每年出版 6 期
ISSN 打印: 2152-5080
ISSN 在线: 2152-5099
Indexed in
WASSERSTEIN METRIC-DRIVEN BAYESIAN INVERSION WITH APPLICATIONS TO SIGNAL PROCESSING
摘要
We present a Bayesian framework based on a new exponential likelihood function driven by the quadratic Wasserstein metric. Compared to conventional Bayesian models based on Gaussian likelihood functions driven by the least-squares norm (L2 norm), the new framework features several advantages. First, the new framework does not rely on the like-lihood of the measurement noise and hence can treat complicated noise structures such as combined additive and multiplicative noise. Second, unlike the normal likelihood function, the Wasserstein-based exponential likelihood function does not usually generate multiple local extrema. As a result, the new framework features better convergence to correct posteriors when a Markov Chain Monte Carlo sampling algorithm is employed. Third, in the particular case of signal processing problems, although a normal likelihood function measures only the amplitude differences between the observed and simulated signals, the new likelihood function can capture both amplitude and phase differences. We apply the new framework to a class of signal processing problems, that is, the inverse uncertainty quantification of waveforms, and demonstrate its advantages compared to Bayesian models with normal likelihood functions.
-
Gelman, A., Carlin, J.B., Stern, H.S., and Rubin, D.B., Bayesian Data Analysis, New York: Chapman and Hall/CRC, 2004.
-
Kaipo, J. and Somersalo, E., Statistical and Computational Inverse Problems, New York: Springer, 2005.
-
Stuart, A.M., Inverse Problems: A Bayesian Perspective, Acta Numer, 19:451-559,2010.
-
Gamerman, D. and Lopes, H.F., Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, New York: Chapman and Hall/CRC, 2006.
-
Villani, C., Topics in Optimal Transportation, Vol. 58, Graduate Studies in Mathematics, Providence, RI: American Mathematical Society, 2003.
-
Villani, C., Optimal Transport: Old and New, Vol. 338, Comprehensive Studies in Mathematics, Berlin: Springer-Verlag, 2009.
-
Engquist,B. andFroese, B.D., Application of the Wasserstein Metric to Seismic Signals, Commun. Math. Sci., 12(5):979-988, 2014.
-
Engquist, B., Brittany, B.F., and Yang, Y., Optimal Transport for Seismic Full Waveform Inversion, Commun. Math. Sci, 14(8):2309-2330,2016.
-
Yang, Y., Engquist, B., Sun, J., and Hamfeldt, B.F., Application of Optimal Transport and the Quadratic Wasserstein Metric to Full-Waveform Inversion, Geophys., 83(1):R43-R62,2018.
-
Engquist, B. and Yang, Y., Seismic Imaging and Optimal Transport, Commun. Inf. Syst., arXiv:1808.04801,2018.
-
Chen, J., Chen, Y., Wu, H., and Yang, D., The Quadratic Wasserstein Metric for Earthquake Location, J. Comput. Phys, 373:188-209,2018.
-
Ballesio, M., Beck, J., Pandey, A., Parisi, L., von Schwerin, E., and Tempone, R., Multilevel Monte Carlo Acceleration of Seismic Wave Propagation under Uncertainty, Numer Anal., arXiv:1810.01710,2018.
-
Metivier, L., Brossier, R., Merigot, Q., Oudet, E., and Virieux, J., Measuring the Misfit between Seismograms Using an Optimal Transport Distance: Application to Full Waveform Inversion, Geophys. J. Int., 205(1):345-377,2016.
-
Metivier, L., Brossier, R., Merigot, Q., Oudet, E., and Virieux, J., An Optimal Transport Approach for Seismic Tomography: Application to 3D Full Waveform Inversion, Inverse Probl, 32(11): 115008,2016.
-
Metivier, L., Allain, A., Brossier, R., Merigot, Q., Oudet, E., and Virieux, J., A Graph-Space Approach to Optimal Transport for Full Waveform Inversion, Soc. Explor. Geophys., pp. 1158-1162,2018.
-
Metivier, L., Allain, A., Brossier, R., Merigot, Q., Oudet, E., and Virieux, J., Optimal Transport for Mitigating Cycle Skipping in Full-Waveform Inversion: A Graph-Space Transform Approach, Geophys., 83(5):R515-R540,2018.
-
Hedjazian, N., Bodin, T., and Metivier, L., An Optimal Transport Approach to Linearized Inversion of Receiver Functions, Geophys. J. Int., 216(1): 130-147,2019.
