×

Multimodal, high-dimensional, model-based, Bayesian inverse problems with applications in biomechanics. (English) Zbl 1407.62076

Summary: This paper is concerned with the numerical solution of model-based, Bayesian inverse problems. We are particularly interested in cases where the cost of each likelihood evaluation (forward-model call) is expensive and the number of unknown (latent) variables is high. This is the setting in many problems in computational physics where forward models with nonlinear PDEs are used and the parameters to be calibrated involve spatio-temporarily varying coefficients, which upon discretization give rise to a high-dimensional vector of unknowns. One of the consequences of the well-documented ill-posedness of inverse problems is the possibility of multiple solutions. While such information is contained in the posterior density in Bayesian formulations, the discovery of a single mode, let alone multiple, poses a formidable computational task. The goal of the present paper is two-fold. On one hand, we propose approximate, adaptive inference strategies using mixture densities to capture multi-modal posteriors. On the other, we extend our work in [“Sparse variational Bayesian approximations for nonlinear inverse problems: applications in nonlinear elastography”, Comput. Methods Appl. Mech. Eng. 299, 215–244 (2016; doi:10.1016/j.cma.2015.10.015)] with regard to effective dimensionality reduction techniques that reveal low-dimensional subspaces where the posterior variance is mostly concentrated. We validate the proposed model by employing importance sampling which confirms that the bias introduced is small and can be efficiently corrected if the analyst wishes to do so. We demonstrate the performance of the proposed strategy in nonlinear elastography where the identification of the mechanical properties of biological materials can inform non-invasive, medical diagnosis. The discovery of multiple modes (solutions) in such problems is critical in achieving the diagnostic objectives.

MSC:

62F15 Bayesian inference
65C05 Monte Carlo methods
92C10 Biomechanics

Software:

PMTK; PRMLT; BayesDA
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Franck, I. M.; Koutsourelakis, P. S., Sparse variational Bayesian approximations for nonlinear inverse problems: applications in nonlinear elastography, Comput. Methods Appl. Mech. Eng., 299, 1511-1517 (2016) · Zbl 1425.65132
[2] Biegler, L.; Biros, G.; Ghattas, O.; Heinkenschloss, M.; Keyes, D.; Mallick, B.; Tenorio, L.; van Bloemen Waanders, B.; Willcox, K.; Marzouk, Y., Large-Scale Inverse Problems and Quantification of Uncertainty, vol. 712 (2011), John Wiley & Sons · Zbl 1203.62002
[3] Gelman, A.; Carlin, J. B.; Stern, H. S.; Rubin, D. B., Bayesian Data Analysis, vol. 2 (2003), Chapman & Hall/CRC: Chapman & Hall/CRC Boca Raton, FL, USA
[4] Oberai, A. A.; Gokhale, N. H.; Goenezen, S.; Barbone, P. E.; Hall, T. J.; Sommer, A. M.; Jiang, J., Linear and nonlinear elasticity imaging of soft tissue in vivo: demonstration of feasibility, Phys. Med. Biol., 54, 5, 1191 (2009)
[5] Ganne-Carrié, N.; Ziol, M.; de Ledinghen, V.; Douvin, C.; Marcellin, P.; Castera, L.; Dhumeaux, D.; Trinchet, J.-C.; Beaugrand, M., Accuracy of liver stiffness measurement for the diagnosis of cirrhosis in patients with chronic liver diseases, Hepatology, 44, 6, 1511-1517 (2006)
