An efficient and accurate parameter identification scheme for inverse Helmholtz problems using SLICM

The inverse Helmholtz problem is crucial in many fields like non-destructive testing and heat conduction analysis, emphasizing the need for efficient numerical solutions. This paper investigates the parameter identification problems of the Helmholtz equation, based on the stabilized Lagrange interpolation collocation method (SLICM) associated with least-squares solution. This method circumvents the limitations of traditional meshfree methods that cannot perform accurate integrations. It offers advantages of high accuracy, good stability, and high computational efficiency, rendering it suitable for solving inverse problems. Additionally, considering potential errors in measurement data, this study employs the least squares method to directly utilize all available information from the measurement data, minimizing errors and avoiding the iterative calculations based on measurement data in the Galerkin methods. To balance the numerical errors among measurement locations, boundaries, and within the domain, this paper studies the optimal weights for the overdetermined system based on the least squares functional obtained through SLICM, achieving a global error balance. Moreover, to further mitigate the noise in measurement data, this paper introduces the Tikhonov regularization technique and selects suitable regularization parameters to process noisy data through the L-curve. Numerical results in 1D, 2D and even 3D complicated domains indicate that SLICM can attain accurate and convergent results in parameter identification, even when the noise level is as high as 10%.