Remarks and Future Directions Sample Clauses

Remarks and Future Directions. We have considered using a weighted-GCV method in Lanczos-hybrid meth- ods for solving large-scale ill-posed problems. The W-GCV method requires choosing yet another parameter, so we proposed and implemented an adap- tive, automatic approach for choosing this parameter. We have demonstrated through a variety of test problems that our approach is effective in stabilizing semi-convergence behavior. Our Matlab implementation used to generate the results presented in this chapter is named HyBR to represent Hybrid Bidiagonalization Regularization. We briefly describe its usage here. The main code is HyBR.m and can be called in the following way: >> [x, output] = HyBR(A, b, P, options); where the required inputs are A and b, as defined in Section 2.1. The optional input variables are P, which is a preconditioner, and options, which is a structure that includes input parameters. A parameter structure containing default parameters can be obtained using the HyBRset.m code. In particular, >> [options] = HyBRset(‘HyBR’); An electronic copy of the Matlab codes can be obtained from the following website: xxxx://xxx.xxxxxx.xxxxx.xxx/software/HyBR . Several open questions remain. With the ability to obtain near opti- mal solutions, Lanczos-hybrid methods should have a significant impact on many applications. Recently, Xxxxxx, Xxxxxx and Espan˜ol [87] suggested a projection-based algorithm that can be implemented for more general regu- larization operators. We can treat this iterative method as a hybrid method LSQR (no regularization) GCV W−GCV with ω=0.01 W−GCV, adaptive ω LSQR (no regularization) GCV W−GCV with ω=0.2 W−GCV, adaptive ω 1 1 0.8 0.8 relative error (Baart) (noise level = 10%) relative error (Heat) (noise level = 10%) 0.6 0.6 0.4 0.4 0.2 0.2 LSQR (no regularization) GCV W−GCV with ω=0.05 W−GCV, adaptive ω LSQR (no regularization) GCV W−GCV with ω=0.4 W−GCV, adaptive ω 0.8 0.8 relative error (Baart) (noise level = 1%) relative error (Heat) (noise level = 1%) 0.6 0.6 0.4 0.4 0.2 0.2 0 10 20 30 40 50 60 70 iteration 0 10 20 30 40 50 60 70 LSQR (no regularization) GCV W−GCV with ω=0.1 W−GCV, adaptive ω LSQR (no regularization) GCV W−GCV with ω=0.8 W−GCV, adaptive ω 1 1 0.8 0.8 relative error (Baart) (noise level = 0.1%) relative error (Heat) (noise level = 0.1%) 0.6 0.6 0.4 0.4 0.2 0.2 Figure 2.9: Relative errors for different noise levels. The plots on the left correspond to the Baart example, and the plots on the right correspond to the Heat example. Optimal c...
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