Abstract This paper proposes a modified Lie-group shooting method to solve multi-dimensional backward heat conduction problems under long time spans. The backward heat conduction problem is renowned for being ill posed because the solutions are generally unstable and highly dependent on the given data. For dealing with those problems, the Lie-group shooting method is one of the most powerful tools to find the unknown initial condition for the backward heat conduction problems in the time domain. In previous studies, the Lie-group shooting method uses the time and spatial semi-discretization technique to change the integration direction of numerical schemes and then increase the time span. However, the conversional Lie-group shooting method cannot get to the core of divergence problems for the backward heat conduction problems, especially the increased computational time. The main reason is that a real single-parameter Lie-group element occurs at zero, and a generalized midpoint Lie-group element is not equivalent to the single-parameter Lie-group element in the Minkowski space. Hence, to overcome the above problems, the relationship of the initial condition, the final condition and a real single-parameter r is assessed. According to the constraint condition of the initial and final condition, a real single-parameter r depends on the time span to maintain the numerical convergence. Again, in order to preserve the same Lie-group property in the time direction, the high-order Lie-group scheme based on the generalized linear group in Euclidean space is introduced, which concurrently satisfies the constraint of the cone structure, the Lie-group and the Lie algebra at each time step. The accuracy and efficiency are validated, even under noisy measurement data, by comparing the estimation results with existing literature.

中文翻译：

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长时间跨度下多维逆向热传导问题的改进李群射击方法

摘要 提出一种改进的李群射击方法来解决长时间跨度下的多维逆向热传导问题。后向热传导问题以不适定而闻名，因为解决方案通常不稳定并且高度依赖于给定的数据。对于处理这些问题，李群射击方法是在时域中寻找反向热传导问题未知初始条件的最有力的工具之一。在以往的研究中，李群射击法是利用时空半离散化技术改变数值方案的积分方向，进而增加时间跨度。但是，转换李群射击方法对于后向热传导问题无法触及发散问题的核心，尤其是计算时间的增加。主要原因是真实的单参数李群元出现在零处，广义中点李群元不等价于闵可夫斯基空间中的单参数李群元。因此，为了克服上述问题，需要评估初始条件、最终条件和真实单参数 r 之间的关系。根据初始条件和最终条件的约束条件，一个真正的单参数r依赖于时间跨度来保持数值收敛。再次，为了在时间方向上保持相同的李群性质，引入了基于欧氏空间广义线性群的高阶李群方案，同时满足锥结构的约束，每个时间步的李群和李代数。通过将估计结果与现有文献进行比较，即使在嘈杂的测量数据下，准确性和效率也得到了验证。