研究目的
To solve the inverse microtomography problem using piecewise uniform regularization in the class of functions with bounded V H variation and to provide a-posteriori error estimates for the approximate solutions.
研究成果
The proposed V H method ensures piecewise uniform convergence for inverse tomography problems with discontinuous solutions, outperforming BV regularization in continuity regions. The new a-posteriori error estimation algorithm is computationally efficient and provides reasonable estimates of solution errors, validated through numerical experiments.
研究不足
The method assumes the exact solution has bounded V H variation and is discontinuous only on a set of zero measure. The computational cost is high, with iterations taking significant time (e.g., 1000 iterations in 593 seconds). The a-posteriori error estimate has a relative accuracy that depends on the choice of n, and it may not be exact.
1：Experimental Design and Method Selection:
The methodology involves using Tikhonov regularization with a special regularizer based on the V H variation norm. The algorithm is designed to ensure piecewise uniform convergence of approximate solutions to the exact solution. A new numerical algorithm for a-posteriori error estimation is proposed, utilizing a function ξn to approximate the error.
2：Sample Selection and Data Sources:
The model problem uses the Shepp-Logan phantom as the exact solution. Data are projections obtained from the Radon transform, with perturbations introduced for error analysis.
3：List of Experimental Equipment and Materials:
A PC with Intel(R) Core(TM) i7-2600 CPU 3.40 GHz and 8GB RAM is used for numerical computations. Software includes MATLAB for matrix operations and optimization.
4：40 GHz and 8GB RAM is used for numerical computations. Software includes MATLAB for matrix operations and optimization.
Experimental Procedures and Operational Workflow:
4. Experimental Procedures and Operational Workflow: The inverse problem is discretized on a grid. The Tikhonov functional is minimized using a conjugate gradients projection method. The regularization parameter is chosen via the discrepancy principle. A-posteriori error estimates are computed by solving an optimization problem to maximize ξn over unit vectors.
5：Data Analysis Methods:
Errors are analyzed using Euclidean and uniform norms. Numerical comparisons are made between V H and BV regularization methods in terms of global and regional errors.
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