Award Abstract # 1845406
CAREER: Fast and Accurate Algorithms for Uncertainty Quantification in Large-Scale Inverse Problems

NSF Org: DMS
Division Of Mathematical Sciences
Recipient: NORTH CAROLINA STATE UNIVERSITY
Initial Amendment Date: January 31, 2019
Latest Amendment Date: July 25, 2023
Award Number: 1845406
Award Instrument: Continuing Grant
Program Manager: Yuliya Gorb
ygorb@nsf.gov
 (703)292-2113
DMS
 Division Of Mathematical Sciences
MPS
 Directorate for Mathematical and Physical Sciences
Start Date: September 1, 2019
End Date: August 31, 2025 (Estimated)
Total Intended Award Amount: $400,000.00
Total Awarded Amount to Date: $400,000.00
Funds Obligated to Date: FY 2019 = $48,820.00
FY 2020 = $98,063.00

FY 2021 = $100,577.00

FY 2022 = $75,367.00

FY 2023 = $77,173.00
History of Investigator:
  • Arvind Saibaba (Principal Investigator)
    asaibab@ncsu.edu
Recipient Sponsored Research Office: North Carolina State University
2601 WOLF VILLAGE WAY
RALEIGH
NC  US  27695-0001
(919)515-2444
Sponsor Congressional District: 02
Primary Place of Performance: North Carolina State University
2311 Stinson Drive 3118
Raleigh
NC  US  27695-8205
Primary Place of Performance
Congressional District:
02
Unique Entity Identifier (UEI): U3NVH931QJJ3
Parent UEI: U3NVH931QJJ3
NSF Program(s): COMPUTATIONAL MATHEMATICS
Primary Program Source: 01001920DB NSF RESEARCH & RELATED ACTIVIT
01002021DB NSF RESEARCH & RELATED ACTIVIT

01002122DB NSF RESEARCH & RELATED ACTIVIT

01002223DB NSF RESEARCH & RELATED ACTIVIT

01002324DB NSF RESEARCH & RELATED ACTIVIT
Program Reference Code(s): 1045, 9263
Program Element Code(s): 127100
Award Agency Code: 4900
Fund Agency Code: 4900
Assistance Listing Number(s): 47.049

ABSTRACT

The need to visualize regions that are impossible to see with the naked eye is pervasive in everyday life. For example, in medicine, accurate visualization of tissue is needed to diagnose and treat tumors. A key step in imaging technologies requires one to solve an inverse problem in order to transform measured data into detailed image reconstructions of the quantities of interest. However, image reconstruction is inherently uncertain, in part, due to noisy measurements from sensors. Ignoring the uncertainty in the imaging process can lead to undesirable outcomes, such as misjudging the location and spread of a suspected tumor. Uncertainty Quantification (UQ) in imaging is in its infancy and hence, the potential for impact in research contributions is high. UQ for imaging is computationally challenging since thousands of inversions are needed beyond the initial inversion to generate accurate statistics of the uncertainty. Current approaches for UQ are inadequate because they either fail to deliver solutions in a reasonable computational time or they lack the applicability across a broad range of imaging technologies.


The project is on the development of fast algorithms for UQ in large-scale inverse problems that are applicable to a broad range of imaging technologies. These algorithms are expected to bring down the computational cost by at least an order of magnitude while maintaining the accuracy of the solutions. More specifically, the project will (1) Advance image reconstruction and UQ techniques for incorporating prior information based on fractional partial differential equation (PDE) and Bayesian level set approaches; and (2) Develop new algorithms and analysis for data-driven dimensionality techniques for UQ in Bayesian inverse problems, using randomized and Krylov subspace methods. The algorithms developed here will be rigorously analyzed and validated on several model problems and applications, including diffuse optical and photoacoustic tomography (in biomedicine) and hydraulic tomography and satellite data fusion (in geoscience). The algorithms developed here are also applicable to other imaging-based inverse problems in biomedicine, geophysics, materials science, etc. Outside of imaging applications, these mathematical advances will be of interest to scientists working in many areas of computational science, for example, fractional partial differential equations (PDEs), model reduction, tensor decompositions, and principal component analysis. Lastly, the PI's education and outreach activities will make UQ and imaging technologies more modular, accessible, and easier to understand for pre-service and early career K-12 educators, undergraduate students, and graduate students. Specifically, the educational program of this project will: (1) Strengthen STEM education through teacher training workshops and practical research experiences for pre-service and early career K-12 teachers, which will result in reproducible teaching modules for use in K-12 education; and (2) Enhance undergraduate and graduate curriculum at North Carolina State University by creating accessible seminar talks and new course content.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH

