Preconditioning. 4.2.1. The fuel tank(s) shall be filled with the specified test fuel. If the existing fuel in the fuel tank(s) does not meet the specifications contained in paragraph 3.4.1. of this annex, the existing fuel shall be drained prior to the fuel fill. The test fuel shall be at a temperature less than or equal to 289 K (+16 °C). For the above operations the evaporative emission control system shall neither be abnormally purged nor abnormally loaded.
Preconditioning. 1 We have seen in Chapter 5 that the nonlinear iterative solver based on the mixed formulation of the Xxxxxxx problem converges – for “academic” test cases – within a small number of iterations and is robust with respect to the regularization parameter ε and the size of the mesh. Also, the solver works well for the non-regularized case ε = 0. The price of solving the mixed formulation as opposed to the formulation in primitive variables is that additional unknowns augment the linear system, that is, the linear systems to be solved at each nonlinear iteration are larger than those for the formulation in primitive variables. Note that our numerical experiments from Chapter 5 are performed on two dimensional test problems on a relatively small scale and a direct method is used for solving the linear systems. Direct methods, however, are usually unsuitable for realistic problems which are normally on a larger scale be- cause of their poor scaling properties and an excessive memory requirement for larger systems. This is in particular the case when solving three dimensional problems, when direct methods become unaffordable even for problems of moderate size. The purpose of this chapter is to introduce a more sophisticated linear solver which is suitable for problems on larger scale, in particular for problems in three dimensions. 1The work presented in this chapter was performed in close collaboration with Panayot Vas- silevski, Center for Applied Scientific Computing, Xxxxxxxx Livermore National Laboratory, Liver- more, CA. More precisely, we will solve linear systems arising from the non-regularized Xxxxxxx problem using a Krylov subspace method. We will introduce a preconditioner which results in a fast and robust convergence of the linear iterative solver within a small number of iterations independent of the mesh size. We begin this chapter by explaining the concept of two important methods in nu- xxxxxxx analysis, multigrid methods and domain decomposition methods. We then provide an overview over the linear systems to be solved and the existing methods for preconditioning them in Section 6.3. In Section 6.4 we suggest using the regularized Xxxxxxx problem as a preconditioner for solving the non-regularized problem. For efficient preconditioning of the linear system, the regularized Xxxxxxx problem needs to be approximated. This can be done in different ways. We present two of these possibilities in this chapter, the inexact factorization of the system in...
Preconditioning. Every preconditioner matrix P for ill-posed inverse problems should satisfy the following general properties: • P is relatively inexpensive to construct, • It is relatively inexpensive to solve linear systems of the form Px = y, • The preconditioned system should satisfy P−1Kr ≈ I, where Kr = Q∗ΦΛQ. The last condition, refers to the rate of convergence, which is faster when more of the large eigenvalues are clustered around 1. The process of image deconvolution requires solving linear systems with highly structured matrices, including circu- lant, Toeplitz and Xxxxxx. Preconditioners for such systems have been widely studied in the literature [26, 80, 81]. In this work we consider a method, where the preconditioner is constructed to be a circulant approximation of matrix K. When the PSF is spatially invariant, then there exist very inexpensive methods [27, 28] (based on FFTs) to construct the circulant preconditioner by solving one of the following minimizations problems min ǁK − PǁF or min ǁK − Pǁ1 (2.16) over all circulant matrices, P. For spatially variant blurs a similar approach is used, however, in that case we construct a preconditioner using a single PSF that is the average of all given PSFs. Although this technique is not optimal, it works well in practice and makes the implementation much easier than creating a separate preconditioner for each PSF [83].
Preconditioning. Cheap approximations of the system matrix are useful for iterative methods that can be accelerated by preconditioners (for an introduction to preconditioning, we recommend the books [42, pp. 275-368], [21, pp. 650-666], and [6, pp. 688-709], as well as the survey [5]). We tested each method as a preconditoner for a preconditioned conjugate gradient least squares (PCGLS) method applied to a 256 × 256 image deconvolution problem with the Satellite Example PSF. For this problem, since each method gave comparable results, we were concerned with how quick the solution was. In particular, we separated out the time to construct the operator from the time required to converge iterations. We present the results for each method in their specific chapter, and summarize the results in Appendix C.
