Manifold construction and parameterization for nonlinear manifold-based model reduction

We present a new manifold construction and parameterization algorithm for model reduction approaches based on projection on manifolds. The new algorithm employs two key ideas: (1) we define an ideal manifold for nonlinear model reduction to be the solution of a set of differential equations with the property that the tangent space at any point on the manifold spans the same subspace as the low-order subspace (e.g., Krylov subspace generated by moment-matching techniques) of the linearized system; (2) we propose the concept of normalized integral curve equations, which are repeatedly solved to identify an almost-ideal manifold. The manifold constructed by our algorithm inherits the important property in that it covers important system responses such as DC and AC responses. It also preserves better local distance metrics on the manifold, thanks to the employment of normalized integral curve equations. To gauge the quality of the resulting manifold, we also derive an error bound of the moments of linearized systems, assuming moment-matching techniques are employed to generate low-order subspaces for linearized systems. The algorithm is also more systematic and generalizable to higher dimensions than the ad hoc procedure in. We illustrate the key ideas through a simple 2-D example. We also combine this new manifold construction and parameterization algorithm with maniMOR to generate reduced models for a quadratic nonlinear system and a CMOS circuit. Simulation results are provided, together with comparisons to full models as well as TPWL reduced models.