173 research outputs found
FATODE: A Library for Forward, Adjoint, and Tangent Linear Integration of ODEs
FATODE is a FORTRAN library for the integration of ordinary differential equations with direct and adjoint sensitivity analysis capabilities.
The paper describes the capabilities, implementation, code organization, and usage of this package.
FATODE implements four families of methods -- explicit Runge-Kutta for nonstiff problems and fully implicit Runge-Kutta, singly diagonally implicit Runge-Kutta, and Rosenbrock for stiff problems.
Each family contains several methods with different orders of accuracy; users can add new methods by simply providing their coefficients.
For each family the forward, adjoint, and tangent linear models are implemented.
General purpose solvers for dense and sparse linear algebra are used; users can easily incorporate problem-tailored linear algebra routines.
The performance of the package is demonstrated on several test problems.
To the best of our knowledge FATODE is the first publicly available general purpose package that offers forward and adjoint sensitivity
analysis capabilities in the context of Runge Kutta methods. A wide range of applications are expected to benefit from its use; examples include parameter estimation,
data assimilation, optimal control, and uncertainty quantification
Comparison of POD reduced order strategies for the nonlinear 2D Shallow Water Equations
This paper introduces tensorial calculus techniques in the framework of
Proper Orthogonal Decomposition (POD) to reduce the computational complexity of
the reduced nonlinear terms. The resulting method, named tensorial POD, can be
applied to polynomial nonlinearities of any degree . Such nonlinear terms
have an on-line complexity of , where is the
dimension of POD basis, and therefore is independent of full space dimension.
However it is efficient only for quadratic nonlinear terms since for higher
nonlinearities standard POD proves to be less time consuming once the POD basis
dimension is increased. Numerical experiments are carried out with a two
dimensional shallow water equation (SWE) test problem to compare the
performance of tensorial POD, standard POD, and POD/Discrete Empirical
Interpolation Method (DEIM). Numerical results show that tensorial POD
decreases by times the computational cost of the on-line stage of
standard POD for configurations using more than model variables. The
tensorial POD SWE model was only slower than the POD/DEIM SWE model
but the implementation effort is considerably increased. Tensorial calculus was
again employed to construct a new algorithm allowing POD/DEIM shallow water
equation model to compute its off-line stage faster than the standard and
tensorial POD approaches.Comment: 23 pages, 8 figures, 5 table
Reverse Engineering of Initial & Boundary Conditions with TELEMAC and Algorithmic Differentiation
Numerical Aspect
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