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Abstract In linear regression, the coefficients are simple to estimate using the least squares method with a known design matrix for the observed measurements. However, real-world applications may encounter complications such as an unknown design matrix and complex-valued parameters. The design matrix can be estimated from prior information but can potentially cause an inverse problem when multiplying by the transpose as it is generally ill-conditioned. This can be combat by adding regularizers to the model but does not always mitigate the issues. Here, we propose our Bayesian approach to a complex-valued latent variable linear model with an application to functional magnetic resonance imaging (fMRI) image reconstruction. The complex-valued linear model and our Bayesian model are evaluated through extensive simulations and applied to experimental fMRI data.
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Recovery of a coefficient in a diffusion equation from large time data
This paper considers the determination of a spatially varying coefficient in a parabolic equation from time trace data. There are many uniqueness theorems known for such problems but the reconstruction step is severally ill-posed: essentially the problem comes down to trying to reconstruct an analytic function from values on a strip. However, we look at an even more restricted data where the measurements are not made on the whole time axis but only for large values adding further to the ill-conditioning situation. In addition, we do not assume the initial state is known. Uniqueness is restored by making changes to the boundary condition, in particular, to the impedance parameter, for each of a series of measurements. We show that an implementation of the above paradigm leads to both uniqueness and an effective reconstruction algorithm. Extension is also made to the case of fractional model and to replacing the parabolic equation with a damped wave equation.
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- Award ID(s):
- 2111020
- PAR ID:
- 10594785
- Publisher / Repository:
- Institute for Physics IOP
- Date Published:
- Journal Name:
- Inverse problems
- Edition / Version:
- 2.0
- Volume:
- 40
- Issue:
- 12
- ISSN:
- 0266-5611
- Page Range / eLocation ID:
- 10.1088/1361-6420/ad904c
- Subject(s) / Keyword(s):
- Inverse problems, coefficient recovery, parabolic, subdiffusion and wave equations.
- Format(s):
- Medium: X Size: 1.74KB Other: .pdf
- Size(s):
- 1.74KB
- Sponsoring Org:
- National Science Foundation
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