[6] Curtis, C.; Shah, S. P.; Chin, S.-F.; Turashvili, G.; Rueda, O. M.; Dunning, M. J.; Speed, D.; Lynch, A. G.; Samarajiwa, S.; Yuan, Y., The genomic and transcriptomic architecture of 2,000 breast tumours reveals novel subgroups, Nature, 486, 7403, 346-352 (2012)
[7] Doyley, M. M., Model-based elastography: a survey of approaches to the inverse elasticity problem, Phys. Med. Biol., 57, 3, Article R35 pp. (2012)
[8] Ophir, J.; Cespedes, I.; Ponnekanti, H.; Yazdi, Y.; Li, X., Elastography: a quantitative method for imaging the elasticity of biological tissues, Ultrason. Imag., 13, 2, 111-134 (1991)
[9] Muthupillai, R.; Lomas, D. J.; Rossman, P. J.; Greenleaf, J. F., Magnetic resonance elastography by direct visualization of propagating acoustic strain waves, Science, 269, 5232, 1854 (1995)
[10] Khalil, A. S.; Chan, R. C.; Chau, A. H.; Bouma, B. E.; Mofrad, M. R.K., Tissue elasticity estimation with optical coherence elastography: toward mechanical characterization of in vivo soft tissue, Ann. Biomed. Eng., 33, 11, 1631-1639 (2005)
[11] (Sarvazyan, A.; Hall, T., Elasticity Imaging, Part I & II. Elasticity Imaging, Part I & II, Curr. Med. Imag. Rev., vols. 7, 8 (2011))
[12] Doyley, M. M.; Parker, K. J., Elastography: general principles and clinical applications, Ultrasound Clin., 9, 1, 1-11 (2014)
[13] Garra, B. S.; Cespedes, E. I.; Ophir, J.; Spratt, S. R.; Zuurbier, R. A.; Magnant, C. M.; Pennanen, M. F., Elastography of breast lesions: initial clinical results, Radiology, 202, 1, 79-86 (1997)
[14] Bamber, J. C.; Barbone, P. E.; Cosgrove, D. O.; Doyely, M. M.; Fuechsel, F. G.; Meaney, P. M.; Miller, N. R.; Shiina, T.; Tranquart, F., Progress in freehand elastography of the breast, IEICE Trans. Inf. Syst., 85, 1, 5-14 (2002)
[15] Thomas, A.; Fischer, T.; Frey, H.; Ohlinger, R.; Grunwald, S.; Blohmer, J.-U.; Winzer, K.-J.; Weber, S.; Kristiansen, G.; Ebert, B., Real-time elastography—an advanced method of ultrasound: first results in 108 patients with breast lesions, Ultrasound Obstet. Gynecol., 28, 3, 335-340 (2006)
[16] Parker, K. J.; Doyley, M. M.; Rubens, D. J., Imaging the elastic properties of tissue: the 20 year perspective, Phys. Med. Biol., 56, 1, Article R1 pp. (2011)
[17] Krouskop, T. A.; Wheeler, T. M.; Kallel, F., Elastic moduli of breast and prostate tissues under compression, Ultrason. Imag., 260-274 (1998)
[18] Hoyt, K.; Castaneda, B.; Zhang, M.; Nigwekar, P.; di Sant’Agnese, P. A.; Joseph, J. V.; Strang, J.; Rubens, D. J.; Parker, K. J., Tissue elasticity properties as biomarkers for prostate cancer, Cancer Biomark., 4, 4/5, 213-225 (2008)
[19] Asbach, P.; Klatt, D.; Hamhaber, U.; Braun, J.; Somasundaram, R.; Hamm, B.; Sack, I., Assessment of liver viscoelasticity using multifrequency MR elastography, Magn. Reson. Med., 60, 2, 373-379 (2008)
[20] Schmitt, C.; Soulez, G.; Maurice, R. L.; Giroux, M.-F.; Cloutier, G., Noninvasive vascular elastography: toward a complementary characterization tool of atherosclerosis in carotid arteries, Ultrasound Med. Biol., 33, 12, 1841-1858 (2007)
