# Detecting Inverse Boundaries by Weighted High-Order Gradient Collocation Method
## 1 Introduction  
The origin of the reproducing kernel (RK) approximation can be dated back to the early development of meshfree methods in the 1970s. Among these methods, smooth particle hydrodynamics (SPH) [1] was the beginning, followed by diffuse element methods (DEM) [2] and lastly by element-free Galerkin methods (EFG) [3] and reproducing kernel particle methods (RKPM) [4; 5]. The aforementioned weak-form-based meshfree methods have been successfully applied to solve partial differential equations (PDEs). Nevertheless, integration in the domain and indirect imposition of boundary conditions make these methods cumbersome. Along the direction of solving PDEs, the strong-form collocation methods were proposed almost at the end of the twentieth century [6]. In particular, the reproducing kernel collocation method (RKCM) has become one promising method with the following merits: localized feature in the approximation and well-conditioned systems in solving PDEs in comparison with global functions adopted in the approximation. The applications of RKCM include solving direct problems such as Poisson problems and elasticity problems [7; 8], where a least-squares formulation is introduced to derive the weighted version of RKCM by minimizing errors on the boundary. As it is known that the inverse problems are generally ill-posed and unstable [9], how to effectively ensure the accuracy of solutions while overcoming the disturbance arising in the measurement isof great importance in computational mechanics [10; 11]. In view of the convergence and stability of the weighted RKCM in solving direct problems, [PERSON] and her co-workers first introduced the weighted RKCM to the inverse analyses including inverse Cauchy problems [12], inverse problems with singularity [13], and inverse elasticity problems [14]. For inverse problems of Cauchy type and elasticity, the weights to be imposed on Dirichlet and Neumann boundary conditions were estimated from the error analysis by using the least-squares functional and equivalent quadrature rule via minimization; furthermore, it was shown that the weights on Dirichlet boundary conditions are different for these two types of problems, and such a difference of weights is consistent with those estimated in the direct problems of Cauchy type and elasticity. Nevertheless, computing the second derivatives of RK shape function in solving second-order PDEs is computationally expensive due to sub-functions composed in the RK shape function. To this end, the gradient reproducing kernel approximation was proposed to avoid calculating direct derivatives twice in solving second-order PDEs [15]. Such an efficient gradient RKCM was recently introduced to investigate multiply connected inverse Cauchy problems [16]; it was shown that the accuracy of approximation can be reached if proper weights are imposed on the boundary conditions, which makes similar comments on the boundary weights as given in the analysis of direct problems. Still, the gradient approximation of RK shape functions requires computing the derivatives once in solving the second-order PDEs. As pointed out in the literature [17], by using the gradient reproducing conditions, the higher-order gradient RK shape functions can be constructed implicitly without taking direct derivatives of RK shape functions, which benefits the second- and fourth-order PDEs. To overcome discretization restriction, the harmonic-enriched reproducing kernel approximation is proposed to solve highly oscillatory PDEs, in which the idea of implicit derivatives of RK shape functions is also adopted [18]. The present work further introduces the higher-order gradient reproducing kernel collocation method (HG-RKCM) in the inverse analysis. Under the least-squares framework, the weights on the boundary conditions are investigated. Then, several complex configurations of multiply connected inverse problems are studied to demonstrate the ability of the method in detecting the unknown boundary.