We generalize the notion of Bézier surfaces and surface splines to Riemannian manifolds. To this end we put forward and compare three possible alternative definitions of Bézier surfaces. We furthermore investigate how to achieve C 0 -and C 1 -continuity of Bézier surface splines. Unlike in Euclidean space and for one-dimensional Bézier splines on manifolds, C 1 -continuity cannot be ensured by simple conditions on the Bézier control points: it requires an adaptation of the Bézier spline evaluation scheme. Finally, we propose an algorithm to optimize the Bézier control points given a set of points to be interpolated by a Bézier surface spline. We show computational examples on the sphere, the special orthogonal group and two Riemannian shape spaces.Keywords: Composite Bézier surface, Riemannian manifold, differentiability conditions, bending energy.Mathematics Subject Classification (2010): 65D05, 65D07, 53C22, 65K10Figure 1: Differentiable Bézier spline surface on the Riemannian space of shells (Section 6.4) interpolating the red shapes. The gray shapes are points on the Bézier surface driven by the control points in green. Their location indicates where in the R 2 domain they are achieved.

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We derive a variational model to fit a composite Bézier curve to a set of data points on a Riemannian manifold. The resulting curve is obtained in such a way that its mean squared acceleration is minimal in addition to remaining close the data points. We approximate the acceleration by discretizing the squared second order derivative along the curve. We derive a closed-form, numerically stable and efficient algorithm to compute the gradient of a Bézier curve on manifolds with respect to its control points, expressed as a concatenation of so-called adjoint Jacobi fields. Several examples illustrate the capabilites and validity of this approach both for interpolation and approximation. The examples also illustrate that the approach outperforms previous works tackling this problem.

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When balanced truncation is used for model order reduction, one has to solve a pair of Lyapunov equations for two Gramians and uses them to construct a reduced-order model. Although advances in solving such equations have been made, it is still the most expensive step of this reduction method. Parametric model order reduction aims to determine reduced-order models for parameter-dependent systems. Popular techniques for parametric model order reduction rely on interpolation. Nevertheless, interpolation of Gramians is rarely mentioned, most probably due to the fact that Gramians are symmetric positive semidefinite matrices, a property that should be preserved by the interpolation method. In this contribution, we propose and compare two approaches for Gramian interpolation. In the first approach, the interpolated Gramian is computed as a linear combination of the data Gramians with positive coefficients. Even though positive semidefiniteness is guaranteed in this method, the rank of the interpolated Gramian can be significantly larger than that of the data Gramians. The second approach aims to tackle this issue by performing the interpolation on the manifold of fixed-rank positive semidefinite matrices. The

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