Rational Krylov subspace (RKS) techniques are well-established and powerful tools for projection-based model reduction of time-invariant dynamic systems. For hyperbolic wavefield problems, such techniques perform well in configurations where only a few modes contribute to the field. RKS methods, however, are fundamentally limited by the Nyquist-Shannon sampling rate, making them unsuitable for the approximation of wavefields in configuration characterized by large travel times and propagation distances, since wavefield responses in such configurations are highly oscillatory in the frequency-domain. To overcome this limitation, we propose to precondition the RKSs by factoring out the rapidly varying frequency-domain field oscillations. The remaining amplitude-functions are generally slowly varying functions of source position and spatial coordinate and allow for a significant compression of the approximation subspace. Our one-dimensional analysis together with numerical experiments for large-scale two-dimensional acoustic models shows superior approximation properties of preconditioned RKS compared with the standard RKS model-order reduction. The preconditioned RKS results in a reduction of the frequency sampling well below the Nyquist-Shannon rate, a weak dependence of the RKS size on the number of inputs and outputs for multiple-input/multiple-output problems, and, most importantly, in a significant coarsening of the finite-difference grid used to generate the RKS. A prototype implementation indicates that the preconditioned RKS algorithm is competitive in the modern high performance computing environment.@en

In this paper we present a structure-preserving model-order reduction technique to efficiently compute electromagnetic wave fields on unbounded domains. As an approximation space, we take the span of the real and imaginary parts of frequency-domain solutions of Maxwell's equations. Reduced-order models for the electromagnetic field belong to this space and the expansion coefficients of these models are determined from a Galerkin condition. We show that the models constructed in this manner are structure-preserving and interpolate the electromagnetic field responses at the expansion frequencies. Moreover, for monostatic field responses (coinciding sources and receivers), the first-order derivative of a reduced-order model with respect to frequency interpolates this first-order derivative of the unreduced monostatic field response as well. A two-dimensional numerical example illustrates the performance of the proposed reduction method.@en

We have developed several Krylov projection-based model-order reduction techniques to simulate electromagnetic wave propagation and diffusion in unbounded domains. Such techniques can be used to efficiently approximate transfer function field responses between a given set of sources and receivers and allow for fast and memory-efficient computation of Jacobians, thereby lowering the computational burden associated with inverse scattering problems. We found how general wavefield principles such as reciprocity, passivity, and the Schwarz reflection principle translate from the analytical to the numerical domain and developed polynomial, extended, and rational Krylov model-order reduction techniques that preserve these structures. Furthermore, we found that the symmetry of the Maxwell equations allows for projection onto polynomial and extended Krylov subspaces without saving a complete basis. In particular, short-term recurrence relations can be used to construct reduced-order models that are as memory efficient as time-stepping algorithms. In addition, we evaluated the differences between Krylov reduced-order methods for the full wave and diffusive Maxwell equations and we developed numerical examples to highlight the advantages and disadvantages of the discussed methods.@en

In this talk we present travel time or asymptotically corrected Krylov subspace methods to efficiently compute time- and frequency-domain wavefields in inhomogeneous structures. Fields characterized by large travel times can be effectively captured, by adding travel time information to parameter dependent Rational Krylov subspaces. This method provides reduced-order models of small order, which can be incorporated in wavefield inversion schemes. Numerical experiments will be presented that illustrate the performance of the method.@en