[21] Hamhaber, U.; Klatt, D.; Papazoglou, S.; Hollmann, M.; Stadler, J.; Sack, I.; Bernarding, J.; Braun, J., In vivo magnetic resonance elastography of human brain at 7 T and 1.5 T, J. Magn. Reson. Imaging, 32, 3, 577-583 (2010)
[22] Ohayon, J.; Finet, G.; Le Floch, S.; Cloutier, G.; Gharib, A. M.; Heroux, J.; Pettigrew, R. I., Biomechanics of atherosclerotic coronary plaque: site, stability and in vivo elasticity modeling, Ann. Biomed. Eng., 42, 2, 269-279 (2014)
[23] Shore, S. W.; Barbone, P. E.; Oberai, A. A.; Morgan, E. F., Transversely isotropic elasticity imaging of cancellous bone, J. Biomech. Eng., 133, 6, Article 061002 pp. (2011)
[24] Barbone, P. E.; Rivas, C. E.; Harari, I.; Albocher, U.; Oberai, A. A.; Zhang, Y., Adjoint-weighted variational formulation for the direct solution of inverse problems of general linear elasticity with full interior data, Int. J. Numer. Methods Eng., 81, 13, 1713-1736 (2010) · Zbl 1183.74091
[25] Oberai, A. A.; Gokhale, N. H.; Doyley, M. M.; Bamber, J. C., Evaluation of the adjoint equation based algorithm for elasticity imaging, Phys. Med. Biol., 49, 13, 2955 (2004)
[26] Doyley, M. M.; Srinivasan, S.; Dimidenko, E.; Soni, N.; Ophir, J., Enhancing the performance of model-based elastography by incorporating additional a priori information in the modulus image reconstruction process, Phys. Med. Biol., 51, 1, 95 (2006)
[27] Arnold, A.; Reichling, S.; Bruhns, O. T.; Mosler, J., Efficient computation of the elastography inverse problem by combining variational mesh adaption and a clustering technique, Phys. Med. Biol., 55, 7, 2035 (2010)
[28] Olson, L. G.; Throne, R. D., Numerical simulation of an inverse method for tumour size and location estimation, Inverse Probl. Sci. Eng., 18, 6, 813-834 (2010) · Zbl 1196.92023
[29] Wang, J.; Zabaras, N., A Bayesian inference approach to the inverse heat conduction problem, Int. J. Heat Mass Transf., 47, 17, 3927-3941 (2004) · Zbl 1070.80002
[30] Wang, J.; Zabaras, N., Hierarchical Bayesian models for inverse problems in heat conduction, Inverse Probl., 21, 1, 183 (2005) · Zbl 1060.62036
[31] Dostert, P.; Efendiev, Y.; Hou, T. Y.; Luo, W., Coarse-gradient Langevin algorithms for dynamic data integration and uncertainty quantification, J. Comput. Phys., 217, 1, 123-142 (2006) · Zbl 1146.76637
[32] Mallat, S., A Wavelet Tour of Signal Processing: The Sparse Way (2008), Academic Press
[33] Koutsourelakis, P.-S., A multi-resolution, non-parametric, Bayesian framework for identification of spatially-varying model parameters, J. Comput. Phys., 228, 17, 6184-6211 (2009) · Zbl 1190.62211
[34] Green, P. J.; , K.; Pereyra, M.; Robert, C. P., Bayesian computation: a summary of the current state, and samples backwards and forwards, Stat. Comput., 25, 4, 835-862 (2015) · Zbl 1331.62017
[35] Roberts, G. O.; Tweedie, R. L., Exponential convergence of Langevin distributions and their discrete approximations, Bernoulli, 341-363 (1996) · Zbl 0870.60027
[36] Roberts, G. O.; Rosenthal, J. S., Optimal scaling for various Metropolis-Hastings algorithms, Stat. Sci., 16, 4, 351-367 (1998) · Zbl 1127.65305
[37] Mattingly, J. C.; Pillai, N. S.; Stuart, A. M., Diffusion limits of the random walk Metropolis algorithm in high dimensions, Ann. Appl. Probab., 22, 3, 881-930 (2012) · Zbl 1254.60081
