Abstract
As efficient separation of variables plays a central role in model reduction for nonlinear and nonaffine parameterized systems, we propose a stochastic discrete empirical interpolation method (SDEIM) for this purpose. In our SDEIM, candidate basis functions are generated through a random sampling procedure, and the dimension of the approximation space is systematically determined by a probability threshold. This random sampling procedure avoids large candidate sample sets for high-dimensional parameters, and the probability based stopping criterion can efficiently control the dimension of the approximation space. Numerical experiments are conducted to demonstrate the computational efficiency of SDEIM, which include separation of variables for general nonlinear functions, e.g., exponential functions of the Karhu nen–Loève (KL) expansion, and constructing reduced order models for FitzHugh–Nagumo equations, where symmetry among limit cycles is well captured by SDEIM.
Highlights
When conducting model reduction for nonlinear and nonaffine parameterized systems [1], separation of variables is an important step
In our stochastic discrete empirical interpolation method (SDEIM), the interpolation is processed through two steps: the first is to randomly select sample points to construct an approximation space for empirical interpolation, and the second is to evaluate if the approximation accuracy meets the probability threshold on additional samples
With a focus on randomized computational methods, we in this paper propose a stochastic discrete empirical interpolation method (SDEIM)
Summary
When conducting model reduction for nonlinear and nonaffine parameterized systems [1], separation of variables is an important step. Rather than reduced basis approximations, we in this paper focus on separation of variables for nonlinear and nonaffine systems, and propose a stochastic discrete empirical interpolation method (SDEIM). In our SDEIM, the interpolation is processed through two steps: the first is to randomly select sample points to construct an approximation space for empirical interpolation, and the second is to evaluate if the approximation accuracy meets the probability threshold on additional samples. These two steps are repeated until the approximation space satisfies the given accuracy and probability requirements.
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