26 research outputs found
Neural Ideal Large Eddy Simulation: Modeling Turbulence with Neural Stochastic Differential Equations
We introduce a data-driven learning framework that assimilates two powerful
ideas: ideal large eddy simulation (LES) from turbulence closure modeling and
neural stochastic differential equations (SDE) for stochastic modeling. The
ideal LES models the LES flow by treating each full-order trajectory as a
random realization of the underlying dynamics, as such, the effect of
small-scales is marginalized to obtain the deterministic evolution of the LES
state. However, ideal LES is analytically intractable. In our work, we use a
latent neural SDE to model the evolution of the stochastic process and an
encoder-decoder pair for transforming between the latent space and the desired
ideal flow field. This stands in sharp contrast to other types of neural
parameterization of closure models where each trajectory is treated as a
deterministic realization of the dynamics. We show the effectiveness of our
approach (niLES - neural ideal LES) on a challenging chaotic dynamical system:
Kolmogorov flow at a Reynolds number of 20,000. Compared to competing methods,
our method can handle non-uniform geometries using unstructured meshes
seamlessly. In particular, niLES leads to trajectories with more accurate
statistics and enhances stability, particularly for long-horizon rollouts.Comment: 18 page
Barriers to sexual health care: a survey of Iranian-American physicians in California, USA
Optimal polynomial smoothers for multigrid V-cycles
The idea of using polynomial methods to improve simple smoother iterations
within a multigrid method for a symmetric positive definite (SPD) system is
revisited. When the single-step smoother itself corresponds to an SPD operator,
there is in particular a very simple iteration, a close cousin of the Chebyshev
semi-iterative method, based on the Chebyshev polynomials of the fourth instead
of first kind, that optimizes a two-level bound going back to Hackbusch. A full
V-cycle bound for general polynomial smoothers is derived using the V-cycle
theory of McCormick. The fourth-kind Chebyshev iteration is quasi-optimal for
the V-cycle bound. The optimal polynomials for the V-cycle bound can be found
numerically, achieving an about 18% lower error contraction factor bound than
the fourth-kind Chebyshev iteration, asymptotically as the number of smoothing
steps goes to infinity. Implementation of the optimized iteration is discussed,
and the performance of the polynomial smoothers are illustrated with a simple
numerical example
Toward robust algebraic multigrid methods for nonsymmetric problems
When analyzing symmetric problems and the methods for solving them, multigrid and algebraic multigrid in particular, one of the primary tools at the analyst's disposal is the energy norm associated with the problem. The lack of this tool is one of the many reasons analysis of nonsymmetric problems and methods for solving them is substantially more difficult than in the symmetric case. We show that there is an analog to the energy norm for a nonsymmetric matrix A, associated with a new absolute value we term the "form" absolute value. This new absolute value can be described as a symmetric positive definite solution to the matrix equation A*|A|-1A = |A|; it exists and is unique in particular whenever A has positive symmetric part. We then develop a novel convergence theory for a general two-level multigrid iteration for any such A, making use of the form absolute value. In particular, we derive a convergence bound in terms of a smoothing property and separate approximation properties for the interpolation and restriction (a novel feature). Finally, we present new algebraic multigrid heuristics designed specifically targeting this new theory, which we evaluate with numerical tests.</p
Toward robust algebraic multigrid methods for nonsymmetric problems
When analyzing symmetric problems and the methods for solving them, multigrid and algebraic multigrid in particular, one of the primary tools at the analyst's disposal is the energy norm associated with the problem. The lack of this tool is one of the many reasons analysis of nonsymmetric problems and methods for solving them is substantially more difficult than in the symmetric case. We show that there is an analog to the energy norm for a nonsymmetric matrix A, associated with a new absolute value we term the "form" absolute value. This new absolute value can be described as a symmetric positive definite solution to the matrix equation A&ast;&verbar;A&verbar;-1A = &verbar;A&verbar;; it exists and is unique in particular whenever A has positive symmetric part. We then develop a novel convergence theory for a general two-level multigrid iteration for any such A, making use of the form absolute value. In particular, we derive a convergence bound in terms of a smoothing property and separate approximation properties for the interpolation and restriction (a novel feature). Finally, we present new algebraic multigrid heuristics designed specifically targeting this new theory, which we evaluate with numerical tests.</p
THE EFFECT OF INTRAPARTICLE DIFFUSION ON RATES OF ADSORPTION IN POROUS SOLIDS
Abstract not availabl