In the rest of this section we give directions for further investigation and mention open problems.
8.2.1 Model
Our method can simply be extended from first-order to second-order systems with symmetric positive definite system matrices. Since the standard Iterative Rational Krylov Algorithm is not applicable to second-order systems, another interpolatory model order reduction method which provides a good choice of interpolation points in frequency space is needed (see e.g., [Wya12]).
It would be nice to test our method on other benchmarks and applications, espe- cially with a larger number of parameters. However, the symmetry requirement is a strong restriction for a system. We only found benchmarks with at most 4 parame- ters, except an example with 7 parameters in [BB15], which is not open source, but property of the Robert Bosch GmbH. Nevertheless, our method is also suitable for higher dimensions.
For non-symmetric systems it is not clear so far how to preserve stability in higher- dimensional parameter spaces. In [GPWL14] a stability-preserving PMOR method for general systems is presented. The idea is to transform the local stable reduced order models such that the symmetric part of the matrix Ar becomes symmet-
ric negative definite. Such systems are called contractive. However, non-negative weighting functions are required to ensure that the symmetric part of the interpo- lated matrix Aer is symmetric negative definite, too. So one cannot interpolate on
sparse grids to deal with higher-dimensional parameter spaces.
A possible remedy to avoid the requirement of non-negative weighting functions might be as follows. First, we split Ar= M + N into a symmetric part M := (Ar+
ATr)/2 and a skew-symmetric part N := (Ar−ATr)/2. Then we interpolate M and N
separately: the matrix−M on the manifold of symmetric positive definite matrices (as Ar in this thesis) and the matrix N by direct matrix interpolation. So the
interpolated matricesM andf N are symmetric negative definite and skew-symmetric,e
respectively. The interpolant of the matrix Aris then defined asAer :=M+f N. Notee
that the symmetric part of Aer is (Aer+AeTr)/2 = (M +f MfT)/2 + (N +e NeT)/2 =M.f
It feels a bit unnatural to interpolate the two parts with two different concepts, but it might be worth a try. Note that the local reduced order models need to be stable, so use e.g., Balanced Truncation, and a mass matrix Er 6= Ir needs to be
inverted to the other side, i.e., Ar E−1r · Ar and Er Ir.
8.2.2 Model Order Reduction
To construct the local reduced order bases we use the H2-optimal frequency inter-
polation points of the center of the parameter space, which we compute with the Iterative Rational Krylov Algorithm, for the whole parameter space.
A combination of interpolation and approximation in frequency space might give reduced order models of a better accuracy over the whole parameter space [BG15]. But to include it into our stability-preserving parametric model order reduction method we would need a symmetric variant of the method proposed in [BG15] and ideally an extension for multi-input, multi-output systems.
To make the local reduced order models compatible, a reference basis is chosen. We take the reduced order basis at the center of the parameter space, i.e., Vref = V`0
with `0such that p`0 = pcenter. Another possibility inspired by [PMEL10, BBBG11]
is to define Vref as the first r singular vectors of Vall= [V1, . . . , VN]. Since Vallcan
become quite large, one should not use the final N that is used for interpolation, but a smaller one that belongs to a coarse grid.
8.2.3 Interpolation
The proposed parametric model order reduction method is not restricted to inter- polation on sparse grids. Any interpolation or even approximation method can be used, but it should be suitable for higher-dimensional spaces. The preservation of stability is not affected.
For very high dimensional parameter spaces energy-norm based sparse grids might be advantageous. One can also think of inventing a completely different sparse grid. However, it is not obvious which norm to take for the cost–benefit optimization.
Currently we treat all entries of the matrices which are interpolated equally and interpolate the matrix Ar regardless of the matrices Br and Cr. However, mini-
mizing the error in the transfer function is not equivalent to minimizingkAr−Aerk
orkA−1
r −Ae−1r k. It is an intriguing question whether it is possible to consider the
matrices Br and Cr when interpolating the matrix Ar.
One possibility could be to interpolate log(−MTArM) instead of log(−Ar) for
an invertible matrix M. This might yield better results if the matrix M is chosen carefully, depending on Br and Cr.
When the dimensions have different importance, the number of interpolation points needed to reach a certain interpolation error could be made smaller by di- mension adaptivity [GG03], i.e., using different refinement levels in each dimension. Before dimension adaptivity can be applied successfully a suitable error estimator that correctly portrays the interplay of the error in the different matrix entries of the three system matrices needs to be constructed.
Spacial adaptivity is only needed if the parameter dependence is very non-smooth in some regions of the parameter space. An error estimator for it might be con- structed from an error estimator for dimension adaptivity.
Literature
A.1 Matrix Basics
In this section we recapitulate some definitions and facts about matrices. These can be found in many textbooks, see for example [GVL89, Lan69, Ste98].
A.1.1 Properties
Definition A.1. A matrix M = [Mij]ni,j=1 ∈ Cn×n is
• diagonal if Mij = 0 for i6= j,
• upper triangular if Mij = 0 for i > j,
• lower triangular if Mij = 0 for i < j.
Definition A.2. A matrix M∈ Cn×n is • symmetric if M = MT, • skew-symmetric if M = −MT, • Hermitian if M = MH, • unitary if MHM = I n, • normal if MHM = MMH.
Every real symmetric matrix is Hermitian.
Every Hermitian matrix and, hence, every real symmetric matrix, is normal. Definition A.3. A matrix M∈ Rn×n is orthogonal if MTM = In.
Every real unitary matrix is orthogonal.
Definition A.4. A matrix M∈ Rn×m has orthonormal columns if MTM = Im.
Definition A.5. A Hermitian matrix M∈ Cn×n is
• positive definite if its eigenvalues are positive, i.e., Λ(M) ⊂ R+,