<p>We investigate numerical aspects of Riemannian interpolation on the Grassmann manifold. Instead of relying on the Riemannian normal coordinates, i.e. the Riemannian exponential and logarithm maps, we approach the interpolation problem with an alternative set of local coordinates and corresponding parameterizations. We show that these coordinates define a second-order retraction. Numerical evaluation does not formally require matrix decompositions. This is an advantage over Riemannian normal coordinates and many other retractions on the Grassmann manifold, especially when derivative data are to be treated. To estimate the interpolation error, we examine the conditioning of the coordinate mappings and state explicit bounds. It turns out that the parameterizations are well-conditioned, but the coordinate charts are generally not. As a remedy, we introduce canonically centered coordinates based on a homogeneous transition to a canonical Stiefel representative of a Grassmann data point chosen to act as the coordinate center. We show that the order of magnitude of the asymptotic interpolation error on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\operatorname {Gr}(n,p)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>Gr</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is the same as in the Euclidean space. Numerical experiments illustrate the findings. The first is academic, where we interpolate a parametric orthogonal projector <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$QQ^T$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Q</mi> <msup> <mi>Q</mi> <mi>T</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>. The <i>Q</i>–factor stems from a parametric compact QR–decomposition. Moreover, we assess the computation time, present an example where closed Riemannian normal coordinates struggle, and conduct an experiment in the context of parametric model reduction of dynamical systems, where we interpolate reduced subspaces that are obtained by proper orthogonal decomposition.</p>

Canonically centered coordinates for Grassmann interpolation: Lagrange, Hermite, and errors

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2026年4月14日

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Rasmus Jensen,
Ralf Zimmermann

摘要

We investigate numerical aspects of Riemannian interpolation on the Grassmann manifold. Instead of relying on the Riemannian normal coordinates, i.e. the Riemannian exponential and logarithm maps, we approach the interpolation problem with an alternative set of local coordinates and corresponding parameterizations. We show that these coordinates define a second-order retraction. Numerical evaluation does not formally require matrix decompositions. This is an advantage over Riemannian normal coordinates and many other retractions on the Grassmann manifold, especially when derivative data are to be treated. To estimate the interpolation error, we examine the conditioning of the coordinate mappings and state explicit bounds. It turns out that the parameterizations are well-conditioned, but the coordinate charts are generally not. As a remedy, we introduce canonically centered coordinates based on a homogeneous transition to a canonical Stiefel representative of a Grassmann data point chosen to act as the coordinate center. We show that the order of magnitude of the asymptotic interpolation error on $\operatorname {Gr}(n,p)$ $Gr (n, p)$ is the same as in the Euclidean space. Numerical experiments illustrate the findings. The first is academic, where we interpolate a parametric orthogonal projector $QQ^T$ $Q Q^{T}$ . The Q–factor stems from a parametric compact QR–decomposition. Moreover, we assess the computation time, present an example where closed Riemannian normal coordinates struggle, and conduct an experiment in the context of parametric model reduction of dynamical systems, where we interpolate reduced subspaces that are obtained by proper orthogonal decomposition.