如何在黎曼流形上以任意Sobolev范数小误差计算测地线长度
How to compute the length of a geodesic on a Riemannian manifold with small error in arbitrary Sobolev norms
AI总结：
通过正则多项式插值流形线元对应的程函方程全局解，提出在任意强Sobolev范数下逼近长度泛函的算法，并利用几何信息获得误差估计，指出算法可精确逼近抛物型偏微分方程解，应用于金融和物理。
摘要：
我们通过正则多项式插值与流形线元 $ds^2=g_{ij}dx^idx^j$ 相关的程函方程的全局解，计算黎曼流形上测地线的长度。我们的算法在任意强Sobolev范数下逼近长度泛函。利用几何信息获得了误差估计。指出了该算法如何用于获得抛物型偏微分方程解的精确逼近，从而在金融和物理中有明显的应用。
英文摘要：
We compute the length of geodesics on a Riemannian manifold by regular polynomial interpolation of the global solution of the eikonal equation related to the line element $ds^2=g_{ij}dx^idx^j$ of the manifold. Our algorithm approximates the length functional in arbitrarily strong Sobolev norms. Error estimates are obtained where the geometric information is used. It is pointed out how the algorithm can be used to get accurate approximation of solutions of parabolic partial differential equations leading obvious applications to finance and physics.
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