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This paper studies an inverse problem of recovering an unknown diffusion coefficient in a parabolic equation. We adopt a total variation regularization method to deal with the ill-posedness. This m...

In solving Bayesian inverse problems, it is often desirable to use a common density parameterization to characterize the prior and posterior. Typically, we seek an optimal approximation of the true posterior from the same distribution family as the prior. As one of the most important classes of distributions in statistics, the exponential family is considered as the parameterization. The optimal parameter values for representing the approximated posterior are achieved by minimizing the deviation between the parameterized density and a homotopy that deforms the prior density into the posterior density. Instead of the usual moment parameters, we introduce a new parameter system in the process, by which we reduce the dimension of the parameter and therefore the computational cost. The parameters are ruled by a system of explicit first-order differential equations. Solving this system over finite “time” interval yields the desired optimal density parameters. This method is proven to be effective using some numerical examples.

We address stabilization problems of networked control systems in the presence of bounded packet losses and time-varying network-induced delays. When the delay is less than the transmission interval, we prove that the stability of the closed-loop system is implied by that of a switched system. Based on theories on switched systems, we present stabilizing conditions by using packet-loss-dependent Lyapunov functions, and design multiple controllers depending on the packet loss and the network-induced delay. Furthermore, based on the stabilizing conditions, we discuss the robustness of the stabilization against the packet loss and the network-induced delay. Several numerical examples and simulations are worked out to demonstrate the effectiveness of the proposed design approach.

In this article, we investigate some Cauchy problems associated with Helmholtz-type equations in rectangle using the method of truncation. Not alike the previous literature, we propose to choose the regularization parameter by an a posteriori rule, with which we can obtain error estimates of Holder-type between the exact solution and their regularized approximation. Numerical simulations show that the proposed scheme is effective and stable.

In this paper, we consider an analytic continuation problem on a strip domain with the data given approximately only on the real axis. The Gauss mollification method is proposed to solve this problem. An a priori error estimate between the exact solution and its regularized approximation is obtained. Moreover, we also propose a new a posteriori parameter choice rule and get a good error estimate. Several numerical examples are provided, which show the method works effectively.

In this paper, we consider the application of the ensemble Kalman filter for some nonlinear operator equations. According to the structural feature of the nonlinear operator, we construct an iteration process and the corresponding linear observation operator. This construction puts our problem into the frame of 3DVar. Based on the linear observation operator, the ensemble Kalman filter approach is adopted to blend the data into the dynamics to obtain an approximation of the unknown parameter. The method is applied to an inverse potential problem and a nonlinear Fredholm integral equation. The numerical results are compared with Bayesian approach, classical regularization methods including Tikhonov regularization and Landweber iteration, which shows that the proposed algorithm is effective and competitive.

The ill-posed problem of attempting to recover the temperature functions from one measured transient data temperature at some interior point of a one-dimensional semi-infinite conductor when the governing linear diffusion equation is of fractional type is discussed. A simple regularization method based on Dirichlet kernel mollification techniques is introduced. We also proposea priorianda posterioriparameter choice rules and get the corresponding error estimate between the exact solution and its regularized approximation. Moreover, a numerical example is provided to verify our theoretical results.

An inverse source identification problem for a time fractional diffusion equation is discussed. The unknown heat source is supposed to be space dependent only. Based on the use of Green’s function, an effective numerical algorithm is developed to recover both the intensities and locations of unknown point sources from final measurements. Numerical results indicate that the proposed method is efficient and accurate.

In this paper, the problem of reconstruction of the solution of nonlinear ill-posed problem F(x)= y by Tikhonov regularization method is considered, where instead of y noisy data yδ with ∥y − yδ∥≤ δ are given and F:D(F)⊂X→Y is a nonlinear operator. A priori parameter choice rule and two a posteriori parameter choice rules are suggested. The order optimal convergence rate is O((−logδ)−p) under logarithmic-type source conditions. Moreover, we apply this method to an inverse boundary identification problem for verifying some of the theoretical results.