical Examples ◽  
Ill Posed ◽  
General Data ◽  
Bound Estimation

Abstract In this manuscript, a purely data-driven statistical regularization method is proposed for extracting the information from big data with randomly distributed noise. Since the variance of the noise may be large, the method can be regarded as a general data preprocessing method in ill-posed problems, which is able to overcome the difficulty that the traditional regularization method is unable to solve, and has superior advantage in computing efficiency. The unique solvability of the method is proved, and a number of conditions are given to characterize the solution. The regularization parameter strategy is discussed, and the rigorous upper bound estimation of the confidence interval of the error in the L 2 L^{2} norm is established. Some numerical examples are provided to illustrate the appropriateness and effectiveness of the method.

scholarly journals Solving inverse problems using data-driven models

Acta Numerica ◽  
2019 ◽  
Vol 28 ◽  
pp. 1-174 ◽  
Author(s):  
Simon Arridge ◽  
Peter Maass ◽  
Ozan Öktem ◽  
Carola-Bibiane Schönlieb
Keyword(s):  
Inverse Problems ◽  
Industrial Applications ◽  
Data Driven ◽  
Analytical Models ◽  
Survey Paper ◽  
Domain Specific ◽  
Combining Data ◽  
Domain Specific Knowledge ◽  
Ill Posed ◽  
Using Data

Recent research in inverse problems seeks to develop a mathematically coherent foundation for combining data-driven models, and in particular those based on deep learning, with domain-specific knowledge contained in physical–analytical models. The focus is on solving ill-posed inverse problems that are at the core of many challenging applications in the natural sciences, medicine and life sciences, as well as in engineering and industrial applications. This survey paper aims to give an account of some of the main contributions in data-driven inverse problems.

scholarly journals Extreme Learning Machines as Encoders for Sparse Reconstruction

2018 ◽  
Author(s):  
Abdullah Al-Mamun ◽  
Chen Lu ◽  
Balaji Jayaraman
Keyword(s):  
Fluid Flows ◽  
Data Driven ◽  
Sparse Reconstruction ◽  
Learning Machines ◽  
Physical Mechanisms ◽  
The Face ◽  
Ill Posed ◽  
Learning Machine ◽  
Proper Orthogonal ◽  
Well Posed

Reconstruction of fine-scale information from sparse data is often needed in practical fluid dynamics where the sensors are typically sparse and yet, one may need to learn the underlying flow structures or inform predictions through assimilation into data-driven models. Given that sparse reconstruction is inherently an ill-posed problem, the most successful approaches encode the physics into an underlying sparse basis space that spans the manifold to generate well-posedness. To achieve this, one commonly uses generic orthogonal Fourier basis or data specific proper orthogonal decomposition (POD) basis to reconstruct from sparse sensor information at chosen locations. Such a reconstruction problem is well-posed as long as the sensor locations are incoherent and can sample the key physical mechanisms. The resulting inverse problem is easily solved using $l_2$ minimization or if necessary, sparsity promoting $l_1$ minimization. Given the proliferation of machine learning and the need for robust reconstruction frameworks in the face of dynamically evolving flows, we explore in this study the suitability of non-orthogonal basis obtained from Extreme Learning Machine (ELM) auto-encoders for sparse reconstruction. In particular, we assess the interplay between sensor quantity and sensor placement for a given system dimension for accurate reconstruction of canonical fluid flows in comparison to POD-based reconstruction.

scholarly journals Extreme Learning Machines as Encoders for Sparse Reconstruction

Fluids ◽  
2018 ◽  
Vol 3 (4) ◽  
pp. 88 ◽  
Author(s):  
S Al Mamun ◽  
Chen Lu ◽  
Balaji Jayaraman
Keyword(s):  
Fluid Flows ◽  
Data Driven ◽  
Sparse Reconstruction ◽  
Learning Machines ◽  
Physical Mechanisms ◽  
The Face ◽  
Ill Posed ◽  
Learning Machine ◽  
Proper Orthogonal ◽  
Well Posed

