Assefaw Gebremedhin, Research, Automatic Differentiation

Many algorithms for nonlinear optimization, solution of differential equations, sensitivity analysis, and uncertainity quantification rely on computation of derivatives. Such algorithms can greatly benefit from Algorithmic (or Automatic) Differentiation (AD), a technology for transforming a computer program for computing a function into a program for computing the function's derivatives. AD furnishes derivatives accurately (i.e. without truncation errors) and with relatively little effort from a user's end, and the time and space complexity of computing derivatives using AD can be bounded by the complexity of evaluating the function itself. For these reasons, AD is a better alternative to other methods of computing derivatives such as finite differencing or symbolic differentiation, or error-prone and tedious hand coding.

Reverse-mode based Hessian computation

Currently together with Mu Wang and Alex Pothen, I am working on new algorithms for Hessian computation that aim at taking full advantage of the potential of the Reverse-mode of AD and the symmetry available in the computational graph of the Hessian.

Sparse computation via compression

Much of my previous work in Automatic Differentiation aimed at exploiting the sparsity (and when applicable, the symmetry) that is inherently available in large derivative matrices -- Jacobians and Hessians -- in order to make their computation efficient in terms of runtime and memory requirement.

A fundamental technique that has proven effective in achieving this objective is computation via compression. Loosely speaking, the idea is to reduce computational cost by calculating sums of columns of a derivative matrix at a time instead of calculating one column at a time. Columns that are to be computed together are determined by exploiting structural properties of the matrix -- in particular, the structural information is used to partition the set of columns into a small number of groups of ``independent'' columns. Here, a variety of matrix partitiong problems arise depending on whether the matrix to be computed is Jacobian (nonsymmetric) or Hessian (symmetric) and the specifics of the computational approach taken. Graph coloring models have proven to be a powerful tool for analyzing the complexity of and designing effective algorithms for these problems since the pioneering works of Coleman and More in the early 1980s.


Below is a catalogued list of papers (recent ones listed first) I have written together with a number of collaborators on graph-theoretic results and innovative algorithms for (a) efficient Reverse-mode based Hessian computation and (b) matrix partitioning (coloring) and related problems in sparse derivative matrix computation. Also incuded is information on the associated serial software package ColPack we developed in support of sparse derivative computation.

Reverse Mode-based Hessian computation Coloring and sparse derivative computation, sequential algorithms Sparsity in Source to Source Transformation AD tools Coloring and sparse derivative computation, parallel algorithms

ColPack is the name of our software package comprising implementations of sequential algorithms for a variety of graph coloring and related problems arising in sparse Jacobian and Hessian computation. Many of the papers listed above, most notably the first paper, discuss the design, analysis, implementation, and performance evaluation of the underlying coloring algorithms in ColPack. In addition to coloring, ColPack has various routines for recovering the numerical values of the entries of a derivative matrix from a compressed representation as well as routines for constructing graph representations of Jacobians and Hessians from sparsity patterns provided in various formats. ColPack is written in C++ heavily using the Standard Template Library. It is designed to be modular, extensible and efficient.

ColPack has been interfaced with the operator overloading based AD tool ADOL-C, which is currently being developed and maintained at Paderborn University, Germany. Recently, ColPack has also been interfaced with the source-to-source transformation AD tool ADIC2, which is developed at Argonne National Laboratory. The ColPack-AD toolkit is being used for end-to-end sparse derivative computation, beginning with automatic detection of the sparsity pattern of the derivative matrix of a function given as a computer program and ending with the recovery of the entries of the original derivative matrix from its compressed representation.

Examples of applications enabled by ColPack together with an AD tool include Simulated Moving Bed optimization in chemical engineering and electric power flow optimization.

ColPack is released for free public use under the GNU Lesser General Public License. Versions subsequent to its first release feature added functionalities and improved performance (code optimizations). For download and other more detailed information on ColPack, visit the Software page.

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