Reduced-Rank Random Fields via BCCB Covariance Representation
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Solution Overview
Problem
Current methods for generating and processing Gaussian random fields are computationally intensive and inefficient, particularly due to the high cost of storing and processing large covariance matrices, which limits their application in uncertainty quantification analysis.
Innovation Solution
A computer-implemented method using block circulant with circulant blocks (BCCB) representation of covariance matrices to reduce redundant computations and achieve data compression, enabling efficient generation and processing of reduced-rank random fields through circulant embedding and 2D fast Fourier transforms.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If standard eigendecomposition or Cholesky factorization is used to generate Gaussian random fields, then the covariance structure is preserved accurately, but the computational cost and storage requirements increase dramatically to O(M³N³)
Solution Approach 1:
The large MN×MN covariance matrix is segmented into M×M block circulant matrices with circulant blocks, where each block corresponds to a spatial location. This segmentation allows the use of fast Fourier transform-based methods that reduce computational complexity from O(M³N³) to O(MN log MN) by exploiting the circulant structure of each block.
Solution Approach 2:
The patent transforms the covariance matrix from its standard form to a block circulant with circulant blocks (BCCB) representation. This parameter transformation enables the use of fast Fourier transform techniques, changing the computational approach from direct matrix decomposition to frequency-domain operations that are significantly more efficient.
2Reliability
If full-rank covariance matrices are stored and processed, then complete statistical information is retained, but storage space and processing time increase significantly
Solution Approach 1:
The patent extracts only the essential spectral components of the covariance matrix by computing its eigenvalue decomposition in the frequency domain. By working with the diagonal spectral representation rather than the full dense matrix, the method retains the necessary statistical information while dramatically reducing storage requirements from O(M²N²) to O(MN).
Solution Approach 2:
The block circulant structure allows the covariance matrix to be represented compactly through its generating vectors or spectral components. Instead of storing the full MN×MN matrix, the patent stores compact representations that can be efficiently expanded when needed, reducing storage while maintaining access to complete statistical information.
3Productivity
If circulant embedding is used to reduce computational complexity to O(MN log MN), then processing speed improves, but the method becomes approximate rather than exact
Solution Approach 1:
The patent performs preliminary spectral decomposition of the covariance matrix to obtain its eigenvalues and eigenvectors in the frequency domain before generating random field samples. This preliminary action creates an exact spectral representation that can be used to generate samples with the precise covariance structure, eliminating the need for approximate circulant embedding while maintaining computational efficiency.
Data Source
AI summary
Method and system are provided for uncertainty quantification analysis with efficient two dimensional (2D) random fields. The method may be used in a modeling system. The method may include providing a covariance matrix of a set of parameters with a covariance function describing an occurring real world phenomenon; and generating a random field based on the covariance matrix including a controlled rank reduction to produce a reduced-rank random field dataset provided as a data structure with data compression. Generating the random field applies rank reduction of a block circulant with circulant blocks (BCCB) representation of a covariance matrix obtained from the covariance matrix using existing symmetry of eigenvalues to eliminate redundant computations. The method may include outputting a reduced-rank random field dataset for use in a user-provided model of the modeling system for uncertainty quantification analysis and simulation of the real world phenomenon.