Reconstruction of fine-scale information from sparse data is often needed in practical fluid dynamics where the sensors are typically sparse and yet, one may need to learn the underlying flow structures or inform predictions through assimilation into data-driven models. Given that sparse reconstruction is inherently an ill-posed problem, the most successful approaches encode the physics into an underlying sparse basis space that spans the manifold to generate well-posedness. To achieve this, one commonly uses a generic orthogonal Fourier basis or a data specific proper orthogonal decomposition (POD) basis to reconstruct from sparse sensor information at chosen locations. Such a reconstruction problem is well-posed as long as the sensor locations are incoherent and can sample the key physical mechanisms. The resulting inverse problem is easily solved using l 2 minimization or if necessary, sparsity promoting l 1 minimization. Given the proliferation of machine learning and the need for robust reconstruction frameworks in the face of dynamically evolving flows, we explore in this study the suitability of non-orthogonal basis obtained from extreme learning machine (ELM) auto-encoders for sparse reconstruction. In particular, we assess the interplay between sensor quantity and sensor placement in a given system dimension for accurate reconstruction of canonical fluid flows in comparison to POD-based reconstruction.

Reconstructing the conductivity profile of a graphite block using inductance spectroscopy with data-driven techniques

2021 ◽  
Vol 63 (2) ◽  
pp. 82-87
Author(s):  
J Hampton ◽  
H Tesfalem ◽  
A Fletcher ◽  
A Peyton ◽  
M Brown
Keyword(s):  
Neural Network ◽  
Electrical Conductivity ◽  
Convolutional Neural Network ◽  
Depth Profile ◽  
Test Data ◽  
Polynomial Regression ◽  
Data Driven ◽  
Radial Depth ◽  
Ill Posed ◽  
Reactor Type

The radial depth profile of the electrical conductivity of the graphite channels in the UK's advanced gas-cooled reactors (AGRs) can be reconstructed and estimated by solving a non-linear optimisation problem using the mutual inductance spectra of a set of coils. This process is slow, as it requires many iterations of a forward solver. Alternatively, a data-driven approach can be used to provide an initial estimate for the optimisation algorithm, reducing the amount of time it takes to solve the ill-posed inverse problem. Two data-driven approaches are compared: multi-variable polynomial regression (MVPR) and a convolutional neural network (CNN). The training data are generated using a finite element (FE) model and superimposed on a noise floor in the interval [20, 60] dB of the weakest amplitude point in the corresponding spectrum. A total of 5000 simulated datasets are generated for training. The results on smoothed test data show that the two models have a comparable mean percentage error norm of 17.8% for the convolutional neural network and 17.3% for multivariable polynomial regression. A further 500 unsmoothed profiles are tested in order to assess the performance of each algorithm on conductivity distributions where the conductivity of each layer is independent of another. The performance of both algorithms is then assessed on reactor-type test data. The results show that the two data-driven algorithms have a comparable performance when estimating the electrical conductivity depth profile of a typical reactor-type distribution, as well as vast deviations. More generally, it is thought that data-driven approaches for depth profiling of some electromagnetic quantity have the potential to be applied to other ill-posed inverse problems where speed is a priority.

Information and estimation in image processing

1987 ◽  
Vol 45 ◽  
pp. 14-17
Author(s):  
B. Roy Frieden
Keyword(s):  
Image Processing ◽  
Optical System ◽  
Large Amplitude ◽  
Communication Theory ◽  
Image Data ◽  
Direct Approach ◽  
Ill Posed

Despite the skill and determination of electro-optical system designers, the images acquired using their best designs often suffer from blur and noise. The aim of an “image enhancer” such as myself is to improve these poor images, usually by digital means, such that they better resemble the true, “optical object,” input to the system. This problem is notoriously “ill-posed,” i.e. any direct approach at inversion of the image data suffers strongly from the presence of even a small amount of noise in the data. In fact, the fluctuations engendered in neighboring output values tend to be strongly negative-correlated, so that the output spatially oscillates up and down, with large amplitude, about the true object. What can be done about this situation? As we shall see, various concepts taken from statistical communication theory have proven to be of real use in attacking this problem. We offer below a brief summary of these concepts.

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