Bayesian inversion techniques for stochastic partial differential equations. (English) Zbl 1538.60124
Summary: We consider the problem of recovering the initial condition for a class of stochastic partial differential equations under Gaussian additive noise. We assume that the covariance operator of the noise is unknown. We develop an adapted Bayesian regularisation strategy, which incorporates the estimation of the unknown parameters into the computation of the initial condition posterior distribution. The proposed method allows estimation of the initial condition curve as well as construction of forecasts of the entire state curve, although the observed data may include only partial observations of the system state. We prove that, under certain conditions, the posterior distribution converges to that under known parameter values when the sample size is large. We also compare the performance of the proposed method to that of Tikhonov regularization on simulated data.
MSC:
| 60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 91G30 | Interest rates, asset pricing, etc. (stochastic models) |
| 62C10 | Bayesian problems; characterization of Bayes procedures |
| 60H50 | Regularization by noise |
| 65J22 | Numerical solution to inverse problems in abstract spaces |
| 91G60 | Numerical methods (including Monte Carlo methods) |
Software:
CMARSReferences:
| [1] | S. A. M. Y. X. Agapiou Stuart Zhang, Bayesian posterior contraction rates for linear severely ill-posed inverse problems, J. Inverse and Ill-posed Problems, 22, 297-321 (2014) · Zbl 1288.62036 · doi:10.1515/jip-2012-0071 |
| [2] | A. C. K. R. T. A. M. J. Apte Jones Stuart Voss, Data assimilation: Mathematical and statistical perspectives, International Journal for Numerical Methods in Fluids, 56, 1033-1046 (2008) · Zbl 1384.62300 · doi:10.1002/fld.1698 |
| [3] | A. K. Bouhamidi Jbilou, Sylvester Tikhonov-regularization methods in image restoration, Journal of Computational and Applied Mathematics, 206, 86-98 (2007) · Zbl 1131.65036 · doi:10.1016/j.cam.2006.05.028 |
| [4] | M. G. B. Buckley Eagleson Silverman, The estimation of residual variance in nonparametric regression, Biometrika, 75, 189-199 (1988) · Zbl 0639.62032 · doi:10.1093/biomet/75.2.189 |
| [5] | R. W. Cameron Martin, Transformations of weiner integrals under translations, Annals of Mathematics, 45, 386-396 (1944) · Zbl 0063.00696 · doi:10.2307/1969276 |
| [6] | R. Carmona and M. Tehranchi, Interest Rate Models: An Infinite Dimensional Stochastic Analysis Perspective, Springer finance, Springer, Berlin; New York, 2006. · Zbl 1124.91030 |
| [7] | R. Cont, Modeling term structure dynamics: An infinite dimensional approach, International Journal of Theoretical and Applied Finance, 08, 357-380 (2005) · Zbl 1113.91020 · doi:10.1142/S0219024905003049 |
| [8] | S. M. A. Cotter Dashti Stuart, Approximation of bayesian inverse problems for pdes, SIAM J. Numer. Anal., 48, 322-345 (2010) · Zbl 1210.35284 · doi:10.1137/090770734 |
| [9] | G. Da Prato, An Introduction to Infinite-Dimensional Analysis, Springer Science & Business Media, 2001. · Zbl 1065.46001 |
| [10] | M. A. Dashti Stuart, The bayesian approach to inverse problems, Handbook of Uncertainty Quantification, 1, 2, 3, 311-428 (2017) |
| [11] | L. Evans, Partial Differential Equations, \(2^{nd}\) edition, American Mathematical Society, 2010. · Zbl 1194.35001 |
| [12] | B. Fitzpatrick, Bayesian analysis in inverse problems, Inverse Problems, 7, 675-702 (1991) · Zbl 0743.35084 · doi:10.1088/0266-5611/7/5/003 |
| [13] | M. L. Fuhry Reichel, A new Tikhonov regularization method, Numerical Algorithms, 59, 433-445 (2012) · Zbl 1236.65038 · doi:10.1007/s11075-011-9498-x |
| [14] | G. H. P. C. D. P. Golub Hansen O’Leary, Tikhonov regularization and total least squares, SIAM J. Matrix Anal. Appl., 21, 185-194 (1999) · Zbl 0945.65042 · doi:10.1137/S0895479897326432 |
| [15] | G. H. Golub and U. Von Matt, Tikhonov Regularization for Large Scale Problems, Springer, Singapore, 1997. |
