Scattered manifold-valued data approximation.

Area of Science:

• Differential Geometry
• Numerical Analysis
• Computer Science

Background:

• Function approximation from scattered data is crucial in various scientific domains.
• Approximating functions mapping Euclidean domains to manifolds presents unique challenges due to manifold curvature and non-Euclidean geometry.
• Existing methods may lack computational efficiency or theoretical guarantees for manifold-valued data.

Purpose of the Study:

• To develop a novel approximation method for functions defined on manifolds using scattered data points.
• To analyze the accuracy and smoothness properties of the proposed approximation technique.
• To investigate the possibility of replacing computationally intensive means with more efficient alternatives.

Main Methods:

• Combining the moving least squares (MLS) method with a generalized mean on manifolds.
• Proving that the proposed approximant inherits the accuracy and smoothness of its linear counterpart.
• Demonstrating the inessentiality of the Karcher mean, allowing for a more general 'center of mass' based on manifold retractions.

Main Results:

• The proposed method achieves approximation orders and smoothness comparable to linear methods.
• The Karcher mean can be replaced by a more computationally efficient generalized mean or center of mass.
• Numerical results validate the theoretical findings, confirming the method's effectiveness.

Conclusions:

• A robust and computationally efficient method for function approximation on manifolds from scattered data has been established.
• The theoretical framework supports the use of generalized means, broadening the applicability of manifold approximation techniques.
• This work contributes to advancing numerical methods for problems involving complex geometric structures.