Dominik Wittwar

Dominik Wittwar, Gabriele Santin, Bernard Haasdonk

In this paper we consider the problem of approximating vector-valued functions over a domain $Ω$. For this purpose, we use matrix-valued reproducing kernels, which can be related to Reproducing kernel Hilbert spaces of vectorial functions and which can be viewed as an extension to the scalar-valued case. These spaces seem promising, when modelling correlations between the target function components, as the components are not learned independently of one another. We focus on the interpolation with such matrix-valued kernels. We derive error bounds for the interpolation error in terms of a generalized power-function and we introduce a subclass of matrix-valued kernels whose power-functions can be traced back to the power-function of scalar-valued reproducing kernels. Finally, we apply these kind of kernels to some artificial data to illustrate the benefit of interpolation with matrix-valued kernels in comparison to a componentwise approach.

Bernard Haasdonk, Boumediene Hamzi, Gabriele Santin et al.

For certain dynamical systems it is possible to significantly simplify the study of stability by means of the center manifold theory. This theory allows to isolate the complicated asymptotic behavior of the system close to a non-hyperbolic equilibrium point, and to obtain meaningful predictions of its behavior by analyzing a reduced dimensional problem. Since the manifold is usually not known, approximation methods are of great interest to obtain qualitative estimates. In this work, we use a data-based greedy kernel method to construct a suitable approximation of the manifold close to the equilibrium. The data are collected by repeated numerical simulation of the full system by means of a high-accuracy solver, which generates sets of discrete trajectories that are then used to construct a surrogate model of the manifold. The method is tested on different examples which show promising performance and good accuracy.

Gabriele Santin, Dominik Wittwar, Bernard Haasdonk

Kernel based regularized interpolation is a well known technique to approximate a continuous multivariate function using a set of scattered data points and the corresponding function evaluations, or data values. This method has some advantage over exact interpolation: one can obtain the same approximation order while solving a better conditioned linear system. This method is well suited also for noisy data values, where exact interpolation is not meaningful. Moreover, it allows more flexibility in the kernel choice, since approximation problems can be solved also for non strictly positive definite kernels. We discuss in this paper a greedy algorithm to compute a sparse approximation of the kernel regularized interpolant. This sparsity is a desirable property when the approximant is used as a surrogate of an expensive function, since the resulting model is fast to evaluate. Moreover, we derive convergence results for the approximation scheme, and we prove that a certain greedy selection rule produces asymptotically quasi-optimal error rates.

Bernard Haasdonk, Boumediene Hamzi, Gabriele Santin et al.

For dynamical systems with a non hyperbolic equilibrium, it is possible to significantly simplify the study of stability by means of the center manifold theory. This theory allows to isolate the complicated asymptotic behavior of the system close to the equilibrium point and to obtain meaningful predictions of its behavior by analyzing a reduced order system on the so-called center manifold. Since the center manifold is usually not known, good approximation methods are important as the center manifold theorem states that the stability properties of the origin of the reduced order system are the same as those of the origin of the full order system. In this work, we establish a data-based version of the center manifold theorem that works by considering an approximation in place of an exact manifold. Also the error between the approximated and the original reduced dynamics are quantified. We then use an apposite data-based kernel method to construct a suitable approximation of the manifold close to the equilibrium, which is compatible with our general error theory. The data are collected by repeated numerical simulation of the full system by means of a high-accuracy solver, which generates sets of discrete trajectories that are then used as a training set. The method is tested on different examples which show promising performance and good accuracy.