发表机构
University of Houston; Tufts University（休斯顿大学; 塔夫茨大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对基于优化的反问题开发TROM框架，用TT - SVD或TT - Cross压缩逼近参数到观测映射，集成到正则化非线性最小二乘公式。通过两个反问题实验评估多种因素影响，结果表明能降成本且在复杂情况下提高鲁棒性。
AI 中文摘要
我们为受参数依赖动力系统控制的基于优化的反问题开发了一个张量降阶建模（TROM）框架。该方法使用TT - SVD或TT - Cross压缩，以张量列格式直接逼近参数到观测的映射，并将结果表示集成到正则化非线性最小二乘公式中。除了加速正向评估外，低秩张量结构用于在降维坐标中重新表述反问题，在不形成完整观测空间雅可比矩阵的情况下组装高斯 - 牛顿量，并在离散参数网格上进行基于TROM的目标最小化。该张量优化步骤既可以作为独立的近似最小化程序，也可以作为后续高斯 - 牛顿求解的数据驱动初始化。针对两个反问题研究了该方法：一个是异质介质中的逆热传导问题，未知参数描述多个低导率夹杂物的位置；另一个是具有高度非凸优化格局的FitzHugh - Nagumo参数估计问题。数值实验评估了ROM近似误差、测量噪声、正则化、初始化、空间离散化和参数维度增加的影响。结果表明，TROM可以以显著降低的在线成本再现全阶反演的行为。实验还表明，降维坐标反演、基于张量的优化和适当的正则化在高维、噪声和强非凸情况下提高了鲁棒性。
英文摘要
We develop a tensor reduced-order modeling (TROM) framework for optimization-based inverse problems governed by parameter-dependent dynamical systems. The approach approximates the parameter-to-observation map directly in tensor-train (TT) format using TT-SVD or TT-Cross and integrates it into a regularized nonlinear least-squares formulation. Beyond accelerating forward evaluations, the low-rank tensor structure reformulates the inverse problem in reduced coordinates, assembles Gauss--Newton quantities without forming the full observation-space Jacobian, and minimizes the TROM objective over the discrete parameter grid. This tensor optimization provides either a stand-alone approximate solution or a data-informed initialization for a subsequent Gauss--Newton solve. We study an inverse heat-transfer problem in a heterogeneous medium, where the parameters describe the locations and radii of low-conductivity inclusions, and a FitzHugh--Nagumo parameter-estimation problem with a highly nonconvex landscape. Numerical experiments assess reduced-order model error, measurement noise, regularization, initialization, spatial discretization, and increasing parameter dimension. The results show that TROM reproduces full-order inversion at substantially reduced online cost. They also demonstrate that reduced-coordinate inversion, tensor-based optimization, and appropriate regularization improve robustness in higher-dimensional, noisy, and strongly nonconvex regimes. For the continuous TROM inverse problem, we develop an error-to-inversion analysis. Under local strong convexity of the regularized FOM objective and parametric smoothness of the FOM observation map, the error between the parameters recovered with the full-order model and TROM is bounded by controlled uniform errors in the surrogate map and its Jacobian, together with local FOM and curvature quantities.
