Abstract:We focus on the nonhomogeneous backward heat problem of finding the initial temperature θ = θ(x, y) = u(x, y, ) such thatwhere Ω = ( , π) × ( , π). In the problem, the source f = f(x, y, t) and the final data h = h(x, y) are determined through random noise data g ij (t) and d ij satisfying the regression modelswhere (X i , Y j ) are grid points of Ω. The problem is severely ill-posed. To regularize the instable solution of the problem, we use the trigonometric least squares method in nonparametric regression associated with the projection method. In addition, convergence rate is also investigated numerically.
In this paper, we deal with the Cauchy problem for the modified Helmholtz equation. We consider two models of data: the bounded variance model and the i.i.d. model. The trigonometric estimators of nonparametric regression is applied to solve the problem. In addition, the general forms of regularization parameter corresponding to the pointwise mean squared error and the mean integrated squared error are discussed in detail. The minimax rate convergence corresponding to the bounded variance model is also presented. In the i.i.d. model, we construct the asymptotic confidence interval for the solution of the problem. Finally, we give some numerical experiments and discuss the obtained results
Let be a bounded domain of R n . In this paper, we consider a final value problem for the nonlinear parabolic equationwhere g, h are given functions and the numbers a, b (b > 0) are modeling parameters. The problem does not fulfill Hadamard's postulates of well posedness: it might not have a solution in the strict sense; its solutions might not be unique or might not depend continuously on the data. Hence, its mathematical analysis is subtle. However, it has many applications in physics and other fields. For this reason, a regularization for the problem is proposed. In our problem, the function f (u) = au − bu 3 is not globally Lipschitz. So, we cannot apply directly recent methods that have been used in ). We have to approximate the function f by a globally Lipschitz function and use an approximate equation to find the regularization solution of the problem. Error estimations between the exact solution and the approximate solution, established from noise data g ε , a δ , b δ , are given.
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