Cauchy Problem For Elliptic Equations Research Articles

On the reconstruction of cardiac transmembrane potential pattern from body surface measurement

Electrocardiographic imaging (ECGI) is a diagnostic tool designed for the noninvasive detection of electrical activity in the heart. Mathematically, this imaging process can be modelled by an inverse problem for the coupled elliptic system with Cauchy data. For the bidomain model describing the electrical activity in the myocardium, we reconstruct the transmembrane potential on the heart surface from ECG recordings. The reconstruction process is split into two steps: firstly computing the electrical potential and current on heart surface from torso surface recordings and then recovering the transmembrane potential on the heart surface. The first step is essentially a Cauchy problem for elliptic equation which is well-known to be severely ill-posed. We realize this step in terms of boundary integral system by a quasi-regularization scheme to deal with the ill-posedness. The second step is implemented by an integral equation of the second kind with nontrivial null space. We remove the non-uniqueness and then provide a numerical scheme by the boundary element method to obtain the transmembrane potential on the heart surface. The regularized solution obtained from noisy measurement data is proven rigorously to be convergent to the exact solution as the measurement error in the torso surface tends to zero. For solving the regularizing system numerically, we establish quadrature formulas for boundary potentials for 3-dimensional bio-tissue in terms of the concept of solid angles. The numerical realizations for different configurations are finally presented to show the validity of the proposed scheme.

Direct numerical algorithm for calculating the heat flux at an inaccessible boundary

Abstract A fast numerical algorithm for solving the Cauchy problem for elliptic equations with variable coefficients in standard calculation domains (rectangles, circles, or rings) is proposed. The algorithm is designed to calculate the heat flux at the inaccessible boundary. It is based on the separation of variables method. This approach employs a finite difference approximation and allows obtaining a solution to a discrete problem in arithmetic operations of the order of N ⁢ ln ⁡ N N\operatorname{ln}N , where 𝑁 is the number of grid points. As a rule, iterative procedures are needed to solve the Cauchy problem for elliptic equations. The currently available direct algorithms for solving the Cauchy problem have been developed only for (Laplace, Helmholtz) operators with constant coefficients and for use of analytical solutions for problems with such operators. A novel feature of the results of the present paper is that the direct algorithm can be used for an elliptic operator with variable coefficients (of a special form). It is important that in this case no analytical solution to the problem can be obtained. The algorithm significantly increases the range of problems that can be solved. It can be used to create devices for determining in real time heat fluxes on the parts of inhomogeneous constructions that cannot be measured. For example, to determine the heat flux on the inner radius of a pipe made of different materials.

Accelerated Dirichlet–Robin alternating algorithms for solving the Cauchy problem for the Helmholtz equation

Abstract The Cauchy problem for Helmholtz equation, for moderate wave number $k^{2}$, is considered. In the previous paper of Achieng et al. (2020, Analysis of Dirichlet–Robin iterations for solving the Cauchy problem for elliptic equations. Bull. Iran. Math. Soc.), a proof of convergence for the Dirichlet–Robin alternating algorithm was given for general elliptic operators of second order, provided that appropriate Robin parameters were used. Also, it has been noted that the rate of convergence for the alternating iterative algorithm is quite slow. Thus, we reformulate the Cauchy problem as an operator equation and implement iterative methods based on Krylov subspaces. The aim is to achieve faster convergence. In particular, we consider the Landweber method, the conjugate gradient method and the generalized minimal residual method. The numerical results show that all the methods work well. In this work, we discuss also how one can approach non-symmetric differential operators by using similar operator equations and model problems which are used for symmetric differential operators.

Least squares formulation for ill-posed inverse problems and applications

In this paper we propose a least squares formulation for ill-posed inverse problems. For example, ill-posed inverse problems in partial differential equations are those such that a solution of inverse problem exists for a smooth data but it does not depend continuously on data, or there is no solution for inverse problem, e.g. the Cauchy problem for elliptic equations and backward solution of parabolic equations. We develop the least squares formulation in which the sum of equations error over domain and data fitting criterion and Tikhonov regularization terms is minimized over entire solutions. In this way we can establish the existence and uniqueness of an inverse solution and establish the continuity of the inverse solution for noisy data in . The method can be applied to a general class of non-linear inverse problems and an operator theoretic stability analysis is developed. We describe how one can apply the method for various PDE inverse problems. Numerical tests using backward heat equation and Cauchy problem for elliptic equations are presented to demonstrate the applicability and performance of the proposed method.

