J. A. Reed, J. Utke, H. S. Abdel-khalik, "Combining Automatic Differentiation Methods for High Dimensional Nonlinear Models," Preprint ANL/MCS-P2014-0112, January 2012. [pdf]
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. In this manuscript, 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.