论文标题
将谐波图的数值算法基准为球体
Benchmarking Numerical Algorithms for Harmonic Maps into the Sphere
论文作者
论文摘要
我们在数值上基准了将谐波图计算为单位球体的方法，尤其关注具有奇异性的谐波图。对于离散化，我们根据拉格朗日有限元进行比较两种不同的方法。虽然第一个方法仅在拉格朗日节点上强制执行单位长度约束，而另一种方法则添加了一个点投影以实现到处的约束。为了解决所得代数问题，我们将不合格的梯度流与Riemannian Trust-Region方法进行了比较。两者都是能量的，并且可以证明可以在全球汇聚到离散的Dirichlet能量的固定点。我们观察到，虽然不合格和符合的离散化都显示出对于平稳问题的相似行为，但不合格的离散化处理奇异性。在求解器侧，二阶信任区域方法在几个步骤后收敛，而梯度流量步骤的数量与反向网格元件直径成比例增加。
We numerically benchmark methods for computing harmonic maps into the unit sphere, with particular focus on harmonic maps with singularities. For the discretization we compare two different approaches, both based on Lagrange finite elements. While the first method enforces the unit-length constraint only at the Lagrange nodes, the other one adds a pointwise projection to fulfill the constraint everywhere. For the solution of the resulting algebraic problems we compare a nonconforming gradient flow with a Riemannian trust-region method. Both are energy-decreasing and can be shown to converge globally to stationary points of the discretized Dirichlet energy. We observe that while the nonconforming and the conforming discretizations both show similar behavior for smooth problems, the nonconforming discretization handles singularities better. On the solver side, the second-order trust-region method converges after few steps, whereas the number of gradient-flow steps increases proportionally to the inverse grid element diameter.
