In this paper we provide a general framework for model reduction methods applied to fluid flow in porous media. Using reduced basis and numerical homogenization techniques we show that the complexity of the numerical ...
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ISBN:
(纸本)9783319399294;9783319399270
In this paper we provide a general framework for model reduction methods applied to fluid flow in porous media. Using reduced basis and numerical homogenization techniques we show that the complexity of the numerical approximation of Stokes flow in heterogeneous media can be drastically reduced. The use of such a computational framework is illustrated at several model problems such as two and three scale porous media.
Optimization algorithms typically perform a series of function evaluations to find an approximation of an optimal point of the objective function. Evaluations can be expensive, e. g., if they depend on the results of ...
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ISBN:
(数字)9783319282626
ISBN:
(纸本)9783319282626;9783319282602
Optimization algorithms typically perform a series of function evaluations to find an approximation of an optimal point of the objective function. Evaluations can be expensive, e. g., if they depend on the results of a complex simulation. When dealing with higher-dimensional functions, the curse of dimensionality increases the difficulty of the problem rapidly and prohibits a regular sampling. Instead of directly optimizing the objective function, we replace it with a sparse grid interpolant, saving valuable function evaluations. We generalize the standard piecewise linear basis to hierarchical B-splines, making the sparse grid surrogate smooth enough to enable gradient-based optimization methods. Also, we use an uncommon refinement criterion due to Novak and Ritter to generate an appropriate sparse grid adaptively. Finally, we evaluate the new method for various artificial and real-world examples.
In this paper we investigate the stability of the space-time discontinuous Galerkin method (STDGM) for the solution of nonstationary, linear convectiondiffusion- reaction problem in time-dependent domains formulated w...
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ISBN:
(纸本)9783319399294;9783319399270
In this paper we investigate the stability of the space-time discontinuous Galerkin method (STDGM) for the solution of nonstationary, linear convectiondiffusion- reaction problem in time-dependent domains formulated with the aid of the arbitrary Lagrangian-Eulerian (ALE) method. At first we define the continuous problem and reformulate it using the ALE method, which replaces the classical partial time derivative with the so called ALE-derivative and an additional convective term. In the second part of the paper we discretize our problem using the space-time discontinuous Galerkin method. The space discretization uses piecewise polynomial approximations of degree p >= 1, in time we use only piecewise linear discretization. Finally in the third part of the paper we present our results concerning the unconditional stability of the method.
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