标题： 量化分层环境中反散射问题的不确定性 显示英文标题

标题： Quantifying uncertainty in inverse scattering problems set in layered environments

摘要： 求解反散射问题的尝试通常会导致需要处理中等到大量偏微分方程的优化和采样问题，这些偏微分方程作为约束条件。 我们在这里关注的是从表面波场测量确定分层介质中的夹杂物，同时量化不确定性并考虑波求解器质量的影响。 夹杂物由描述其材料特性和形状的少数几个参数来表征。 我们设计了算法，通过使用贝叶斯正则化和波约束来优化代价泛函，从而估计最可能的配置。 特别是，我们基于算法微分和自适应有限元网格开发了一种自动化的Levenberg-Marquardt-Fletcher型方案，用于时间相关波动方程约束（夹杂物随时间变化）。 在单频率合成测试中，对于噪声水平增加的情况，该方案在几次迭代内收敛。 为了获得其他可能的高概率配置和不对称效应的整体视图，我们求助于可并行化的仿射不变马尔可夫链蒙特卡洛方法，这需要解决数百万个波问题。 这迫使我们使用固定的网格。 尽管最优配置保持相似，但我们遇到了额外的高概率夹杂物，受先验信息、噪声水平和分层结构的影响，可以通过考虑更多频率来减少这种影响。 我们在一个简单的非反射边界条件下分析了自适应网格和固定网格在计算中的影响，在截断分层域中，我们建立了适定性和收敛性结果。 显示英文摘要

摘要： The attempt to solve inverse scattering problems often leads to optimization and sampling problems that require handling moderate to large amounts of partial differential equations acting as constraints. We focus here on determining inclusions in a layered medium from the measurement of wave fields on the surface, while quantifying uncertainty and addressing the effect of wave solver quality. Inclusions are characterized by a few parameters describing their material properties and shapes. We devise algorithms to estimate the most likely configurations by optimizing cost functionals with Bayesian regularizations and wave constraints. In particular, we design an automatic Levenberg-Marquardt-Fletcher type scheme based on the use of algorithmic differentiation and adaptive finite element meshes for time dependent wave equation constraints with changing inclusions. In synthetic tests with a single frequency, this scheme converges in few iterations for increasing noise levels. To attain a global view of other possible high probability configurations and asymmetry effects we resort to parallelizable affine invariant Markov Chain Monte Carlo methods, at the cost of solving a few million wave problems. This forces the use of prefixed meshes. While the optimal configurations remain similar, we encounter additional high probability inclusions influenced by the prior information, the noise level and the layered structure, effect that can be reduced by considering more frequencies. We analyze the effect on the calculations of working with adaptive and fixed meshes, under a simple choice of non-reflecting boundary conditions in truncated layered domains for which we establish wellposedness and convergence results.