[38] Pillai, N. S.; Stuart, A. M.; Thiéry, A. H., Optimal scaling and diffusion limits for the Langevin algorithm in high dimensions, Ann. Appl. Probab., 22, 6, 2320-2356 (2012) · Zbl 1272.60053
[39] Lee, H. K.; Higdon, D. M.; Bi, Z.; Ferreira, M. A.; West, M., Markov random field models for high-dimensional parameters in simulations of fluid flow in porous media, Technometrics, 44, 3, 230-241 (2002)
[40] Holloman, C. H.; Lee, H. K.; Higdon, D. M., Multi-resolution Genetic Algorithms and Markov Chain Monte Carlo (2002), Duke University, Technical report
[41] Chopin, N.; Lelièvre, T.; Stoltz, G., Free energy methods for Bayesian inference: efficient exploration of univariate Gaussian mixture posteriors, Stat. Comput., 22, 4, 897-916 (2012) · Zbl 1252.62015
[42] Del Moral, P.; Doucet, A.; Jasra, A., Sequential Monte Carlo samplers, J. R. Stat. Soc., Ser. B, Stat. Methodol., 68, 3, 411-436 (2006) · Zbl 1105.62034
[43] Del Moral, P.; Doucet, A.; Jasra, A., On adaptive resampling strategies for sequential Monte Carlo methods, Bernoulli, 18, 1, 252-278 (2012) · Zbl 1236.60072
[44] Cui, T.; Martin, J.; Marzouk, Y. M.; Solonen, A.; Spantini, A., Likelihood-informed dimension reduction for nonlinear inverse problems, Inverse Probl., 30, 11, Article 114015 pp. (2014) · Zbl 1310.62030
[45] Spantini, A.; Solonen, A.; Cui, T.; Martin, J.; Tenorio, L.; Marzouk, Y., Optimal low-rank approximations of Bayesian linear inverse problems, SIAM J. Sci. Comput., 37, 6, A2451-A2487 (2015) · Zbl 1325.62060
[46] Cui, T.; Marzouk, Y.; Willcox, K., Scalable posterior approximations for large-scale Bayesian inverse problems via likelihood-informed parameter and state reduction, J. Comput. Phys., 315, 363-387 (2016) · Zbl 1349.65189
[47] Cui, T.; Law, K. J.; Marzouk, Y. M., Dimension-independent likelihood-informed MCMC, J. Comput. Phys., 304, 109-137 (2016) · Zbl 1349.65009
[48] Chatelin, S.; Bernal, M.; Deffieux, T.; Papadacci, C.; Flaud, P.; Nahas, A.; Boccara, C.; Gennisson, J.-L.; Tanter, M.; Pernot, M., Anisotropic polyvinyl alcohol hydrogel phantom for shear wave elastography in fibrous biological soft tissue: a multimodality characterization, Phys. Med. Biol., 59, 22, 6923 (2014)
[49] Fang, Q.; Moore, R. H.; Kopans, D. B.; Boas, D. A., Compositional-prior-guided image reconstruction algorithm for multi-modality imaging, Biomed. Opt. Express, 1, 1, 223-235 (2010)
[50] Fromageau, J.; Gennisson, J.-L.; Schmitt, C.; Maurice, R. L.; Mongrain, R.; Cloutier, G., Estimation of polyvinyl alcohol cryogel mechanical properties with four ultrasound elastography methods and comparison with gold standard testings, IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 54, 3, 498-509 (2007)
[51] Gill, J.; Casella, G., Dynamic tempered transitions for exploring multimodal posterior distributions, Polit. Anal., 12, 4, 425-443 (2004)
[52] Li, W.; Lin, G., An adaptive importance sampling algorithm for Bayesian inversion with multimodal distributions, J. Comput. Phys., 294, 173-190 (2015) · Zbl 1349.62025
[53] Feroz, F.; Hobson, M. P., Multimodal nested sampling: an efficient and robust alternative to Markov Chain Monte Carlo methods for astronomical data analyses, Mon. Not. R. Astron. Soc., 384, 2, 449-463 (2008)
