Hierarchical Tucker tensor optimization - applications to 4D seismic data interpolation

TitleHierarchical {Tucker} tensor optimization - applications to {4D} seismic data interpolation
Publication TypeSINBAD Presentation
AuthorsCurt Da Silva, Felix J. Herrmann
Year of Publication2013

In this work, we develop optimization algorithms on the manifold of Hierarchical Tucker (HT) tensors, an extremely efficient format for representing high-dimensional tensors exhibiting particular low-rank structure. With some minor alterations to existing theoretical developments, we develop an optimization framework based on the geometric understanding of HT tensors as a smooth manifold, a generalization of smooth curves/surfaces. Building on the existing research of solving optimization problems on smooth manifolds, we develop Steepest Descent and Conjugate Gradient methods for HT tensors. The resulting algorithms converge quickly, are immediately parallelizable, and do not require the computation of SVDs. We also derive efficient Gauss-Newton based algorithms which converge much faster than standard, first-order methods. We also extend ideas about favourable sampling conditions for missing-data recovery from the field of Matrix Completion to Tensor Completion and demonstrate how the organization of data can affect the success of recovery. As a result, if one has data with randomly missing source pairs, using these ideas, coupled with an efficient solver, one can interpolate large-scale seismic data volumes with missing sources and/or receivers by exploiting the multidimensional dependencies in the data. We are able to recover data volumes amidst extremely high subsampling ratios (in some cases, > 75%) using this approach and we demonstrate our recovery on a synthetic 5D data set provided to us by BG.

Keywords3D data interpolation, hierarchical tucker, Presentation, private, SINBAD, SINBADSPRING2013, SLIM, structured tensor
Citation Keydasilva2013SINBADSPRhtuck