Department of Mathematical Sciences
New Jersey Institute of Technology
office: CULM 518
email: christin [at] njit [dot] edu
Multiscale Modeling and Inverse Problems
My research focuses on inverse problems for multiscale partial differential equations (PDEs) in which solution data is used to determine coefficients in the equation. PDE-constrained inverse problems can pose a huge computational challenge, in particular when the coefficients are of multiscale form. The mathematical analysis is based on homogenization theory for PDEs and applied harmonic analysis in the form of information theory. The numerical analysis involves the design of algorithms that incorporate techniques from multiscale computation to forward modeling. This research can be applied to microstructure recovery problems arising in exploration seismology, medical imaging, and underwater acoustics.
- C. Frederick and B. Engquist, Numerical methods for multiscale inverse problems, Communications in Mathematical Sciences 15: 305-328, 2017.
- C. Frederick, K. Ren, S. Vallelian, Image reconstruction in quantitative PAT with the simplified P2 approximation, SIAM J. Imaging Sci., 11(4), 28472876., 2018.
- B. Engquist, C. Frederick, Q. Huynh, H-M. Zhou, Seabed identification in sonar imaging via Helmholtz simulations and discrete optimization, J. Comput. Physics 338: 477-492, 2017.
- C. Frederick, S. Villar, Z-H.
classification using physics-based modeling and machine learning, The
Journal of the Acoustical Society of America 148, 859 (2020).
Sampling Theory and Riesz bases of exponentials
These studies establish the close relation between computational grids in multiscale modeling and sampling strategies developed in information theory. We consider multiscale functions of the type that are studied in averaging and homogenization theory and in multiscale modeling. Using results regarding inverse Vandermonde matrices, we prove that under certain band limiting conditions these multiscale functions can be uniquely and stably recovered from non-uniform samples of optimal rate. It is also interesting to extend this study to multivariate bandlimited functions and higher dimensional sampling lattices.
- B. Engquist and C. Frederick, Nonuniform sampling and multiscale computation, Multiscale Modeling and Simulation, 12-4, pp. 1890-1901, 2014.
- C. Frederick, An L2−stability estimate for periodic nonuniform sampling in higher dimensions, Linear Algebra and its Applications, 555: 361-372, 2018.
- C. Frederick and K. Okoudjou.
Duality for Riesz Bases of Exponentials on Multi-tiles. Appl. Comput. Harmon. Anal., vol. 51, pp. 104-117, 2021.
- C. Frederick and K. Yacoubou Djima.
Explicit Constructions of Duals for
Riesz Bases of Exponentials via Sampling. 2020
- C. Frederick and A. Mayeli.
spectral pairs and exponential bases. 2020
Applications in Engineering and Science