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 Reconstruction of basal properties in ice sheets using iterative inverse methods

2012 ◽  
Vol 58 (210) ◽  
pp. 795-808 ◽  
Author(s):  
Marijke Habermann ◽  
David Maxwell ◽  
Martin Truffer
Keyword(s):  
Inverse Problems ◽  
Input Data ◽  
Large Scale ◽  
Gradient Methods ◽  
Inverse Methods ◽  
Model Parameters ◽  
Discrepancy Principle ◽  
Regularization Techniques ◽  
Ill Posed ◽  
Nonlinear Conjugate Gradient Methods

AbstractInverse problems are used to estimate model parameters from observations. Many inverse problems are ill-posed because they lack stability, meaning it is not possible to find solutions that are stable with respect to small changes in input data. Regularization techniques are necessary to stabilize the problem. For nonlinear inverse problems, iterative inverse methods can be used as a regularization method. These methods start with an initial estimate of the model parameters, update the parameters to match observation in an iterative process that adjusts large-scale spatial features first, and use a stopping criterion to prevent the overfitting of data. This criterion determines the smoothness of the solution and thus the degree of regularization. Here, iterative inverse methods are implemented for the specific problem of reconstructing basal stickiness of an ice sheet by using the shallow-shelf approximation as a forward model and synthetically derived surface velocities as input data. The incomplete Gauss-Newton (IGN) method is introduced and compared to the commonly used steepest descent and nonlinear conjugate gradient methods. Two different stopping criteria, the discrepancy principle and a recent- improvement threshold, are compared. The IGN method is favored because it is rapidly converging, and it incorporates the discrepancy principle, which leads to optimally resolved solutions.

Lagrange principle and its regularization as a theoretical basis of stable solving optimal control and inverse problems

2021 ◽  
pp. 151-171
Author(s):  
Mikhail I. Sumin
Keyword(s):  
Optimal Control ◽  
Inverse Problems ◽  
Theoretical Basis ◽  
Optimization Problems ◽  
Existence Theorems ◽  
General Structure ◽  
Approximate Solutions ◽  
Lagrange Principle ◽  
Ill Posed ◽  
Final Observation

The paper is devoted to the regularization of the classical optimality conditions (COC) — the Lagrange principle and the Pontryagin maximum principle in a convex optimal control problem for a parabolic equation with an operator (pointwise state) equality-constraint at the final time. The problem contains distributed, initial and boundary controls, and the set of its admissible controls is not assumed to be bounded. In the case of a specific form of the quadratic quality functional, it is natural to interpret the problem as the inverse problem of the final observation to find the perturbing effect that caused this observation. The main purpose of regularized COCs is stable generation of minimizing approximate solutions (MAS) in the sense of J. Warga. Regularized COCs are: 1) formulated as existence theorems of the MASs in the original problem with a simultaneous constructive representation of specific MASs; 2) expressed in terms of regular classical Lagrange and Hamilton–Pontryagin functions; 3) are sequential generalizations of the COCs and retain the general structure of the latter; 4) “overcome” the ill-posedness of the COCs, are regularizing algorithms for solving optimization problems, and form the theoretical basis for the stable solving modern meaningful ill-posed optimization and inverse problems.

Trefftz Functions Applied to Direct and Inverse Non-Fourier Heat Conduction Problems

2014 ◽  
Vol 136 (9) ◽  
Author(s):  
Krzysztof Grysa ◽  
Artur Maciag ◽  
Justyna Adamczyk-Krasa
Keyword(s):  
Boundary Condition ◽  
Heat Conduction ◽  
Inverse Problems ◽  
Approximate Method ◽  
Heat Conduction Equation ◽  
Input Data ◽  
Direct And Inverse Problems ◽  
Ill Posed ◽  
Fourier Heat Conduction ◽  
Boundary Inverse Problems

The paper presents a new approximate method of solving non-Fourier heat conduction problems. The approach described here is suitable for solving both direct and inverse problems. The way of generating Trefftz functions for non-Fourier heat conduction equation has been shown. Obtained functions have been used for solving direct and boundary inverse problems (identification of boundary condition). As a rule, inverse problems are ill-posed. Therefore, each method of solving these problems has to be checked according to disturbance of the input data. Presented examples confirm high usability of the presented approach for solving direct and inverse non-Fourier heat conduction problems.

scholarly journals Shape Optimization by Pursuing Diffeomorphisms

2015 ◽  
Vol 15 (3) ◽  
pp. 291-305 ◽  
Author(s):  
Ralf Hiptmair ◽  
Alberto Paganini
Keyword(s):  
Finite Element ◽  
Shape Optimization ◽  
Optimization Problems ◽  
Approximation Properties ◽  
The Finite Element Method ◽  
Finite Element Discretization ◽  
Element Discretization ◽  
B Spline ◽  
Ill Posed ◽  
Well Posed

AbstractWe consider PDE constrained shape optimization in the framework of finite element discretization of the underlying boundary value problem. We present an algorithm tailored to preserve and exploit the approximation properties of the finite element method, and that allows for arbitrarily high resolution of shapes. It employs (i) B-spline based representations of the deformation diffeomorphism, and (ii) superconvergent domain integral expressions for the shape gradient. We provide numerical evidence of the performance of this method both on prototypical well-posed and ill-posed shape optimization problems.

Inverse Problems in Heat Transfer: New Trends on Solution Methodologies and Applications

2010 ◽  
Author(s):  
Helcio R. B. Orlande
Keyword(s):  
Inverse Problems ◽  
Research Work ◽  
Practical Interest ◽  
Physical Problem ◽  
Analysis And Design ◽  
Oil Pipelines ◽  
Regularization Techniques ◽  
Ill Posed ◽  
Well Posed

Systematic methods for the solution of inverse problems have developed significantly during the last twenty years and have become a powerful tool for analysis and design in engineering. Inverse analysis is nowadays a common practice in which the groups involved with experiments and numerical simulation synergistically collaborate throughout the research work, in order to obtain the maximum of information regarding the physical problem under study. Inverse problems are mathematically classified as ill-posed, that is, their solutions do not satisfy either one of the requirements of existence, uniqueness or stability. The solution approaches generally consist of the reformulation of the inverse problem in terms of an approximate well-posed problem. In this paper we briefly review various approaches for the solution of inverse problems, including those based on classical regularization techniques and those based on the Bayesian statistics. Applications of inverse problems are then presented for cases of practical interest, such as the characterization of non-homogeneous materials and the prediction of the temperature field in oil pipelines.

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.

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.

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