我们关注土壤高保真亚表面成像，这在地质勘察和地球物理勘探中普遍存在。具体而言，我们尝试利用表面测量数据，反演半无限三维任意非均匀介质中Lame参数的空间分布。我们通过介质的完整波形响应推导反演问题，采用偏微分方程约束优化方法直接在时域内求解，以最小化选定测量点的介质响应与试分布Lame参数计算响应之间的误差。我们讨论了使所提反演方案具有算法鲁棒性的策略。为将计算域限制在感兴趣的大小，我们采用完美匹配层（PML）。为求解正向问题，我们使用一种 recently developed 的混合有限元方法，其中PML的位移-应力公式与内部域的标准位移-only 公式耦合，从而得到计算上高效的方法。时间积分通过使用显式Runge-Kutta方案实现，这在并行计算机上的大规模问题中尤为合适。我们验证了通过所提方案获得的材料梯度的准确性，并报告了数值结果，展示了对于平滑和尖锐的剖面，Lame参数的成功重建。
英文摘要：
We are concerned with high-fidelity subsurface imaging of the soil, which commonly arises in geotechnical site characterization and geophysical explorations. Specifically, we attempt to image the spatial distribution of the Lame parameters in semi-infinite, three-dimensional, arbitrarily heterogeneous formations, using surficial measurements of the soil's response to probing elastic waves. We use the complete waveform response of the medium to derive the inverse problem, by using a partial-differential-equation (PDE)-constrained optimization approach, directly in the time-domain, to minimize the misfit between the observed response of the medium at select measurement locations, and a computed response corresponding to a trial distribution of the Lame parameters. We discuss strategies that lend algorithmic robustness to our proposed inversion scheme. To limit the computational domain to the size of interest, we employ perfectly-matched-layers (PMLs).
In order to resolve the forward problem, we use a recently developed hybrid finite element approach, where a displacement-stress formulation for the PML is coupled to a standard displacement-only formulation for the interior domain, thus leading to a computationally cost-efficient scheme. Time-integration is accomplished by using an explicit Runge-Kutta scheme, which is well-suited for large-scale problems on parallel computers.
We verify the accuracy of the material gradients obtained via our proposed scheme, and report numerical results demonstrating successful reconstruction of the two Lame parameters for both smooth and sharp profiles.
An adaptive finite element PML method for the elastic wave scattering problem in periodic structures
Beijing University of Posts and Telecommunications（北京邮电大学）；Purdue University（普渡大学）；Jilin University（吉林大学）；Academy of Mathematics and System Sciences, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
Convergence of the PML method for thermoelastic wave scattering problems
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；School of Mathematics and Statistics, Beijing Jiaotong University（北京交通大学数学与统计学院）；SKLMS（数学力学研究所）
A pure source transfer domain decomposition method for Helmholtz equations in unbounded domain
提出了一种求解亥姆霍兹散射问题的有界区域截断完美匹配层（PML）近似的纯源转移区域分解方法（PSTDDM）。该方法是对[Z.陈和X。Xiang，SIAM J. Numer.肛门，51（2013），pp. 2331- 2356]。STDDM将区域分解为$N$非重叠层后，由源传递和波展开两个步骤串联而成，依次求解相邻层上的$N-1$截断PML问题和$N-2$截断半空间PML问题。而PSTDDM仅由两个相反方向上的两个并行源传输步骤组成，并且在每个步骤中连续解决两个相邻层上的截断PML问题$N-1$。这样的修改的一个好处是，在两个相邻层上的截断PML问题可以通过PSTDDM沿着平行于层的方向进一步解决。因此，我们得到了一个分块PSTDDM上的分解组成的$N^2$平方，这减少了子域问题的大小，更适合于大规模的问题。在波数为常数的情况下，证明了逐层PSTDDM和逐块PSTDDM的收敛性。数值算例表明，PSTDDM对常波数的离散Helmholtz方程有很好的逼近，可以作为预处理GMRES方法中的一个有效的预处理子用于求解常波数和非均匀波数的离散Helmholtz方程.
Convergence of the PML solution for elastic wave scattering by biperiodic structures
Beijing University of Posts and Telecommunications（北京邮电大学）；Purdue University（普渡大学）；Jilin University（吉林大学）；Academy of Mathematics and System Sciences, CAS（中国科学院数学与系统科学研究院；中国科学院大学）；University of Chinese Academy of Sciences
Convergence of the PML method for scattering problems in poroelastic media
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；Beijing Key Laboratory of Biological Big Data and Topological Statistics, School of Mathematics and Statistics, Beijing Jiaotong University（北京交通大学数学与统计学院）；SKLMS（数学力学科学研究院）
Wavenumber-explicit stability and preasymptotic error analysis of UPML finite element method for obstacle scattering problems
School of Mathematical Science, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；State Key Laboratory of Mathematical Sciences, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所数学科学国家重点实验室）
