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The h-p version of the continuous Petrov-Galerkin time stepping
method is analyzed for nonlinear initial value problems. An
L∞-error bound explicit with respect to the local
discretization and regularity parameters is derived. Numerical examples are
provided to illustrate the theoretical results.
Conjugate symplectic eigenvalue problems arise in solving discrete linear-quadratic optimal control problems and discrete algebraic Riccati equations. In this article, backward errors of approximate pairs of conjugate symplectic matrices are obtained from their properties. Several numerical examples are given to illustrate the results.
A weakly over-penalized symmetric interior penalty method is applied to solve elliptic eigenvalue problems. We derive a posteriori error estimator of residual type, which proves to be both reliable and efficient in the energy norm. Some numerical tests are provided to confirm our theoretical analysis.
An inverse diffraction problem is considered. Both classical Tikhonov regularisation and a slow-evolution-from-the-continuation-boundary (SECB) method are used to solve the ill-posed problem. Regularisation error estimates for the two methods are compared, and the SECB method is seen to be an improvement on the classical Tikhonov method. Two numerical examples demonstrate their feasibility and efficiency.
Image registration is an ill-posed problem that has been studied widely in recent
years. The so-called curvature-based image registration method is one of the
most effective and well-known approaches, as it produces smooth solutions and
allows an automatic rigid alignment. An important outstanding issue is the
accurate and efficient numerical solution of the Euler-Lagrange system of two
coupled nonlinear biharmonic equations, addressed in this article. We propose a
fourth-order compact (FOC) finite difference scheme using a splitting operator
on a 9-point stencil, and discuss how the resulting nonlinear discrete system
can be solved efficiently by a nonlinear multi-grid (NMG) method. Thus after
measuring the h-ellipticity of the nonlinear discrete operator involved by a
local Fourier analysis (LFA), we show that our FOC finite difference method is
amenable to multi-grid (MG) methods and an appropriate point-wise smoothing
procedure. A high potential point-wise smoother using an outer-inner iteration
method is shown to be effective by the LFA and numerical experiments. Real
medical images are used to compare the accuracy and efficiency of our approach
and the standard second-order central (SSOC) finite difference scheme in the
same NMG framework. As expected for a higher-order finite difference scheme, the
images generated by our FOC finite difference scheme prove significantly more
accurate than those computed using the SSOC finite difference scheme. Our
numerical results are consistent with the LFA analysis, and also demonstrate
that the NMG method converges within a few steps.
Convergence analysis is presented for recently proposed multistep schemes, when applied to a special type of forward-backward stochastic differential equations (FB-SDEs) that arises in finance and stochastic control. The corresponding k-step scheme admits a k-order convergence rate in time, when the exact solution of the forward stochastic differential equation (SDE) is given. Our analysis assumes that the terminal conditions and the FBSDE coefficients are sufficiently regular.