Abstract. We develop a local regularization theory for the nonlinear inverse autoconvolution problem. Unlike classical regularization techniques such as Tikhonov regularization, this theory provides regularization methods that preserve the causal nature of the autoconvolution problem, allowing for fast sequential numerical solution ( O(rN 2 − r 2 N ) flops where r N for the method discussed in this paper as applied to the nonlinear problem; in comparison, the cost for Tikhonov regularization applied to a general linear problem is O(N 3 ) flops). We prove the convergence of the regularized solutions to the true solution as the noise level in the data shrinks to zero and supply convergence rates for the case of both L 2 and continuous data. We propose several regularization methods and provide a theoretical basis for their convergence; of note is that this class of methods does not require an initial guess of the unknown solution. Our numerical results confirm effectiveness of the methods, with results comparing favorably to numerical examples found in the literature for the autoconvolution problem (e.g., [13] for examples using Tikhonov regularization with total variation constraints, and [16] for examples using the method of Lavrent'ev); this especially seems to be true when it comes to the recovery of sharp features in the unknown solution. We also show the effectiveness of our method in cases not covered by the theory.
Local regularization methods allow for the application of sequential solution techniques for the solution of Volterra problems, retaining the causal structure of the original Volterra problem and leading to fast solution techniques. Stability and convergence of these methods was shown to hold on a large class of linear Volterra problems, i.e., the class of ν-smoothing problems for ν = 1, 2, . . . in Lamm (2005 Inverse Problems 21 785-803). In this paper, we enlarge the family of convergent local regularization methods to include sequential versions of classical regularization methods such as sequential Tikhonov regularization. In fact, sequential Tikhonov regularization was considered earlier by Lamm and Eldén (1997 SIAM J. Numer. Anal. 34 1432-50) but there the theory was limited to the class of discretized one-smoothing Volterra problems. An interesting feature of sequential classical regularization methods is that they involve two regularization parameters: the usual local regularization parameter r controls the size of the local problem while a second parameter α controls the amount of regularization to be applied in each subproblem. This approach suggests a wavelet type of regularization method with the parameter r controlling spatial resolution and α controlling frequency resolution. In this paper, we also show how the 'future polynomial regularization' method of Cinzori ( 2004Inverse Problems 20 1791-806) can be viewed as a special case of the general framework of Lamm (2005) in the 1-smoothing case. In addition, we extend the results of Lamm (2005) to nonlinear Volterra problems of Hammerstein type and give numerical results to illustrate the effectiveness of the method in this case.
Recent studies reveal that Allee effect may play important roles in the growth of tumor. We present one of the first mathematical models of avascular tumor that incorporates the weak Allee effect. The model considers the densities of tumor cells in three stages: proliferating cells, quiescent cells, and necrotic cells. We investigate how Allee effect impacts the growth of the avascular tumor. We also investigate the effect of apoptosis of proliferating cells and necrosis of quiescent cells. The system is numerically solved in 2D using different sets of parameters. We show that Allee effect and apoptosis play important roles in the growth of tumor and the formation of necrotic core.
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