In this paper, an orthogonal basis for the space of linear splines based on linear B-splines is introduced. Some properties and modifications of this basis are investigated, then operational matrices of integration in collocation points are obtained using stable formula. Theoretical considerations are discussed. Applications to the numerical solutions for some linear and nonlinear inverse problems are given, in which the approximations are obtained using the first and second integrals of orthogonal linear splines that lead to an efficient solution procedure. For solving linear problems, the proposed method is combined with the Tikhonov regularization. Also, the trust-region-dogleg method is used for nonlinear equations. Numerical results validate the effectiveness of the orthogonal linear spline basis for linear and nonlinear inverse problems.

在本文中，引入了基于线性B样条的线性样条空间的正交基。研究了该基的一些性质和修改，然后利用稳定公式得到了配置点积分运算矩阵。讨论了理论考虑。给出了某些线性和非线性逆问题的数值解的应用，其中使用正交线性样条的一阶和二阶积分获得近似值，从而产生有效的求解过程。为了解决线性问题，所提出的方法与吉洪诺夫正则化相结合。此外，信赖域狗腿法用于非线性方程。数值结果验证了正交线性样条基对线性和非线性反问题的有效性。