[54] Tsilifis, P.; Bilionis, I.; Katsounaros, I.; Zabaras, N., Computationally efficient variational approximations for Bayesian inverse problems, J. Verif. Valid. Uncertain. Quantificat., 1, 3, Article 031004 pp. (2016)
[55] Reynolds, D. A.; Quatieri, T. F.; Dunn, R. B., Speaker verification using adapted Gaussian mixture models, Digit. Signal Process., 10, 1-3, 19-41 (2000)
[56] Jain, A. K.; Murty, M. N.; Flynn, P. J., Data clustering: a review, ACM Comput. Surv., 31, 3, 264-323 (1999)
[57] Choudrey, R. A.; Roberts, S. J., Variational mixture of Bayesian independent component analyzers, Neural Comput., 15, 1, 213-252 (2003), wOS:000180281100009 · Zbl 1031.68107
[58] Beal, M. J., Variational Algorithms for Approximate Bayesian Inference (2003), University of London
[59] Kuusela, M.; Raiko, T.; Honkela, A.; Karhunen, J., A gradient-based algorithm competitive with variational Bayesian EM for mixture of Gaussians, (2009 International Joint Conference on Neural Networks (2009), IEEE), 1688-1695
[60] Jin, B., A variational Bayesian method to inverse problems with impulsive noise, J. Comput. Phys., 231, 2, 423-435 (2012) · Zbl 1243.65115
[61] Kennedy, M. C.; O’Hagan, A., Bayesian calibration of computer models, J. R. Stat. Soc., Ser. B, Stat. Methodol., 63, 3, 425-464 (2001) · Zbl 1007.62021
[62] Higdon, D.; Gattiker, J.; Williams, B.; Rightley, M., Computer model calibration using high-dimensional output, J. Am. Stat. Assoc., 103, 482, 570-583 (2008) · Zbl 1469.62414
[63] Bayarri, M. J.; Berger, J. O.; Paulo, R.; Sacks, J.; Cafeo, J. A.; Cavendish, J.; Lin, C.-H.; Tu, J., A framework for validation of computer models, Technometrics, 49, 2, 138-154 (2007)
[64] Berliner, L. M.; Jezek, K.; Cressie, N.; Kim, Y.; Lam, C. Q.; van der Veen, C. J., Modeling dynamic controls on ice streams: a Bayesian statistical approach, J. Glaciol., 54, 187, 705-714 (2008)
[65] Koutsourelakis, P.-S., A novel Bayesian strategy for the identification of spatially varying material properties and model validation: an application to static elastography, Int. J. Numer. Methods Eng., 91, 3, 249-268 (2012) · Zbl 1246.74021
[66] Strong, M.; Oakley, J. E., When is a model good enough? Deriving the expected value of model improvement via specifying internal model discrepancies, SIAM/ASA J. Uncertain. Quantificat., 2, 1, 106-125 (2014) · Zbl 1351.60133
[67] Sargsyan, K.; Najm, H. N.; Ghanem, R., On the statistical calibration of physical models, Int. J. Chem. Kinet., 47, 4, 246-276 (2015)
[68] Murphy, K. P., Machine Learning: A Probabilistic Perspective (2012), MIT Press · Zbl 1295.68003
[69] Antoniak, C. E., Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems, Ann. Stat., 1152-1174 (1974) · Zbl 0335.60034
[70] MacEachern, S. N.; Müller, P., Estimating mixture of Dirichlet process models, J. Comput. Graph. Stat., 7, 2, 223-238 (1998)
[71] Wellman, P.; Howe, R. D.; Dalton, E.; Kern, K. A., Breast Tissue Stiffness in Compression Is Correlated to Histological Diagnosis (1999), Harvard BioRobotics Laboratory, Technical Report
[72] Calvetti, D.; Somersalo, E., Hypermodels in the Bayesian imaging framework, Inverse Probl., 24, 3, Article 034013 pp. (2008) · Zbl 1137.62062
