The Bayesian Approach to Inverse Robin Problems

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS
Aksel K. Rasmussen, Fanny Seizilles, Mark Girolami, Ieva Kazlauskaite
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引用次数: 0

Abstract

SIAM/ASA Journal on Uncertainty Quantification, Volume 12, Issue 3, Page 1050-1084, September 2024.
Abstract.In this paper, we investigate the Bayesian approach to inverse Robin problems. These are problems for certain elliptic boundary value problems of determining a Robin coefficient on a hidden part of the boundary from Cauchy data on the observable part. Such a nonlinear inverse problem arises naturally in the initialization of large-scale ice sheet models that are crucial in climate and sea-level predictions. We motivate the Bayesian approach for a prototypical Robin inverse problem by showing that the posterior mean converges in probability to the data-generating ground truth as the number of observations increases. Related to the stability theory for inverse Robin problems, we establish a logarithmic convergence rate for Sobolev-regular Robin coefficients, whereas for analytic coefficients we can attain an algebraic rate. The use of rescaled analytic Gaussian priors in posterior consistency for nonlinear inverse problems is new and may be of separate interest in other inverse problems. Our numerical results illustrate the convergence property in two observation settings.
逆向罗宾问题的贝叶斯方法
SIAM/ASA《不确定性量化期刊》,第12卷,第3期,第1050-1084页,2024年9月。 摘要.本文研究了逆罗宾问题的贝叶斯方法。这些问题是某些椭圆边界值问题,即根据可观测部分的考奇数据确定边界隐藏部分的罗宾系数。这种非线性逆问题自然出现在对气候和海平面预测至关重要的大尺度冰盖模型的初始化过程中。我们通过证明随着观测数据数量的增加,后验均值在概率上会收敛于数据生成的基本真相,从而激发了针对典型的罗宾反问题的贝叶斯方法。与罗宾逆问题的稳定性理论相关,我们确定了 Sobolev 规则罗宾系数的对数收敛率,而对于解析系数,我们可以达到代数收敛率。在非线性逆问题的后验一致性中使用重标度解析高斯前验是一种新方法,可能对其他逆问题也有意义。我们的数值结果说明了在两种观测环境下的收敛特性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
ACS Applied Bio Materials
ACS Applied Bio Materials Chemistry-Chemistry (all)
CiteScore
9.40
自引率
2.10%
发文量
464
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