We consider a regularization concept for the solution of ill--posed operator equations, where the operator is composed of a continuous and a discontinuous operator. A particular application is level set regularization, where we develop a novel concept of minimizers. The proposed level set regularization is capable of handling changing topologies. A functional analytic framework explaining the splitting of topologies is given. The asymptotic limit of the level set regularization method is an evolution process, which is implemented numerically and the quality of the proposed algorithm is demonstrated by solving an inverse source problem.
In this paper we propose a novel regularization strategy for the local discontinuous Galerkin method to solve the Hamilton-Jacobi equation in the context of level-set reinitialization. The novel regularization idea works in analogy to shock-capturing schemes for discontinuous Galerkin methods, which are based on finite volume sub-cells. In this spirit, the local discontinuous Galerkin method is combined with an upwind/downwind finite volume sub-cell discretization, which is applied in areas of low regularity. To ensure the applicability on unstructured meshes, the finite volume discretization …
A balancing domain decomposition by constraints preconditioner for a hybridizable discontinuous Galerkin discretization of an elliptic optimal control problem
Identifying the discontinuous diffusion coefficient in an elliptic equation with observation data of the gradient of the solution is an important nonlinear and ill-posed inverse problem. Models with total variational (TV) regularization have been widely studied for this problem, while the theoretically required nonsmoothness property of the TV regularization and the hidden convexity of the models are usually sacrificed when numerical schemes are considered in the literature. In this paper, we show that the favorable nonsmoothness and convexity properties can be entirely kept if the well-known …
In this paper, we study a fractional-order variant of the asymptotical regularization method, called {\it Fractional Asymptotical Regularization (FAR)}, for solving linear ill-posed operator equations in a Hilbert space setting. We assign the method to the general linear regularization schema and prove that under certain smoothness assumptions, FAR with fractional order in the range $(1,2)$ yields an acceleration with respect to comparable order optimal regularization methods. Based on the one-step Adams-Moulton method, a novel iterative regularization scheme is developed for the numerical rea…
In this paper, we consider the $α\| \cdot\|_{\ell_1}-β\| \cdot\|_{\ell_2}$ sparsity regularization with parameter $α\geqβ\geq0$ for nonlinear ill-posed inverse problems. We investigate the well-posedness of the regularization. Compared to the case where $α>β\geq0$, the results for the case $α=β\geq0$ are weaker due to the lack of coercivity and Radon-Riesz property of the regularization term. Under certain condition on the nonlinearity of $F$, we prove that every minimizer of $ α\| \cdot\|_{\ell_1}-β\| \cdot\|_{\ell_2}$ regularization is sparse. For the case $α>β\geq0$, if the exact solution i…
In this article we develop and analyze novel iterative regularization techniques for the solution of systems of nonlinear ill--posed operator equations. The basic idea consists in considering separately each equation of this system and incorporating a loping strategy. The first technique is a Kaczmarz-type method, equipped with a novel stopping criteria. The second method is obtained using an embedding strategy, and again a Kaczmarz-type approach. We prove well-posedness, stability and convergence of both methods.
