645 research outputs found
Efficient Higher Order Derivatives of Objective Functions Composed of Matrix Operations
This paper is concerned with the efficient evaluation of higher-order
derivatives of functions that are composed of matrix operations. I.e., we
want to compute the -th derivative tensor , where is given as an algorithm that
consists of many matrix operations. We propose a method that is a combination
of two well-known techniques from Algorithmic Differentiation (AD): univariate
Taylor propagation on scalars (UTPS) and first-order forward and reverse on
matrices. The combination leads to a technique that we would like to call
univariate Taylor propagation on matrices (UTPM). The method inherits many
desirable properties: It is easy to implement, it is very efficient and it
returns not only but yields in the process also the derivatives
for . As performance test we compute the gradient
% and the Hessian by a combination of forward
and reverse mode of f(X) = \trace (X^{-1}) in the reverse mode of AD for . We observe a speedup of about 100 compared to
UTPS. Due to the nature of the method, the memory footprint is also small and
therefore can be used to differentiate functions that are not accessible by
standard methods due to limited physical memory
On the efficient computation of high-order derivatives for implicitly defined functions
Scientific studies often require the precise calculation of derivatives. In
many cases an analytical calculation is not feasible and one resorts to
evaluating derivatives numerically. These are error-prone, especially for
higher-order derivatives. A technique based on algorithmic differentiation is
presented which allows for a precise calculation of higher-order derivatives.
The method can be widely applied even for the case of only numerically
solvable, implicit dependencies which totally hamper a semi-analytical
calculation of the derivatives. As a demonstration the method is applied to a
quantum field theoretical physical model. The results are compared with
standard numerical derivative methods.Comment: 11 pages, 4 figures, to appear in Comput. Phys. Commu
Validated Computation of the Local Truncation Error of Runge-Kutta Methods with Automatic Differentiation
International audienceIn this paper, we propose a novel approach to bound the local truncation error based on the order condition which is usable for explicit and implicit Runge-Kutta methods
Total Generalized Variation for Manifold-valued Data
In this paper we introduce the notion of second-order total generalized
variation (TGV) regularization for manifold-valued data in a discrete setting.
We provide an axiomatic approach to formalize reasonable generalizations of TGV
to the manifold setting and present two possible concrete instances that
fulfill the proposed axioms. We provide well-posedness results and present
algorithms for a numerical realization of these generalizations to the manifold
setup. Further, we provide experimental results for synthetic and real data to
further underpin the proposed generalization numerically and show its potential
for applications with manifold-valued data
Automating embedded analysis capabilities and managing software complexity in multiphysics simulation part I: template-based generic programming
An approach for incorporating embedded simulation and analysis capabilities
in complex simulation codes through template-based generic programming is
presented. This approach relies on templating and operator overloading within
the C++ language to transform a given calculation into one that can compute a
variety of additional quantities that are necessary for many state-of-the-art
simulation and analysis algorithms. An approach for incorporating these ideas
into complex simulation codes through general graph-based assembly is also
presented. These ideas have been implemented within a set of packages in the
Trilinos framework and are demonstrated on a simple problem from chemical
engineering
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