| [16] | S. Gugushvili, A. Van Der Vaart and D. Yan, Bayesian linear inverse problems in regularity scales, in Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, 56 (2020), 2081-2107. · Zbl 1460.62059 |
| [17] | R. D. A. Heath Jarrow Morton, Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation, Econometrica, 60, 77-105 (192) · Zbl 0751.90009 |
| [18] | T. Helin, On infinite-dimensional hierarchical probability models in statistical inverse problems, Inverse Problems & Imaging, 3, 567-597 (2009) · Zbl 1232.62047 · doi:10.3934/ipi.2009.3.567 |
| [19] | T. M. Helin Burger, Maximum a posterior probability estimates in infinite-dimensional bayesian inverse problems, Inverse Problems, 31, 085009 (2015) · Zbl 1325.62058 · doi:10.1088/0266-5611/31/8/085009 |
| [20] | F. M. Hildebrandt Trabs, Parameter estimation for spdes based on discrete observations in time and space, Electronic Journal of Statistics, 15, 2716-2776 (2021) · Zbl 1471.62276 · doi:10.1214/21-ejs1848 |
| [21] | J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, Applied Mathematical Sciences, 160. Springer-Verlag, New York, 2005. · Zbl 1068.65022 |
| [22] | M. E. S. Lassas Saksman Siltanen, Discretization-invariant bayesian inversion and besov space priors, Inverse Problems and Imaging, 3, 87-122 (2009) · Zbl 1191.62046 · doi:10.3934/ipi.2009.3.87 |
| [23] | M. Musiela and M. Rutkowski, Martingale Methods in Financial Modelling, Springer-Verlag, New York, 2005. · Zbl 1058.60003 |
| [24] | R. M. Pourgholi Rostamian, A numerical technique for solving IHCPs using Tikhonov regularization method, Applied Mathematical Modelling, 34, 2102-2110 (2010) · Zbl 1193.80028 · doi:10.1016/j.apm.2009.10.022 |
| [25] | S. Reich and P. Rozdeba, Posterior contraction rates for non-parametric state and drift estimation, preprint, arXiv: 2003.09219. |
| [26] | L. M. S. M. Roininen Girolami Lasanen Markkanen, Hyperpriors for matérn fields with applications in bayesian inversion, Inverse Problems and Imaging, 13, 1-29 (2019) · Zbl 1454.60068 · doi:10.3934/ipi.2019001 |
| [27] | C. C. Schillings Schwab, Sparsity in Bayesian inversion of parametric operator equations, Inverse Problems, 30, 065007 (2014) · Zbl 1291.65033 · doi:10.1088/0266-5611/30/6/065007 |
| [28] | A. M. Stuart, The bayesian approach to inverse problems, preprint, arXiv: 1302.6989. |
| [29] | A. Tarantola, Inverse Problem Theory: Methods for Data Fitting and Model Parameter Estimation, SIAM, 1987. · Zbl 0875.65001 |
| [30] | P. G.-W. A. Taylan Weber Beck, New approaches to regression by generalized additive models and continuous optimization for modern applications in finance, science and technology, Optimization, 56, 675-698 (2007) · Zbl 1123.62055 · doi:10.1080/02331930701618740 |
| [31] | L. Tenorio, Statistical regularization of inverse problems, SIAM Review, 43, 347-366 (2001) · Zbl 0976.65114 · doi:10.1137/S0036144500358232 |
| [32] | E. B. A. A. M. G.-W. Tirkolaee Goli Faridnia Soltani Weber, Multi-objective optimization for the reliable pollution-routing problem with cross-dock selection using pareto-based algorithms, Journal of Cleaner Production, 276, 122927 (2020) |
| [33] | E. B. A. P. F. Tirkolaee Goli Ghasemi Goodarzian, Designing a sustainable closed-loop supply chain network of face masks during the covid-19 pandemic: Pareto-based algorithms, Journal of Cleaner Production, 333, 130056 (2022) |
| [34] | M. D. P. A. E. J. P. Vauhkonen Vadász Karjalainen Somersalo Kaipio, Tikhonov regularization and prior information in electrical impedance tomography, IEEE Transactions on Medical Imaging, 17, 285-293 (1998) |
| [35] | G.-W. I. G. P. F. Weber Batmaz Köksal Taylan Yerlikaya-Özkurt, Cmars: A new contribution to nonparametric regression with multivariate adaptive regression splines supported by continuous optimization, Inverse Problems in Science and Engineering, 20, 371-400 (2012) · Zbl 1254.65020 · doi:10.1080/17415977.2011.624770 |
| [36] | F. C.-L. Yang Fu, A simplified Tikhonov regularization method for determining the heat source, Applied Mathematical Modelling, 34, 3286-3299 (2010) · Zbl 1201.65177 · doi:10.1016/j.apm.2010.02.020 |
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.
