likelihood-informed dimension reduction

As a combined result of the smoothness of forward models, the regularity imposed by prior assumptions and the noise in incomplete data, the information update from the prior distribution (left figure) to the posterior distribution (right figure) may be confined to a relatively low-dimensional parameter subspace (indicated by red arrows).

We work on likelihood-informed subspace (LIS) to identify this subspace for breaking the curse of dimensionality. See my recent presentation for details. For more information, see the work on building dimension-robust MCMC samplers using LIS [1,2], its optimality in linear problems [3,4], and error analysis for general problems [5-8].

_{[1] Cui, T., Martin, J., Marzouk, Y. M., Solonen, A., & Spantini, A. (2014). Likelihood-informed dimension reduction for nonlinear inverse problems. Inverse Problems, 30(11), 114015. https://doi.org/10.1088/0266-5611/30/11/114015}

_{[2] Cui, T., Law, K. J. H., & Marzouk, Y. M. (2016). Dimension-independent likelihood-informed MCMC. Journal of Computational Physics, 304(1), 109–137. https://doi.org/10.1016/j.jcp.2015.10.008}

_{[3] Spantini, A., Solonen, A., Cui, T., Martin, J., Tenorio, L., & Marzouk, Y. (2015). Optimal low-rank approximations of Bayesian linear inverse problems. SIAM Journal on Scientific Computing, 37(6), A2451–A2487. https://doi.org/10.1137/140977308}

_{[4] Spantini, A., Cui, T., Willcox, K., Tenorio, L., & Marzouk, Y. (2017). Goal-oriented optimal approximations of Bayesian linear inverse problems. SIAM Journal on Scientific Computing, 39(5), S167–S196. https://doi.org/10.1137/16M1082123}

_{[5] Cui, T., & Zahm, O. (2021). Data-free likelihood-informed dimension reduction of Bayesian inverse problems. Inverse Problems, 37(4), 045009. https://doi.org/10.1088/1361-6420/abeafb}

_{[6] Cui, T., & Tong, X. T. (2022). A unified performance analysis of likelihood-informed subspace methods. Bernoulli, 28(4), 2788–2815. https://doi.org/10.3150/21-BEJ1437}

_{[7] Zahm, O., Cui, T., Law, K., Spantini, A., & Marzouk, Y. (2022). Certified dimension reduction in nonlinear Bayesian inverse problems. Mathematics of Computation, 91(336), 1789–1835. https://doi.org/10.1090/mcom/3737}

_{[8] Cui, T., Tong, X. T., & Zahm, O. (2022). Prior normalization for certified likelihood-informed subspace detection of Bayesian inverse problems. Inverse Problems, 38(12), 124002. https://doi.org/10.1088/1361-6420/ac9582}