Preconditioning. Now we illustrate that preconditioning can significantly accelerate conver- gence of the iterative method. We only show results for the Gaussian blurred image with synthetic boundary conditions; similar results can be obtained with motion blur. Synthetic BC deblurred image zoom: table zoom: face Synthetic BC with preconditioning Figure 2.17: Deblurring results on Gaussian blurred “Xxxxxxx” with syn- thetic boundary conditions. The first row is obtained without precondition- ing at the 500th iteration; the second row is obtained with preconditioning at the 20th iteration. Table 2.6: PSNRs of deblurring results with and without preconditioning. Blurred image Synthetic Synthetic with preconditioning PSNR 24.5681 28.7532 29.6790 iteration - 500 20 Figure 2.18: Plot of the deblurring errors with and without preconditioning. The deblurring results with and without preconditioning are shown in Figure 2.17, the corresponding PSNRs are shown in Table 2.6, and the er- ror plots are shown in Figure 2.18. Recall that without preconditioning, the minimum error is not yet attained even at 500th iteration. With precondi- tioning, the relative error drops very quickly, attaining its minimum at 20th iteration before increasing a little, and then levels off. In addition, we obtain a higher PSNR and recover more details, e.g. the texture of the chair.
Preconditioning. The choice of the preconditiong matrix M is crucial for solving the optimization problem (5.18). If we do not have a preconditioner or we choose the preconditioner M as identity, the original Hessian for the weighted least squares problem is expressed as
Preconditioning. The reordering method performs nearly identically to the truncation method in the preconditioning test. The setup of the reordering method, which requires computing the approximate TSVD, took very slightly longer, and both methods took the same number of iterations to converge. Computing the TSVD took on average 1.22 seconds. In total, completing the 7 iterations took 0.202 seconds, for a total computation time from start to finish of 1.43 seconds. This total time is only .10 second slower than the truncation method, which is not significant enough to conclude the reordering method will consistently be slower for this test problem. So, like the truncation method, the reordering method is considerably faster than not using a preconditioner but slower than the baseline method, with the most significant difference in the setup time. As with the truncation method, the reordering method may be preferable over the baseline in cases when multiple right-hand sides are used, or when K is poorly approximated by K1. Appendix C details the differences between the performance of each preconditioner. No preconditioner Reordering Iterations 345 7 Setup time (sec) 0.0 1.22 Calculate time (sec) 6.04 0.208 Total time (sec) 6.04 1.43 Table 5.1: Reordering Method Timings for PCGLS. A summary of the timing results for the PCLGS problem, with the reordering method used as a preconditioner, is shown. The reordering method saves time and takes fewer iterations compared to an unpreconditioned system. The reordering method is an effective preconditioner overall, reducing the number of iterations and time taken to converge compared to not using a preconditioner.
Preconditioning. Unlike the baseline method, the truncation method does not construct a full set of singular vectors and values by default. For preconditioning, this can cause prob- lems. For example if we use a low-rank preconditioner M to precondition a full rank system Kx + e = d, we get the system MKx + Me = Md. The product MK has a lower rank than K, making this problem under-determined. Similarly, right-preconditioning causes the restored solution to exist in a smaller subspace than the original problem. When using a low rank preconditioner, the rank must be high enough to capture most of the information about the problem. With this restriction, we used a rank of 1520 for this problem, split into A = 38 and m = 40. The rest of the parameters for this example are identical to the baseline method. It took 1.10 seconds on average to construct the truncated preconditioner, and the iterations took a total of 0.179 seconds to converge, with 7 iterations re- quired for convergence. So, in total, the PCG method converged in 1.28 seconds for the truncation method. This total time is slower than the using the baseline as a pre- conditioner but faster than no preconditioner. The time taken on iterations is faster for the truncation method than the baseline method, which indicates that the trun- cation method may be more useful in cases where there are many right-hand sides. Table 4.1 shows the comparative times for no preconditioner versus the truncation method. See Appendix C for a full comparison between all methods. No preconditioner Truncated Iterations 345 7 Setup time (sec) 0.0 1.10 Calculate time (sec) 6.04 0.179 Total time (sec) 6.04 1.28 Table 4.1: Truncation Preconditioner Timings. A summary of the timing results for the PCLGS problem, with the truncation method used as a preconditioner, is shown. The truncation method saves time and takes fewer iterations compared to an unpreconditioned system. Despite the added computational cost compared to the baseline method, the trun- cation method is an effective preconditioner. In particular, it dramatically decreases the number of iterations required for convergence.
