Solving physics-driven inverse problems via structured least squares. Murray-Bruce, J. & Dragotti, P. L. In 2016 24th European Signal Processing Conference (EUSIPCO), pages 331-335, Aug, 2016.
Solving physics-driven inverse problems via structured least squares [pdf]Paper  doi  abstract   bibtex   
Numerous physical phenomena are well modeled by partial differential equations (PDEs); they describe a wide range of phenomena across many application domains, from modeling EEG signals in electroencephalography to, modeling the release and propagation of toxic substances in environmental monitoring. In these applications it is often of interest to find the sources of the resulting phenomena, given some sparse sensor measurements of it. This will be the main task of this work. Specifically, we will show that finding the sources of such PDE-driven fields can be turned into solving a class of well-known multi-dimensional structured least squares problems. This link is achieved by leveraging from recent results in modern sampling theory - in particular, the approximate Strang-Fix theory. Subsequently, numerical simulation results are provided in order to demonstrate the validity and robustness of the proposed framework.
@InProceedings{7760264,
  author = {J. Murray-Bruce and P. L. Dragotti},
  booktitle = {2016 24th European Signal Processing Conference (EUSIPCO)},
  title = {Solving physics-driven inverse problems via structured least squares},
  year = {2016},
  pages = {331-335},
  abstract = {Numerous physical phenomena are well modeled by partial differential equations (PDEs); they describe a wide range of phenomena across many application domains, from modeling EEG signals in electroencephalography to, modeling the release and propagation of toxic substances in environmental monitoring. In these applications it is often of interest to find the sources of the resulting phenomena, given some sparse sensor measurements of it. This will be the main task of this work. Specifically, we will show that finding the sources of such PDE-driven fields can be turned into solving a class of well-known multi-dimensional structured least squares problems. This link is achieved by leveraging from recent results in modern sampling theory - in particular, the approximate Strang-Fix theory. Subsequently, numerical simulation results are provided in order to demonstrate the validity and robustness of the proposed framework.},
  keywords = {electroencephalography;environmental monitoring (geophysics);inverse problems;least squares approximations;partial differential equations;signal sampling;toxicology;physics-driven inverse problem;partial differential equation;PDE;EEG signal modeling;electroencephalography;toxic substance propagation;environmental monitoring;sparse sensor measurement;multidimensional structured least square problem;sampling theory;approximate Strang-Fix theory;numerical simulation;Green's function methods;Sensors;Mathematical model;Inverse problems;Europe;Signal processing;Brain modeling;Spatiotemporal sampling;sensor networks;inverse source problems;structured least squares;Prony's method;finite rate of innovation (FRI)},
  doi = {10.1109/EUSIPCO.2016.7760264},
  issn = {2076-1465},
  month = {Aug},
  url = {https://www.eurasip.org/proceedings/eusipco/eusipco2016/papers/1570255808.pdf},
}
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