Constructing a variational quasi‐reversibility method for a Cauchy problem for elliptic equations

In the recent developments of regularization theory for inverse and ill‐posed problems, a variational quasi‐reversibility (QR) method has been designed to solve a class of time‐reversed quasi‐linear parabolic problems. Known as a PDE‐based approach, this method relies on adding a suitable perturbing operator to the original problem and consequently, on gaining the corresponding fine stabilized operator, which leads us to a forward‐like problem. In this work, we establish new conditional estimates for such operators to solve a prototypical Cauchy problem for elliptic equations. This problem is based on the stationary case of the inverse heat conduction problem, where one wants to identify the heat distribution in a certain medium, given the partial boundary data. Using the new QR method, we obtain a second‐order initial value problem for a wave‐type equation, whose weak solvability can be deduced using a priori estimates and compactness arguments. Weighted by a Carleman‐like function, a new type of energy estimates is explored in a variational setting when we investigate the Hölder convergence rate of the proposed scheme. Besides, a linearized version of this scheme is analyzed. Numerical examples are provided to corroborate our theoretical analysis.

Analysis of Dirichlet\u2013Robin Iterations for Solving the Cauchy Problem for Elliptic Equations

The Cauchy problem for general elliptic equations of second order is considered. In a previous paper (Berntsson et al. in Inverse Probl Sci Eng 26(7):1062–1078, 2018), it was suggested that the alternating iterative algorithm suggested by Kozlov and Maz’ya can be convergent, even for large wavenumbers k^2, in the Helmholtz equation, if the Neumann boundary conditions are replaced by Robin conditions. In this paper, we provide a proof that shows that the Dirichlet–Robin alternating algorithm is indeed convergent for general elliptic operators provided that the parameters in the Robin conditions are chosen appropriately. We also give numerical experiments intended to investigate the precise behaviour of the algorithm for different values of k^2 in the Helmholtz equation. In particular, we show how the speed of the convergence depends on the choice of Robin parameters.

Generalized-Fractional Tikhonov-Type Method for the Cauchy Problem of Elliptic Equation

This article researches an ill-posed Cauchy problem of the elliptic-type equation. By placing the a-priori restriction on the exact solution we establish conditional stability. Then, based on the generalized Tikhonov and fractional Tikhonov methods, we construct a generalized-fractional Tikhonov-type regularized solution to recover the stability of the considered problem, and some sharp-type estimates of convergence for the regularized method are derived under the a-priori and a-posteriori selection rules for the regularized parameter. Finally, we verify that the proposed method is efficient and acceptable by making the corresponding numerical experiments.

An implicit iterative method for numerical solution of the Cauchy problem for elliptic equations

We propose and study some implicit gradient two-layer iterative method for numerical solution of the Cauchy problem for elliptic equations. The results of computational experiments are presented.

An Implicit Iterative Method for Numerical Solution of the Cauchy Problem for Elliptic Equations

An Implicit Iterative Method for Numerical Solution of the Cauchy Problem for Elliptic Equations

A finite difference method for the very weak solution to a Cauchy problem for an elliptic equation

Abstract We introduce the concept of very weak solution to a Cauchy problem for elliptic equations. The Cauchy problem is regularized by a well-posed non-local boundary value problem whose solution is also understood in a very weak sense. A stable finite difference scheme is suggested for solving the non-local boundary value problem and then applied to stabilizing the Cauchy problem. Some numerical examples are presented for showing the efficiency of the method.

Regularization method for an ill-posed Cauchy problem for elliptic equations

Abstract The paper is devoted to investigating a Cauchy problem for homogeneous elliptic PDEs in the abstract Hilbert space given by u ′′ ⁢ ( t ) - A ⁢ u ⁢ ( t ) = 0 {u^{\prime\prime}(t)-Au(t)=0} , 0 < t < T {0<t<T} , u ⁢ ( 0 ) = φ {u(0)=\varphi} , u ′ ⁢ ( 0 ) = 0 {u^{\prime}(0)=0} , where A is a positive self-adjoint and unbounded linear operator. The problem is severely ill-posed in the sense of Hadamard [23]. We shall give a new regularization method for this problem when the operator A is replaced by A α = A ⁢ ( I + α ⁢ A ) - 1 {A_{\alpha}=A(I+\alpha A)^{-1}} and u ⁢ ( 0 ) = φ {u(0)=\varphi} is replaced by a nonlocal condition. We show the convergence of this method and we construct a family of regularizing operators for the considered problem. Convergence estimates are established under a priori regularity assumptions on the problem data. Some numerical results are given to show the effectiveness of the proposed method.

A new general filter regularization method for Cauchy problems for elliptic equations with a locally Lipschitz nonlinear source

Up to now, studies on the semi-linear Cauchy problem for elliptic partial differential equations needed to assume that the source term present in the governing equation is a global Lipschitz function. The current paper is the first investigation to not only the more general but also the more practical case of interest when the source term is only a local Lipschitz function. In such a situation, the methods of solution from the previous studies with a global Lipschitz source term are not directly applicable and therefore, novel ideas and techniques need to be developed to tackle the local Lipschitz nonlinearity. This locally Lipschitz source arises in many applications of great physical interest governed by, for example, the sine-Gordon, Lane–Emden, Allen–Cahn and Liouville equations. The inverse problem is severely ill-posed in the sense of Hadamard by violating the continuous dependence upon the input Cauchy data. Therefore, in order to obtain a stable solution we consider theoretical aspects of regularization of the problem by a new generalized filter method. Under some priori assumptions on the exact solution, we prove and obtain rigorously convergence estimates.