In this paper, we study the convergence of adaptive mixed interior penalty discontinuous Galerkin method for H(curl)-elliptic problems. We first get the mixed model of H(curl)-elliptic problem by introducing a new intermediate variable. Then we discuss the continuous variational problem and discrete variational problem, which based on interior penalty discontinuous Galerkin approximation. Next, we construct the corresponding posteriori error indicator, and prove the contraction of the summation of the energy error and the scaled error indicator. At last, we confirm and illustrate the theoretic…
Recently, inverse problems have attracted more and more attention in computational mathematics and become increasingly important in engineering applications. After the discretization, many of inverse problems are reduced to linear systems. Due to the typical ill-posedness of inverse problems, the reduced linear systems are often ill-posed, especially when their scales are large. This brings great computational difficulty. Particularly, a small perturbation in the right side of an ill-posed linear system may cause a dramatical change in the solution. Therefore, regularization methods should be …
In this paper, we consider the inverse source problem for the time-fractional diffusion equation, which has been known to be an ill-posed problem. To deal with the ill-posedness of the problem, we propose to transform the problem into a regularized problem with L^2 and total variational (TV) regularization terms. Differing from the classical Tikhonov regularization with L^2 penalty terms, the TV regularization is beneficial for reconstructing discontinuous or piecewise constant solutions. The regularized problem is then approximated by a fully discrete scheme. Our theoretical results include: …
We propose a discontinuous Galerkin(DG) method to approximate the elliptic interface problem on unfitted mesh using a new approximation space. The approximation space is constructed by patch reconstruction with one degree of freedom per element. The optimal error estimates in both L2 norm and DG energy norm are obtained, without the typical constraints for DG method on how the interface intersects to the elements in the mesh. Other than enjoying the advantages of DG method, our method may achieve even better efficiency than the conforming finite element method, as illustrated by numerical exam…
We discuss the mathematical modeling and numerical discretization of transport problems on one-dimensional networks. Suitable coupling conditions are derived that guarantee conservation of mass across network junctions and dissipation of a mathematical energy which allows to prove existence of unique solutions. We then consider the space discretization by a hybrid discontinuous Galerkin method which provides a suitable upwind mechanism to handle the transport problem and allows to incorporate the coupling conditions in a natural manner. In addition, the method inherits mass conservation and st…
A singularly perturbed parabolic problem of convection-diffusion type with a discontinuous initial condition is examined. An analytic function is identified which matches the discontinuity in the initial condition and also satisfies the homogenous parabolic differential equation associated with the problem. The difference between this analytical function and the solution of the parabolic problem is approximated numerically, using an upwind finite difference operator combined with an appropriate layer-adapted mesh. The numerical method is shown to be parameter-uniform. Numerical results are pre…
In part I we introduced modified Landweber-Kaczmarz methods and have established a convergence analysis. In the present work we investigate three applications: an inverse problem related to thermoacoustic tomography, a nonlinear inverse problem for semiconductor equations, and a nonlinear problem in Schlieren tomography. Each application is considered in the framework established in the previous part. The novel algorithms show robustness, stability, computational efficiency and high accuracy.
This paper describes and compares some structure preserving techniques for the solution of linear discrete ill-posed problems with the t-product. A new randomized tensor singular value decomposition (R-tSVD) with a t-product is presented for low tubal rank tensor approximations. Regularization of linear inverse problems by truncated tensor eigenvalue decomposition (T-tEVD), truncated tSVD (T-tSVD), randomized T-tSVD (RT-tSVD), t-product Golub-Kahan bidiagonalization (tGKB) process, and t-product Lanczos (t-Lanczos) process are considered. A solution method that is based on reusing tensor Krylo…
In this paper we consider a level set reinitialization technique based on a high-order, local discontinuous Galerkin method on unstructured triangular meshes. A finite volume based subcell stabilization is used to improve the nonlinear stability of the method. Instead of the standard hyperbolic level set reinitialization, the flow of time Eikonal equation is discretized to construct an approximate signed distance function. Using the Eikonal equation removes the regularization parameter in the standard approach which allows more predictable behavior and faster convergence speeds around the inte…
