Optimization Tools for Inverse Problems Using the Nonlinear L- and A-curve

We consider a new idea for solving Tikhonov regularized discretized ill-posed problems. The optimization problem is formulated as a nonlinear least squares problems containing the Tikhonov regularization parameter λ. In order to find the size of the regularization parameter and attain good convergence in the optimization method we use the nonlinear Land a-curve. The nonlinear L-curve is a direct generalization of the linear L-curve and can be used to find a good regularized solution. The a-curve is the Tikhonov function as a function of the regularization parameter and is most useful in monitoring the global convergence of the method. Our model algorithm for solving the Tikhonov problem is to use a linearization around the best attained point xk (possibly given by the nonlinear L-curve) giving a linear Land a-curve. Following the trajectory of the solution to this linear problem the new point chosen is the one that gives sufficient decrease in the size of the residual. NOMENCLATURE λ The regularization parameter. α Step length in optimization method. xc The center for the regularization. xk Approximation of the Tikhonov problem at iteration k. t(x);y(x) Size of the residual and solution. Address all correspondence to this author. J(x) The Jacobian ∂ f=∂x. tk;yk; fk;Jk Abbreviations for t(xk);y(xk); f (xk) and ∂ f ∂x (xk). λ̄k The regularization parameter used as an upper limit for the choice of regularization parameter in step k. t̄k; ȳk The point on the linear L-curve minimizing determining λ̄k. INTRODUCTION We consider nonlinear equations of the form f (x) = 0; f : R ! R : (1) In our case (1) is a discrete version of an ill-posed infinite dimensional problem. Characteristic for such ill-posed problems are that the singular values of the Jacobian J = ∂ f=∂x decrease rapidly to zero without any useful gap. This fact prevents the efficient use of standard methods such as the Gauss-Newton method. Therefore, we will use the Tikhonov problem min x T (x;λ); T (x;λ) = t(x)+λy(x) λ 0 (2)