[73] Neal, R. M.; Hinton, G. E., A view of the EM algorithm that justifies incremental, sparse, and other variants, (Learning in Graphical Models (1998), Springer), 355-368 · Zbl 0916.62019
[74] Titsias, M.; Lázaro-Gredilla, M., Doubly stochastic variational Bayes for non-conjugate inference, (Proceedings of the 31st International Conference on Machine Learning. Proceedings of the 31st International Conference on Machine Learning, ICML-14 (2014)), 1971-1979
[75] Peierls, R., On a minimum property of the free energy, Phys. Rev., 54, 11, 918-919 (1938) · Zbl 0020.08403
[76] Opper, M.; Saad, D., Advanced Mean Field Methods: Theory and Practice (2001), MIT Press · Zbl 0994.68172
[77] Papadimitriou, D. I.; Giannakoglou, K. C., Direct, adjoint and mixed approaches for the computation of Hessian in airfoil design problems, Int. J. Numer. Methods Fluids, 56, 10, 1929-1943 (2008) · Zbl 1141.76058
[78] Wen, Z.; Yin, W., A feasible method for optimization with orthogonality constraints, Math. Program., 142, 1-2, 397-434 (2013), wOS:000330038100014 · Zbl 1281.49030
[79] Bishop, C. M., Pattern Recognition and Machine Learning (2006), Springer · Zbl 1107.68072
[80] Robert, C., The Bayesian Choice: From Decision-Theoretic Foundations to Computational Implementation (2007), Springer Science & Business Media · Zbl 1129.62003
[81] Liu, J. S., Monte Carlo Strategies in Scientific Computing (2001), Springer Science & Business Media · Zbl 0991.65001
[82] Holzapfel, G. A., Nonlinear Solid Mechanics, vol. 24 (2000), Wiley: Wiley Chichester · Zbl 0980.74001
[83] Gladilin, E.; Eils, R., Nonlinear elastic model for image registration and soft tissue simulation based on piecewise St. Venant-Kirchhoff material approximation, Proc. SPIE, 6914, Article 69142O pp. (2008)
[84] Yanovsky, I.; Le Guyader, C.; Leow, A.; Toga, A.; Thompson, P.; Vese, L., Unbiased volumetric registration via nonlinear elastic regularization, (2nd MICCAI Workshop on Mathematical Foundations of Computational Anatomy (2008))
[85] Liu, T.; Babaniyi, O. A.; Hall, T. J.; Barbone, P. E.; Oberai, A. A., Noninvasive in-vivo quantification of mechanical heterogeneity of invasive breast carcinomas, PLoS ONE, 10, 7, Article e0130258 pp. (2015)
[86] Dobrescu, R., Diagnosis of breast cancer from mammograms by using fractal measures, Int. J. Med. Imag., 1, 2, 32 (2013)
[87] Rangayyan, R. M.; El-Faramawy, N. M.; Desautels, J. L.; Alim, O. A., Measures of acutance and shape for classification of breast tumors, IEEE Trans. Med. Imaging, 16, 6, 799-810 (1997)
[88] Robert, C.; Casella, G., Monte Carlo Statistical Methods (2013), Springer Science & Business Media
[89] Bardsley, J. M., Gaussian Markov random field priors for inverse problems, Inverse Probl. Imaging, 7, 2, 397-416 (2013) · Zbl 1268.65011
[90] MacKay, D. J., Bayesian nonlinear modeling for the prediction competition, ASHRAE Trans., 100, 2, 1053-1062 (1994)
[91] Dempster, A.; Laird, N.; Rubin, D., Maximum likelihood from incomplete data via EM algorithm, J. R. Stat. Soc. B, 39, 1, 1-38 (1977), wOS:A1977DM46400001 · Zbl 0364.62022
[92] Liu, J. S.; Chen, R., Blind deconvolution via sequential imputations, J. Am. Stat. Assoc., 90, 430, 567-576 (1995) · Zbl 0826.62062
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.