FAST WAVE PROPAGATION BY MODEL ORDER REDUCTION

Large scale wave propagation simulation is currently achievable in reasonable turnaround times by using distributed computing in multiple cpu clusters. However, if one needs to perform many such simulations, as is the case in optimization, tomography, or seismic imaging, then the resources required are still prohibitive. Model order reduction of large dynamical systems has been successfully used in several application domains to paliate that problem and in this paper we explore one of its manifestations, Proper Orthogonal Decomposition, for wave propagation. We describe the method and show how it can be easily interfaced with two different high fidelity simulators. We exemplify its use on several problems of increasing complexity and size. AMS subject classifications. 65M99 1. Introduction. There are many applications that require the repeated trans ient simu- lation of acoustic, elastic or electromagnetic wave propag ation. To name a few: structural analysis, blast on structures, vibrations of Navy vessels, sonar, design of piezoelectric trans- ducers for medical ultrasound, medical imaging and therapeutics uses of ultrasound, earth seismic imaging for the oil industry and earthquake seismology, optimization driven by sim- ulation for material identification and optimal design. As s uch, any significant improvement in the performance of numerical simulators would be very important. Model Order Reduction (MOR) refers to a collection of techniques to reduce the number of degrees of freedom of the very large scale dynamical systems that result after space dis- cretization of time-dependent partial differential equat ions. Some of these techniques have been successfully employed in the simulation of VLSI circuits, computational fluid mechan- ics, real-time control, heat conduction and other problems (1, 3, 5, 7). Not much has been done for wave propagation, although it does not seem that there are fundamental difficulties for its application (2). However, since none of these techniques are trivial to inter face with existing large scale high fidelity codes, it is important to be able to select wisel y the correct approach in order to minimize development costs. At this time we have centered our attention on the class of methods that go by the name of Proper Orthogonal Decomposition (POD). We start from the premise that it is possible to run a few full simulations with in the domain of interest. POD uses snapshots from these simulations to form an orthogonal basis for the solution space. This can be thought of as a problem-dependent modal decomposition, as opposite to the use of artificial basis functions (Fourier expansions, wavelet s). By using truncated Singular Value Decompositions it is possible to reduce even further the siz e of this basis without sacrificing accuracy and also to prevent the introduction of high freque ncy noise. The dynamic behav- ior of a new problem is calculated by solving projected collocation equations for the time dependent coefficients of a linear combination of the natura l basis functions. A different class of methods, tailored to problems where even a few high fidelity sim- ulations are not an option, is based on Krylov subspace machinery for large-scale matrix computations (5). These methods generate reduced-order models that are in a certain sense optimal, directly from the large-scale data matrices descr ibing the given linear system. In-