Combining automatic differentiation methods for high-dimensional nonlinear models

Abstract

Earlier work has shown that the efficient subspace method can be employed to reduce the effective size of the input data stream for high-dimensional models when the effective rank of the first-order sensitivity matrix is orders of magnitude smaller than the size of the input data. Here, the method is extended to handle nonlinear models, where the evaluation of higher-order derivatives is important but also challenging because the number of derivatives increases exponentially with the size of the input data streams. A recently developed hybrid approach is employed to combine reverse-mode automatic differentiation to calculate first-order derivatives and perform the required reduction in the input data stream, followed by forward-mode automatic differentiation to calculate higher-order derivatives with respect only to the reduced input variables. Three test cases illustrate the viability of the approach. © 2012 Springer-Verlag.