Comparative analysis of the numerical methods for 3D continuation problem for parabolic equation with data on the part of the boundary

The problem of continuation of the solution of a three-dimensional parabolic equation with data given on a time-like surface is investigated. Two numerical methods for solving the continuation problem are compared: the finite-difference scheme inversion and the solution of inverse problem by gradient method. The functional gradient formula is obtained. The results of numerical calculations are presented.

A Posteriori Fractional Tikhonov Regularization Method for the Problem of Analytic Continuation

In this paper, the numerical analytic continuation problem is addressed and a fractional Tikhonov regularization method is proposed. The fractional Tikhonov regularization not only overcomes the difficulty of analyzing the ill-posedness of the continuation problem but also obtains a more accurate numerical result for the discontinuity of solution. This article mainly discusses the a posteriori parameter selection rules of the fractional Tikhonov regularization method, and an error estimate is given. Furthermore, numerical results show that the proposed method works effectively.

Analytic continuation of noisy data using Adams Bashforth residual neural network

<p style='text-indent:20px;'>We propose a data-driven learning framework for the analytic continuation problem in numerical quantum many-body physics. Designing an accurate and efficient framework for the analytic continuation of imaginary time using computational data is a grand challenge that has hindered meaningful links with experimental data. The standard Maximum Entropy (MaxEnt)-based method is limited by the quality of the computational data and the availability of prior information. Also, the MaxEnt is not able to solve the inversion problem under high level of noise in the data. Here we introduce a novel learning model for the analytic continuation problem using a Adams-Bashforth residual neural network (AB-ResNet). The advantage of this deep learning network is that it is model independent and, therefore, does not require prior information concerning the quantity of interest given by the spectral function. More importantly, the ResNet-based model achieves higher accuracy than MaxEnt for data with higher level of noise. Finally, numerical examples show that the developed AB-ResNet is able to recover the spectral function with accuracy comparable to MaxEnt where the noise level is relatively small.</p>

Regularization of a continuation problem for electrodynamic equations

The problem of continuation of a solution of electrodynamic equations from the time-like half-plane S=\{x\in\mathbb{R}^{3}\mid x_{3}=0\} inside the half-space \mathbb{R}^{3}_{+}=\{x\in\mathbb{R}^{3}\mid x_{3}>0\} is considered. A regularization method for a solution of this problem with approximate data is proposed, and the convergence of this method for the class of functions that are analytic with respect to space variables is stated.

Space time stabilized finite element methods for a unique continuation problem subject to the wave equation

We consider a stabilized finite element method based on a spacetime formulation, where the equations are solved on a global (unstructured) spacetime mesh. A unique continuation problem for the wave equation is considered, where a noisy data is known in an interior subset of spacetime. For this problem, we consider a primal-dual discrete formulation of the continuum problem with the addition of stabilization terms that are designed with the goal of minimizing the numerical errors. We prove error estimates using the stability properties of the numerical scheme and a continuum observability estimate, based on the sharp geometric control condition by Bardos, Lebeau and Rauch. The order of convergence for our numerical scheme is optimal with respect to stability properties of the continuum problem and the interpolation errors of approximating with polynomial spaces. Numerical examples are provided that illustrate the methodology.

Approximate Schur-Block ILU Preconditioners for Regularized Solution of Discrete Ill-Posed Problems

High order iterative methods with a recurrence formula for approximate matrix inversion are proposed such that the matrix multiplications and additions in the calculation of matrix polynomials for the hyperpower methods of orders of convergence p=4k+3, where k≥1 is integer, are reduced through factorizations and nested loops in which the iterations are defined using a recurrence formula. Therefore, the computational cost is lowered from κ=4k+3 to κ=k+4 matrix multiplications per step. An algorithm is proposed to obtain regularized solution of ill-posed discrete problems with noisy data by constructing approximate Schur-Block Incomplete LU (Schur-BILU) preconditioner and by preconditioning the one step stationary iterative method. From the proposed methods of approximate matrix inversion, the methods of orders p=7,11,15,19 are applied for approximating the Schur complement matrices. This algorithm is applied to solve two problems of Fredholm integral equation of first kind. The first example is the harmonic continuation problem and the second example is Phillip’s problem. Furthermore, experimental study on some nonsymmetric linear systems of coefficient matrices with strong indefinite symmetric components from Harwell-Boeing collection is also given. Numerical analysis for the regularized solutions of the considered problems is given and numerical comparisons with methods from the literature are provided through tables and figures.