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Antil, Harbir and Saibaba, Arvind K "Randomized reduced basis methods for parameterized fractional elliptic PDEs" Finite Elements in Analysis and Design , v.227 , 2023 https://doi.org/10.1016/j.finel.2023.104046 Citation Details
Antil, Harbir and Saibaba, Arvind K. "Efficient Algorithms for Bayesian Inverse Problems with WhittleMatérn Priors" SIAM Journal on Scientific Computing , 2023 https://doi.org/10.1137/22M1494397 Citation Details
Hallman, Eric and Ipsen, Ilse C. and Saibaba, Arvind K. "Monte Carlo Methods for Estimating the Diagonal of a Real Symmetric Matrix" SIAM Journal on Matrix Analysis and Applications , v.44 , 2023 https://doi.org/10.1137/22M1476277 Citation Details
Kilmer, Misha E. and Saibaba, Arvind K. "Structured Matrix Approximations via Tensor Decompositions" SIAM Journal on Matrix Analysis and Applications , v.43 , 2022 https://doi.org/10.1137/21M1418290 Citation Details
Majumder, Suman and Guan, Yawen and Reich, Brian J. and Saibaba, Arvind K. "Kryging: geostatistical analysis of large-scale datasets using Krylov subspace methods" Statistics and Computing , v.32 , 2022 https://doi.org/10.1007/s11222-022-10104-3 Citation Details
Pasha, Mirjeta and Saibaba, Arvind K. and Gazzola, Silvia and Español, Malena I. and Sturler, Eric de "A computational framework for edge-preserving regularization in dynamic inverse problems" ETNA - Electronic Transactions on Numerical Analysis , v.58 , 2022 https://doi.org/10.1553/etna_vol58s486 Citation Details
Reese, William and Hart, Joseph and Waanders, Bart_van Bloemen and Perego, Mauro and Jakeman, John D and Saibaba, Arvind K "HYPERDIFFERENTIAL SENSITIVITY ANALYSIS IN THE CONTEXT OF BAYESIAN INFERENCE APPLIED TO ICE-SHEET PROBLEMS" International Journal for Uncertainty Quantification , v.14 , 2024 https://doi.org/10.1615/Int.J.UncertaintyQuantification.2023047605 Citation Details
Saibaba, Arvind K. and Chung, Julianne and Petroske, Katrina "Efficient Krylov subspace methods for uncertainty quantification in large Bayesian linear inverse problems" Numerical Linear Algebra with Applications , v.27 , 2020 https://doi.org/10.1002/nla.2325 Citation Details
Saibaba, Arvind K. and Hart, Joseph and van Bloemen Waanders, Bart "Randomized algorithms for generalized singular value decomposition with application to sensitivity analysis" Numerical Linear Algebra with Applications , v.28 , 2021 https://doi.org/10.1002/nla.2364 Citation Details
Saibaba, Arvind K. and Minster, Rachel and Kilmer, Misha E. "Efficient randomized tensor-based algorithms for function approximation and low-rank kernel interactions" Advances in Computational Mathematics , v.48 , 2022 https://doi.org/10.1007/s10444-022-09979-7 Citation Details

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