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<p>The computational challenges inherent in multilinear systems present significant obstacles in contemporary engineering and scientific applications, where traditional matrix-based iterative approaches are inadequate for computing tensor inverses and solving multilinear equations. This study addresses these limitations by explicitly developing novel projection-based iterative methodologies for multilinear systems within the context of the <i>t</i>-product, including tensor-based implementations of the biconjugate gradient method, the squared conjugate gradient method, and the stabilized biconjugate gradient method. An innovative tensor-based iterative representation is introduced to compute the Moore–Penrose inverse of tensors, accompanied by the design of efficient and robust computational algorithms for these methodologies via the <i>t</i>-product. The computational efficiency and practical advantages of these approaches are validated by a few numerical examples. In addition, an application is demonstrated to solve color image restoration problems.</p>

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Projection-based iterative methods for solving multilinear systems

  • 期刊论文

  • 发表日期：

    2026年6月2日

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  • Krushnachandra Panigrahy,
  • Ratikanta Behera,
  • Jajati Keshari Sahoo

摘要

The computational challenges inherent in multilinear systems present significant obstacles in contemporary engineering and scientific applications, where traditional matrix-based iterative approaches are inadequate for computing tensor inverses and solving multilinear equations. This study addresses these limitations by explicitly developing novel projection-based iterative methodologies for multilinear systems within the context of the t-product, including tensor-based implementations of the biconjugate gradient method, the squared conjugate gradient method, and the stabilized biconjugate gradient method. An innovative tensor-based iterative representation is introduced to compute the Moore–Penrose inverse of tensors, accompanied by the design of efficient and robust computational algorithms for these methodologies via the t-product. The computational efficiency and practical advantages of these approaches are validated by a few numerical examples. In addition, an application is demonstrated to solve color image restoration problems.