The mountain-pass theorem is a remarkable result that forms the basis for the calculation of transition states in biology and chemistry. The mountain-pass theorem is also a fundamental tool in nonlinear analysis where it is used to prove existence results for variational problems in infinite-dimensional dynamical systems.
In this talk we describe the background for the mountain-pass theorem, and propose an algorithm -- the elastic string algorithm -- for the computation of mountain-passes in finite-dimensional problems. We analyze the convergence and numerical performance of this algorithm for benchmark problems in chemistry and infinite-dimensional variational problems, and show that any limit limit point of the elastic string algorithm is a path that crosses a critical point at which the Hessian matrix is not positive definite.