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

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&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

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&amp;ast;&amp;verbar;A&amp;verbar;-1A = &amp;verbar;A&amp;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