A numerical method for a Cauchy problem for elliptic partial differential equations

Han, Weimin; Huang, Jianguo; Kazmi, Kamran; Chen, Yu

doi:10.1088/0266-5611/23/6/008

Cited by 9 publications

(8 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In general, problem (8)-(9) may not admit a solution, and if solution exists it may be not unique. However, we can show the following two theorems about the solution of the control problem which are similar to the results in [24]. Theorem 2.2.…”

supporting

confidence: 69%

See 1 more Smart Citation

PDE-constrained optimal control approach for the approximation of an inverse Cauchy problem

Chang¹,

Gong²,

Sun³

et al. 2015

IPI

View full text Add to dashboard Cite

This paper concerns the approximation of a Cauchy problem for the elliptic equation. The inverse problem is transformed into a PDE-constrained optimal control problem and these two problems are equivalent under some assumptions. Different from the existing literature which is also based on the optimal control theory, we consider the state equation in the sense of very weak solution defined by the transposition technique. In this way, it does not need to impose any regularity requirement on the given data. Moreover, this method can yield theoretical analysis simply and numerical computation conveniently. To deal with the ill-posedness of the control problem, Tikhonov regularization term is introduced. The regularized problem is well-posed and its solution converges to the non-regularized counterpart as the regularization parameter approaches zero. We establish the finite element approximation to the regularized control problem and the convergence of the discrete problem is also investigated. Then we discuss the first order optimality condition of the control problem further and obtain an efficient numerical scheme for the Cauchy problem via the adjoint state equation. The paper is ended with numerical experiments.

show abstract

supporting

confidence: 69%

“…They also proved that problem (4)-(6) is equivalent to the original problem (1) under some regularity assumptions on the data, and thus the ill-posedness of the original problem is inherited. Subsequently, Han et al [24] improved the result of [12] by introducing a regularization term to deal with the ill-posedness, which minimizes…”

mentioning

confidence: 99%

PDE-constrained optimal control approach for the approximation of an inverse Cauchy problem

Chang¹,

Gong²,

Sun³

et al. 2015

IPI

View full text Add to dashboard Cite

show abstract

“…When α goes to zero, the error will blow up. But, it is well known that the penalty parameter α is vary small in many applications, such as the regularization process of some ill‐posed problems (see e.g., ). In this case, the error estimate in Theorem is not valid.…”

Section: The a Priori Error Estimatesmentioning

confidence: 99%

Numerical analysis for the approximation of optimal control problems with pointwise observations

Chang

Gong

Yan

2013

Math Methods in App Sciences

View full text Add to dashboard Cite

Communicated by Q. WangIn this paper, we study the numerical methods for optimal control problems governed by elliptic PDEs with pointwise observations of the state. The first order optimality conditions as well as regularities of the solutions are derived. The optimal control and adjoint state have low regularities due to the pointwise observations. For the finite dimensional approximation, we use the standard conforming piecewise linear finite elements to approximate the state and adjoint state variables, whereas variational discretization is applied to the discretization of the control. A priori and a posteriori error estimates for the optimal control, the state and adjoint state are obtained. Numerical experiments are also provided to confirm our theoretical results.of elliptic equations with measure data presented in [16] and [17], a priori error estimates for the control, the state, and the adjoint state are derived. Moreover, the a posteriori error estimates of residual type are presented, which can be applied as indicators for the mesh refinement in adaptive finite element methods. Numerical examples are also included to illustrate the theoretical results. It should be pointed out that our theoretical and numerical results are still valid when the penalty parameter˛is very small, which often appears in many applications, for example, inverse problems.Our paper is organized as follows. In the next section, we present the mathematical setting and formulate the optimal control problems in detail. Finite element approximation to optimal control problems is also presented. In Sections 3 and 4, we prove the a priori error and a posteriori error estimates, respectively. We finally present numerical results, which confirm our analytical findings in section 5.where B is the linear operator from U to L 2 . /, U is the control space. Here, B can be an identity operator such that Bu D u or Bu D ! u where ! is an open subset of and ! the characteristic function of the subset !. z D .z 1 , , z m / 2 R m is given, and X j (j D 1, , m) are space points strictly contained in . Moreover, U ad U is a closed convex subset of typeAccording to the assumptions on and A, for each Bu 2 L 2 . /, we have y.x; u/ 2 H 2 . / \ H 1 0 . /, thus y.x; u/ 2 C. / by imbedding theorem and the aforementioned optimal control problem is well-defined. From standard arguments ([18]), we can prove that the

show abstract

“…Some are based on standard Galerkin formulations, but rely on structured meshes or a special form of the continuous problem for stability [15,24,25]. Some use the above mentioned regularization techniques to ensure stability [3,4,6,7,13] a related approach is to recast the problem as a minimization problem [11,18,17], possibly with regularization.…”

Section: Introductionmentioning

confidence: 99%

A stabilized nonconforming finite element method for the elliptic Cauchy problem

Burman

2016

Math. Comp.

View full text Add to dashboard Cite

In this paper we propose a nonconforming finite element method for the solution of the ill-posed elliptic Cauchy problem. We prove error estimates using continuous dependence estimates in the L 2 -norm. The effect of perturbations in data on the estimates is investigated. The recently derived framework from [8,9] is extended to include the case of nonconforming approximation spaces and we show that the use of such spaces allows us to reduce the amount of stabilization necessary for convergence, even in the case of ill-posed problems.2) where a(u, w) = Ω ∇u · ∇w dx, and l(w) := Ω f w dx + Γ C ψ w ds.

show abstract

A numerical method for a Cauchy problem for elliptic partial differential equations

Cited by 9 publications

References 20 publications

PDE-constrained optimal control approach for the approximation of an inverse Cauchy problem

PDE-constrained optimal control approach for the approximation of an inverse Cauchy problem

Numerical analysis for the approximation of optimal control problems with pointwise observations

A stabilized nonconforming finite element method for the elliptic Cauchy problem

Contact Info

Product

Resources

About