Randomized tensor decomposition for large-scale data assimilation problems

Abstract:

Ensembled-based data assimilation is commonly used for reservoir characterization and history matching with uncertainty quantification in energy resources engineering, carbon dioxide sequestration and groundwater. Data assimilation methods are computationally demanding for high-dimensional model and data spaces. This study develops a new computationally efficient technique for large-scale data assimilation problems based on randomized high-order singular value decomposition (HOSVD). Tensors are multiway arrays that generalize matrices to multiple dimensions; therefore, they provide a natural solution to store and represent three-dimensional geological models and measurements. To take advantage of the inherent multidimensional structure of tensors, the HOSVD method performs the orthogonal decomposition of the data in the high-order space, whereas in the original SVD the data must be stored in column vectors before decomposition and the information about high-dimensional correlations is lost. However, in complex geological models with a large number of geophysical observations, the dimension of tensors becomes very large. To improve the computational efficiency and reduce the memory usage, before the tensor decomposition with HOSVD, we propose to reduce the dimension of tensors by the randomized linear algebra algorithm, which guarantees that most information of the original tensor is preserved with high probability in the reduced tensor. Then, the data assimilation is performed efficiently and accurately in the low-dimensional model and data space. To valid the method, we apply it to a three-dimensional synthetic case for CO₂ storage in deep saline aquifers.