An official website of the United States government
Abstract Computational inverse problems utilize a finite number of measurements to infer a discrete approximation of the unknown parameter function. With motivation from the setting of PDE-based optimization, we study the unique reconstruction of discretized inverse problems by examining the positivity of the Hessian matrix. What is the reconstruction power of a fixed number of data observations? How many parameters can one reconstruct? Here we describe a probabilistic approach, and spell out the interplay of the observation size (r) and the number of parameters to be uniquely identified (m). The technical pillar here is the random sketching strategy, in which the matrix concentration inequality and sampling theory are largely employed. By analyzing a randomly subsampled Hessian matrix, we attain a well-conditioned reconstruction problem with high probability. Our main theory is validated in numerical experiments, using an elliptic inverse problem as an example.
more »
« less
Discrete inverse problems with internal functionals *
Abstract We study the problem of finding the resistors in a resistor network from measurements of the power dissipated by the resistors under different loads. We give sufficient conditions for local uniqueness, i.e. conditions that guarantee that the linearization of this non-linear inverse problem admits a unique solution. Our method is inspired by a method to study local uniqueness of inverse problems with internal functionals in the continuum, where the inverse problem is reformulated as a redundant system of differential equations. We use our method to derive local uniqueness conditions for other discrete inverse problems with internal functionals including a discrete analogue of the inverse Schrödinger problem and problems where the resistors are replaced by impedances and dissipated power at the zero and a positive frequency are available. Moreover, we show that the dissipated power measurements can be obtained from measurements of thermal noise induced currents.
more »
« less
- PAR ID:
- 10595482
- Publisher / Repository:
- IOP Publishing
- Date Published:
- Journal Name:
- Inverse Problems
- Volume:
- 41
- Issue:
- 4
- ISSN:
- 0266-5611
- Page Range / eLocation ID:
- 045004
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract In this paper, we introduce a bilevel optimization framework for addressing inverse mean-field games, alongside an exploration of numerical methods tailored for this bilevel problem. The primary benefit of our bilevel formulation lies in maintaining the convexity of the objective function and the linearity of constraints in the forward problem. Our paper focuses on inverse mean-field games characterized by unknown obstacles and metrics. We show numerical stability for these two types of inverse problems. More importantly, we, for the first time, establish the identifiability of the inverse mean-field game with unknown obstacles via the solution of the resultant bilevel problem. The bilevel approach enables us to employ an alternating gradient-based optimization algorithm with a provable convergence guarantee. To validate the effectiveness of our methods in solving the inverse problems, we have designed comprehensive numerical experiments, providing empirical evidence of its efficacy.more » « less
-
Abstract BackgroundComputed tomography (CT) reconstruction problems are always framed as inverse problems, where the attenuation map of an imaged object is reconstructed from the sinogram measurement. In practice, these inverse problems are often ill‐posed, especially under few‐view and limited‐angle conditions, which makes accurate reconstruction challenging. Existing solutions use regularizations such as total variation to steer reconstruction algorithms to the most plausible result. However, most prevalent regularizations rely on the same priors, such as piecewise constant prior, hindering their ability to collaborate effectively and further boost reconstruction precision. PurposeThis study aims to overcome the aforementioned challenge a prior previously limited to discrete tomography. This enables more accurate reconstructions when the proposed method is used in conjunction with most existing regularizations as they utilize different priors. The improvements will be demonstrated through experiments conducted under various conditions. MethodsInspired by the discrete algebraic reconstruction technique (DART) algorithm for discrete tomography, we find out that pixel grayscale values in CT images are not uniformly distributed and are actually highly clustered. Such discovery can be utilized as a powerful prior for CT reconstruction. In this paper, we leverage the collaborative filtering technique to enable the collaboration of the proposed prior and most existing regularizations, significantly enhancing the reconstruction accuracy. ResultsOur experiments show that the proposed method can work with most existing regularizations and significantly improve the reconstruction quality. Such improvement is most pronounced under limited‐angle and few‐view conditions. Furthermore, the proposed regularization also has the potential for further improvement and can be utilized in other image reconstruction areas. ConclusionsWe propose improving the performance of iterative CT reconstruction algorithms by applying the collaborative filtering technique along with a prior based on the densely clustered distribution of pixel grayscale values in CT images. Our experimental results indicate that the proposed methodology consistently enhances reconstruction accuracy when used in conjunction with most existing regularizations, particularly under few‐view and limited‐angle conditions.more » « less
-
The majority of iterative algorithms for CT reconstruction rely on discrete-to-discrete modeling, where both the sinogram measurements and image to be estimated are discrete arrays. However, tomographic projections are ideally modeled as line integrals of a continuous attenuation function, i.e., the true inverse problem is discrete-to-continuous in nature. Recently, coordinate-based neural networks (CBNNs), also known as implicit neural representations, have gained traction as a flexible type of continuous domain image representation in a variety of inverse problems arising in computer vision and computational imaging. Using standard neural network training techniques, a CBNN can be fit to measurements to give a continuous domain estimate of the image. In this study, we empirically investigate the potential of CBNNs to solve the continuous domain inverse problems in CT imaging. In particular, we experiment with reconstructing an analytical phantom from its ideal sparse-view sinogram measurements. Our results illustrate that reconstruction with a CBNN are more accurate than filtered back projection and algebraic reconstruction techniques at a variety of resolutions, and competitive with total variation regularized iterative reconstruction.more » « less
-
Abstract We present an approach to shape optimization problems that uses an unfitted finite element method (FEM). The domain geometry is represented, and optimized, using a (discrete) level set function and we consider objective functionals that are defined over bulk domains. For a discrete objective functional, defined in the unfitted FEM framework, we show that theexactdiscrete shape derivative essentially matches the shape derivative at the continuous level. In other words, our approach has the benefits of both optimize-then-discretize and discretize-then-optimize approaches. Specifically, we establish the shape Fréchet differentiability of discrete (unfitted) bulk shape functionals using both the perturbation of the identity approach and direct perturbation of the level set representation. The latter approach is especially convenient for optimizing with respect to level set functions. Moreover, our Fréchet differentiability results hold foranypolynomial degree used for the discrete level set representation of the domain. We illustrate our results with some numerical accuracy tests, a simple model (geometric) problem with known exact solution, as well as shape optimization of structural designs.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript
- Journal Article:
- https://doi.org/10.1088/1361-6420/adbd69
