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<p>The alternating direction method of multipliers (ADMM) has been widely used for various separable convex optimization problems in different applications. When considering some complicated optimization problems arising in scientific computing areas, it is necessary to inexactly solve the subproblem in the ADMM to reduce vast computations. In this work, we propose and analyze a new inexact ADMM algorithm nested with some easily applicable inexactness criterion. Under the same conditions ensuring the linear convergence of the exactly-solved ADMM, we prove that the proposed inexactness criterion can guarantee the linear convergence of our inexact ADMM. Then we apply the algorithm to two nonlinear inverse problems in elliptic equations and present the specific implementation details. Numerical results for identifying discontinuous coefficients in elliptic equations are reported to demonstrate the feasibility and efficiency of the proposed inexact ADMM algorithm.</p>

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An Inexact ADMM with Linear Convergence and its Application to Inverse Coefficient Problems

  • 期刊论文

  • 发表日期：

    2026年3月19日

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  • Tiancan Feng,
  • Wenyi Tian

摘要

The alternating direction method of multipliers (ADMM) has been widely used for various separable convex optimization problems in different applications. When considering some complicated optimization problems arising in scientific computing areas, it is necessary to inexactly solve the subproblem in the ADMM to reduce vast computations. In this work, we propose and analyze a new inexact ADMM algorithm nested with some easily applicable inexactness criterion. Under the same conditions ensuring the linear convergence of the exactly-solved ADMM, we prove that the proposed inexactness criterion can guarantee the linear convergence of our inexact ADMM. Then we apply the algorithm to two nonlinear inverse problems in elliptic equations and present the specific implementation details. Numerical results for identifying discontinuous coefficients in elliptic equations are reported to demonstrate the feasibility and efficiency of the proposed inexact ADMM algorithm.