Applied Math Seminar: Interpolation of Manifold-Valued Functions via the Polar Decomposition

Seminar | November 1 | 4-5 p.m. | 891 Evans Hall

 Evan Gawlik, University of California, San Diego

 Department of Mathematics

Manifold-valued data and manifold-valued functions play an important role in a wide variety of applications, including mechanics, computer vision and graphics, medical imaging, and numerical relativity. This talk will describe a family of interpolation operators for manifold-valued functions, with an emphasis on functions taking values in symmetric spaces and Lie groups. A key role in our construction is played by the polar decomposition – the well-known factorization of a real nonsingular matrix into the product of a symmetric positive-definite matrix times an orthogonal matrix – and its generalization to Lie groups. We demonstrate that this factorization can be leveraged to carry out a number of seemingly disparate tasks, including the design of finite elements for numerical relativity, the interpolation of subspaces for reduced-order modeling, and the approximation of acceleration-minimizing curves on the special orthogonal group.