A modified integral equation method of the nonlinear elliptic equation with globally and locally Lipschitz source

The paper is devoted to investigating a Cauchy problem for nonlinear elliptic PDEs in the abstract Hilbert space. The problem is hardly solved by computation since it is severely ill-posed in the sense of Hadamard. We shall use a modified integral equation method to regularize the nonlinear problem with globally and locally Lipschitz source terms. Convergence estimates are established under priori assumptions on exact solution. A numerical test is provided to illustrate that the proposed method is feasible and effective. These results extend some earlier works on a Cauchy problem for elliptic equations

A new regularization method for Cauchy problem of elliptic equation

In this paper, we study a Cauchy problem of an elliptic equation in a multi-dimensional case. Such a problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. To overcome the ill-posedness of the model problem, a new regularization method is applied to solve it. The corresponding error estimates between the exact solution and its regularization approximation are given. Finally, two numerical experiments are provided to illustrate the main results.

Modified Tikhonov Method for Cauchy Problem of Elliptic Equation with Variable Coefficients

A Cauchy problem for the elliptic equation with variable coefficients is considered. This problem is severely ill-posed. Then, we need use the regularization techniques to overcome its ill-posedness and get a stable numerical solution. In this paper, we use a modified Tikhonov regularization method to treat it. Under the a-priori bound assumptions for the exact solution, the convergence estimates of this method are established. Numerical results show that our method works well.

A novel mixed group preserving scheme for the inverse Cauchy problem of elliptic equations in annular domains

In this paper, the inverse Cauchy problems for elliptic equations, including the Laplace equation, the Poisson equation, and the Helmholtz equation, defined in annular domains are investigated. When the outer boundary of an annulus is imposed by overspecified boundary data, we seek unknown data in the inner boundary through a combination of the spring-damping regularization method (SDRM) and the mixed group-preserving scheme (MGPS). Several numerical examples are examined to show that the MGPS plus the SDRM can overcome the ill-posed behavior of this highly ill-conditioned inverse Cauchy problem. The presently proposed novel algorithm has good efficiency and stability against the disturbance from large random noise even up to 50%, and the computational cost of MGPS is very time saving.

Remarks on certain singular perturbations with ill-posed limit in shell theory and elasticity

Some problems of elasticity and shell theory are considered. Thecommon feature of these problems is the presence of a smallparameter $\varepsilon$. If $\varepsilon>0$ the correspondingequations are elliptic and the boundary conditions satisfy theShapiro - Lopatinsky condition. When $\varepsilon=0$, this conditionis violated and the problem can be non-solvable in the distributionspaces. The rather difficult passing to the limit is studied usingthe related Cauchy problem for elliptic equations. This approachallows to show that the most important is the transition zone wherethe frequencies $|\xi|\asymp \log (\varepsilon^{-1})$.

A quasi-boundary-value method for the Cauchy problem for elliptic equations with nonhomogeneous Neumann data

Abstract A Cauchy problem for elliptic equations with nonhomogeneous Neumann data in a cylindrical domain is investigated in this paper. For the theoretical aspect the a-priori and a-posteriori parameter choice rules are suggested and the corresponding error estimates are obtained. About the numerical aspect, for a simple case results given by two methods based on the discrete Sine transform and the finite difference method are presented; an idea of left-preconditioned GMRES (Generalized Minimum Residual) method is proposed to deal with the high dimensional case to save the time; a view of dealing with a general domain is suggested. Some ill-posed problems regularized by the quasi-boundary-value method are listed and some rules of this method are suggested.

A Variational Method and Approximations of a Cauchy Problem for Elliptic Equations

A Cauchy problem for general elliptic second-order linear partial differential equations in which the Dirichlet data in H1/2(Γ1 U Γ3) is assumed available on a larger part of the boundary Γ of the bounded domain Ω than the boundary portion Γ1 on which the Neumann data is is investigated using a conjugate gradient method. We obtain an approximation to the solution of the Cauchy problem by minimizing a certain discrete functional and interpolating using the finite diference or boundary element method. The minimization involves solving equations obtained by discretising mixed boundary value problems for the same operator and its adjoint. It is proved that the solution of the discretised optimization problem converges to the continuous one, as the mesh size tends to zero. Numerical results are presented and discussed.

A non-local boundary value problem method for the Cauchy problem for elliptic equations

Let H be a Hilbert space with norm ‖ ⋅ ‖, A:D(A) ⊂ H → H a positive definite, self-adjoint operator with compact inverse on H, and T and ϵ given positive numbers. The ill-posed Cauchy problem for elliptic equations is regularized by the well-posed non-local boundary value problem with a ⩾ 1 being given and α > 0 the regularization parameter. A priori and a posteriori parameter choice rules are suggested which yield order-optimal regularization methods. Numerical results based on the boundary element method are presented and discussed.