论文标题
部分可观测时空混沌系统的无模型预测
Solving forward and inverse problems in a non-linear 3D PDE via an asymptotic expansion based approach
论文作者
论文摘要
本文涉及使用渐近扩张来有效地解决涉及非线性奇异扰动的时间依赖性反应的有效解决的前进和反问题。通过与过渡层区域中的局部坐标一起使用渐近扩展，我们证明了具有尖锐的过渡层的平滑溶液的存在和唯一性，用于三维部分偏微分方程。此外，借助渐近扩展，为相应的逆源问题得出了一个简化的模型，该模型接近整个区域的原始逆问题，除了狭窄的过渡层。我们表明，当测量数据包含噪声时，这种简化不会降低反转结果的准确性。基于这个简单的反转模型，提出了一种渐近膨胀正则化算法，用于在三维情况下有效地解决逆源问题。模型问题显示了提出的数值方法的可行性。
This paper concerns the use of asymptotic expansions for the efficient solving of forward and inverse problems involving a nonlinear singularly perturbed time-dependent reaction--diffusion--advection equation. By using an asymptotic expansion with the local coordinates in the transition-layer region, we prove the existence and uniqueness of a smooth solution with a sharp transition layer for a three-dimensional partial differential equation. Moreover, with the help of asymptotic expansion, a simplified model is derived for the corresponding inverse source problem, which is close to the original inverse problem over the entire region except for a narrow transition layer. We show that such simplification does not reduce the accuracy of the inversion results when the measurement data contain noise. Based on this simpler inversion model, an asymptotic-expansion regularization algorithm is proposed for efficiently solving the inverse source problem in the three-dimensional case. A model problem shows the feasibility of the proposed numerical approach.
