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In this article, we introduce a novel direct sampling indicator function for solving the inverse source problem (ISP) associated with the Helmholtz equation. The proposed method is rigorously evaluated through numerical analysis of both singular and general sources, providing a comprehensive assessment of its performance. Several numerical examples are given to validate the theoretical estimations. [ABSTRACT FROM AUTHOR]

Background Purpose Methods and Materials Results Conclusion Intensity‐modulated proton therapy (IMPT) optimizes spot intensities and position, providing better conformability. However, the successful application of IMPT is dependent upon addressing the challenges posed by range and setup uncertainties. In order to address the uncertainties in IMPT, robust optimization is essential.This study aims to develop a novel fast algorithm for robust optimization of IMPT with minimum monitor unit (MU) constraint.The study formulates a robust optimization problem and proposes a novel, fast algorithm based on the alternating direction method of multipliers (ADMM) framework. This algorithm enables distributed computation and parallel processing. Ten clinical cases were used as test scenarios to evaluate the performance of the proposed approach. The robust optimization method (RBO‐NEW) was compared with plans that only consider nominal optimization using CTV (NMO‐CTV) without handling uncertainties and PTV (NMO‐PTV) to handle the uncertainties, as well as with conventional robust‐optimized plans (RBO‐CONV). Dosimetric metrics, including D95, homogeneity index, and Dmean, were used to evaluate the dose distribution quality. The area under the root‐mean‐square dose (RMSD)–volume histogram curves (AUC) and dose–volume histogram (DVH) bands were used to evaluate the robustness of the treatment plan. Optimization time cost was also assessed to measure computational efficiency.The results demonstrated that the RBO plans exhibited better plan quality and robustness than the NMO plans, with RBO‐NEW showing superior computational efficiency and plan quality compared to RBO‐CONV. Specifically, statistical analysis results indicated that RBO‐NEW was able to reduce the computational time from 389.70±207.40$389.70\pm 207.40$ to 228.60±123.67$228.60\pm 123.67$ s (p<0.01$p<0.01$) and reduce the mean organ‐at‐risk (OAR) dose from 9.38±12.80$9.38\pm 12.80$% of the prescription dose to 9.07±12.39$9.07\pm 12.39$% of the prescription dose (p<0.05$p<0.05$) compared to RBO‐CONV.This study introduces a novel fast robust optimization algorithm for IMPT treatment planning with minimum MU constraint. Such an algorithm is not only able to enhance the plan's robustness and computational efficiency without compromising OAR sparing but also able to improve treatment plan quality and reliability. [ABSTRACT FROM AUTHOR]