| Citation: |
Yu Shen, Xiangtuan Xiong. A SIMPLIFIED ITERATIVE ALGORITHM WITH RELAXATION FACTOR FOR SOLVING AN ANNULAR INVERSE ELECTROCARDIOGRAPHY PROBLEM[J]. Journal of Applied Analysis & Computation, 2027, 17(2): 1454-1480. doi: 10.11948/20260174
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A SIMPLIFIED ITERATIVE ALGORITHM WITH RELAXATION FACTOR FOR SOLVING AN ANNULAR INVERSE ELECTROCARDIOGRAPHY PROBLEM
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Abstract
This paper presents a simplified iterative algorithm with a relaxation factor for solving the Cauchy problem of Laplace equation in an annular domain, with applications to inverse electrocardiography. Building upon the classical Kozlov-Maz’ya iterative framework, we introduce a new simplified iteration algorithm with the relaxation factor to accelerate convergence. Theoretical analysis and numerical experiments demonstrate that the proposed method maintains computational efficiency while significantly improving reconstruction accuracy and numerical stability.
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References
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Yu Shen, Xiangtuan Xiong. A SIMPLIFIED ITERATIVE ALGORITHM WITH RELAXATION FACTOR FOR SOLVING AN ANNULAR INVERSE ELECTROCARDIOGRAPHY PROBLEM[J]. Journal of Applied Analysis & Computation, 2027, 17(2): 1454-1480. doi: 10.11948/20260174
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Figure 1.
The input data g(θ) for Example 4.1.
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Figure 2.
The convergence error vs number of iterations for Example 4.1.
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Figure 3.
(3a) The residual norm during the iterative process with ε = 0.01 for Example 4.1; (3b) reconstruction result with ε = 0.01 where RRMSE=0.0131 and iterative step is k = 3 for Example 4.1.
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Figure 4.
(4a) The residual norm during the iterative process with ε = 0.05 for Example 4.1; (4b) reconstruction result with ε = 0.05 where RRMSE=0.0486 and iterative step is k = 2 for Example 4.1.
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Figure 5.
(5a) The convergence error during the iterative process with ε = 0.1 for Example 4.1; (5b) reconstruction result with ε = 0.10 where RRMSE=0.0967 and iterative step is k = 2 for Example 4.1.
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Figure 6.
The input data g(θ) for Example 4.2.
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Figure 7.
The residual norm during the iterative process for Example 4.2 with relaxation factor c = −0.5 under different noise-amplitude parameters: (7a) ε = 0.01, (7b) ε = 0.05, (7c) ε = 0.10. Stopping iterations: k = 36, 23, 18, respectively.
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Figure 8.
Final relative error (RRMSE) as a function of the relaxation factor c for different noise-amplitude parameters: (8a) ε = 0.01, (8b) ε = 0.05, (8c) ε = 0.10.
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Figure 9.
Results for Example 4.2 with relaxation factor c = −1.8. (a,c,e) Convergence error versus iteration number for ε = 0.01, 0.05, 0.10; (b,d,f) corresponding reconstructed epicardial potential distributions at the optimal stopping iterations.
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Figure 10.
The input data g(θ) for Example 4.3.
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Figure 11.
Results for Example 4.3 with relaxation factor c = −1.3. (a,c,e) Convergence error versus iteration number for ε = 0.01, 0.05, 0.10; (b,d,f) corresponding reconstructed epicardial potential distributions at the optimal stopping iterations.