-
Bayes, T., Price, R., and Canton, J., An Essay towards Solving a Problem in the Doctrine of Chances. By the Late Rev. Mr. Bayes, F.R.S. Communicated by Mr. Price, in a Letter to John Canton, A.M.F.R., Philos. Trans. R S. London, 53:370-418, 1763.
-
Goodman, J., Speckle Phenomena in Optics: Theory and Applications, Englewood, CO: Roberts and Company Publ., 2007.
-
Papadakis, N., Optimal Transport for Image Processing, PhD, Universite de Bordeaux, 2015.
-
Kolouri, S., Zou, Y., and Rohde, G.K., Sliced Wasserstein Kernels for Probability Distributions, Proc. of 2016IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 5258-5267,2016.
-
Esfahani, P.M. and Kuhn, D., Data-Driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations, Math. Prog;, 171:115-166,2017.
-
Benamou, J.D., Froese, B.D., and Oberman, A.M., Numerical Solution of the Optimal Transportation Problem Using the Monge-Ampere Equation, J. Comput. Phys, 260:107-126,2014.
-
Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., and Teller, E., Equation of State Calculations by Fast Computing Machines, J. Chem. Phys, 21:1087-1092,1953.
-
Hastings, W.K., Monte Carlo Sampling Methods Using Markov Chains and Their Applications, Biometrika, 57:97-109,1970.
-
Chib, S. and Greenberg, E., Understanding the Metropolis-Hastings Algorithm, Am. Stat., 49:327-335,1995.
-
Clayton, R. and Engquist, B., Absorbing Boundary Conditions for Acoustic and Elastic Wave Equations, Bull. Seismol. Soc. Am., 67:1529-1540,1977.
-
Nilsson, S., Petersson, N.A., Sjogren, B., and Kreiss, H.O., Stable Difference Approximations for the Elastic Wave Equation in Second Order Formulation, SIAM J. Numer. Anal, 45:1902-1936,2007.
-
Sjogreen, B. and Petersson, N.A., Source Estimation by Full Wave Form Inversion, J. Sci. Comput:., 59:247-276,2014.
-
Long, Q., Motamed, M., and Tempone, R., Fast Bayesian Optimal Experimental Design for Seismic Source Inversion, Comput. Methods Appl. Mech. Eng., 291:123-145,2015.
-
Motamed, M., Nobile, F., and Tempone, R., A Stochastic Collocation Method for the Second Order Wave Equation with a Discontinuous Random Speed, Numer. Math., 123:493-536,2013.
-
Appelo, D., Hagstrom, T., and Vargas, A., Hermite Methods for the Scalar Wave Equation, SIAM J. Sci. Comput, 40(6):A3902-A3927,2018.
-
Blei, D.M., Kucukelbir, A., and McAuliffe, J.D., Variational Inference: A Review for Statisticians, J. Am. Stat. Assoc., 112:859-877,2017.
-
Chu Ning, Hou Yaochun, Wu Dazhuan, Djafari Ali Mohammed, A Variational Bayesian Inference with Small Dataset for High-Precision Infrared Thermal Imaging, 2019 6th International Conference on Systems and Informatics (ICSAI), 2019. Crossref
-
Dunlop Matt, Yang Yunan, New likelihood functions and level-set prior for Bayesian full-waveform inversion, SEG Technical Program Expanded Abstracts 2020, 2020. Crossref
-
Latz Jonas, Madrigal-Cianci Juan P., Nobile Fabio, Tempone Raúl, Generalized parallel tempering on Bayesian inverse problems, Statistics and Computing, 31, 5, 2021. Crossref
-
Dunlop Matthew M., Yang Yunan, Stability of Gibbs Posteriors from the Wasserstein Loss for Bayesian Full Waveform Inversion, SIAM/ASA Journal on Uncertainty Quantification, 9, 4, 2021. Crossref
-
Scarinci Andrea, Marzouk Youssef, Gu Chen, Fehler Michael, Waheed Umair bin, Kaka Sanlinn, Dia Ben M., Transport Lagrangian misfit measures and velocity model uncertainty in Bayesian moment tensor inversion, First International Meeting for Applied Geoscience & Energy Expanded Abstracts, 2021. Crossref
-
Tamang Sagar K., Ebtehaj Ardeshir, Zou Dongmian, Lerman Gilad, Regularized variational data assimilation for bias treatment using the Wasserstein metric, Quarterly Journal of the Royal Meteorological Society, 146, 730, 2020. Crossref
-
Motamed Mohammad, A hierarchically low-rank optimal transport dissimilarity measure for structured data, BIT Numerical Mathematics, 2022. Crossref