Error Estimates for Arnoldo-Tikhonov Regularization for Ill-Posed Operator Equations
Institute for Industrial Mathematics, Kepler University（工业数学研究所，开普勒大学）；Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences（约翰·拉德纳计算与应用数学研究所，奥地利科学院）；Department of Mathematical Sciences, Kent State University（凯特州立大学数学科学系）
A singularly perturbed parabolic problem of convection-diffusion type with a discontinuous initial condition is examined. A particular complimentary error function is identified which matches the discontinuity in the initial condition. The difference between this analytical function and the solution of the parabolic problem is approximated numerically. A co-ordinate transformation is used so that a layer-adapted mesh can be aligned to the interior layer present in the solution. Numerical analysis is presented for the associated numerical method, which establishes that the numerical method is a…
Discontinuous Galerkin method for an integro-differential equation modeling dynamic fractional order viscoelasticity
Chalmers University of Technology and University of Gothenburg（查尔姆斯理工大学和哥德堡大学）；Technical University of Gabrovo（加布罗沃理工大学）；University of Kurdistan（库尔德斯坦大学）
We consider spectral mixed discontinuous Galerkin finite element discretizations of the Lamé system of linear elasticity in polyhedral domains in $\mathbb{R}^3$. In order to resolve possible corner, edge, and corner-edge singularities, anisotropic geometric edge meshes consisting of hexahedral elements are applied. We perform a computational study on the discrete inf-sup stability of these methods, and especially focus on the robustness with respect to the Poisson ratio close to the incompressible limit (i.e. the Stokes system). Furthermore, under certain realistic assumptions (for analytic da…
We introduce a local adaptive discontinuous Galerkin method for convection-diffusion-reaction equations. The proposed method is based on a coarse grid and iteratively improves the solution's accuracy by solving local elliptic problems in refined subdomains. For purely diffusion problems, we already proved that this scheme converges under minimal regularity assumptions [A. Abdulle and G.Rosilho de Souza, ESAIM: M2AN, 53(4):1269--1303, 2019]. In this paper, we provide an algorithm for the automatic identification of the local elliptic problems' subdomains employing a flux reconstruction strategy…
In this article we investigate the connection between regularization theory for inverse problems and dynamic programming theory. This is done by developing two new regularization methods, based on dynamic programming techniques. The aim of these methods is to obtain stable approximations to the solution of linear inverse ill-posed problems. We follow two different approaches and derive a continuous and a discrete regularization method. Regularization properties for both methods are proved as well as rates of convergence. A numerical benchmark problem concerning integral operators with convolut…
Shallow neural network yields regularization for ill-posed inverse problems
School of Mathematics and Statistics, Beijing Institute of Technology（北京理工大学数学与统计学院）；MSU-BIT-SMBU Joint Research Center of Applied Mathematics, Shenzhen MSU-BIT University（深圳北理莫斯科大学应用数学联合研究中心）
Two-parameter regularization of ill-posed spherical pseudo-differential equations in the space of continuous functions
Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University（中山大学）；Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences（奥地利科学院约翰·拉顿计算与应用数学研究所）；School of Mathematics and Statistics, University of New South Wales（新南威尔士大学数学与统计学院）
Regularization matrices for discrete ill-posed problems in several space-dimensions
University School（大学学校）；Geomathematics Key Laboratory of Sichuan, College of Management Science, Chengdu University of Technology（四川地球数学重点实验室，成都理工大学管理学院）；Department of Mathematics, SAPIENZA Università di Roma（罗马智慧大学数学系）；Department of Mathematical Sciences, Kent State University（肯特州立大学数理科学系）
In this paper we investigate the problem of recovering the source term in an elliptic system from a measurement of the state on a part of the boundary. For the particular interest in reconstructing probably discontinuous sources, we use the standard least squares method with the total variation regularization. The finite element method is then applied to discretize the minimization problem, we show the stability and the convergence of this technique. Furthermore, we have proposed an algorithm to stably solve the minimization problem. We prove the iterate sequence generated by the derived algor…
This paper provides a new regularization method which is particularly suitable for linear exponentially ill-posed problems. Under logarithmic source conditions (which have a natural interpretation in terms of Sobolev spaces in the aforementioned context), concepts of qualifications as well as order optimal rates of convergence are presented. Optimality results under general source conditions expressed in terms of index functions are also studied. Finally, numerical experiments on three test problems attest the better performance of the new method compared to the well known Tikhonov method in i…
