Rob Remis

Vladimir Druskin, Rob Remis

The Krylov subspace projection approach is a well-established tool for the reduced order modeling of dynamical systems in the time domain. In this paper, we address the main issues obstructing the application of this powerful approach to the time-domain solution of exterior wave problems. We use frequency independent perfectly matched layers to simulate the extension to infinity. Pure imaginary stretching functions based on Zolotarev's optimal rational approximation of the square root are implemented leading to perfectly matched layers with a controlled accuracy over a complete spectral interval of interest. A new Krylov-based solution method via stability-corrected operator exponents is presented which allows us to construct reduced-order models (ROMs) that respect the delicate spectral properties of the original scattering problem. The ROMs are unconditionally stable and are based on a renormalized bi-Lanczos algorithm. We give a theoretical foundation of our method and illustrate its performance through a number of numerical examples in which we simulate 2D electromagnetic wave propagation in unbounded domains, including a photonic waveguide example. The new algorithm outperforms the conventional finite-difference time domain method for problems on large time intervals.

Vladimir Druskin, Rob Remis, Mikhail Zaslavsky et al.

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 2D acoustic models show 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 (MIMO) 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.

Kirsten Koolstra, Jeroen van Gemert, Peter Börnert et al.

Purpose: Design of a preconditioner for fast and efficient parallel imaging and compressed sensing reconstructions. Theory: Parallel imaging and compressed sensing reconstructions become time consuming when the problem size or the number of coils is large, due to the large linear system of equations that has to be solved in l_1 and l_2-norm based reconstruction algorithms. Such linear systems can be solved efficiently using effective preconditioning techniques. Methods: In this paper we construct such a preconditioner by approximating the system matrix of the linear system, which comprises the data fidelity and includes total variation and wavelet regularization, by a matrix with the assumption that is a block circulant matrix with circulant blocks. Due to its circulant structure, the preconditioner can be constructed quickly and its inverse can be evaluated fast using only two fast Fourier transformations. We test the performance of the preconditioner for the conjugate gradient method as the linear solver, integrated into the Split Bregman algorithm. Results: The designed circulant preconditioner reduces the number of iterations required in the conjugate gradient method by almost a factor of~5. The speed up results in a total acceleration factor of approximately 2.5 for the entire reconstruction algorithm when implemented in MATLAB, while the initialization time of the preconditioner is negligible. Conclusion: The proposed preconditioner reduces the reconstruction time for parallel imaging and compressed sensing in a Split Bregman implementation and can easily handle large systems since it is Fourier-based, allowing for efficient computations. Key words: preconditioning; compressed sensing; Split Bregman; parallel imaging