Studying Kolmogorov flow with persistent homology

Transcription

1 Studying Kolmogorov flow with persistent homology ATHDDA, University of Victoria, Victoria, BC August 21, 215

2 A model for turbulence in 2D: Kolmogorov Flow. The velocity field u(x, y, t) is given by where: u t + βu u = 1 ρ p + ν 2 u αu + f, (1) ρ is the fluid density p is the pressure ν is the viscosity u =. f = χ sin(κy)ˆx is the forcing that drives the flow β and α are parameters to take into account 3D effects (commonly present in experiments)

3 Useful to re-write Equation (1) in terms of the z-component of the vorticity field ω = ( u) ˆk, a scalar field, to get ω t + βu ω = ν 2 ω αω + χκ cos(κy). (2) For this study, we make the following choices: β =.83 ν = m 2 /s α =.63 s 1 ρ = 959 kg/m 3 λ = 2π/κ =.254 m The strength of the forcing is then parameterized by a dimensionless parameter called the Reynolds number, λ Re = 3 χ 8ν 2.

4 ω

5 Impose periodic boundary conditions in both the x and y directions: ω(x, y) = ω(x + L x, y) ω(x, y) = ω(x, y + L y ) where L x =.85 m and L y = 4λ =.116 m are the dimensions of the domain in the x and y directions, respectively. Equation (2) with these boundary conditions is invariant under any combination of the following coordinate transformations: Translation along x: T δx (x, y) = (x + δx, y) Reflection about the x axis followed by half-period shift along y: D(x, y) = ( x, y + λ/2) Rotation by π: R(x, y) = ( x, y) seen as a rotation by π about the z-axis (vorticity axis)

6 Consequence: each solution to Equation (2) corresponds to a set of solutions which are dynamically equivalent ω (a) (b) (c) Figure: One of these things just doesn t belong...

7 Challenge: identify solutions that are symmetry-related to study relative equilibria (REQ) and relative periodic orbits (RPO). Two approaches: Fourier methods Persistent homology Two examples:

8 REQ: Weakly turbulent regime at Re = R R 1 R Projection onto the real parts of the three dominant Fourier modes. Gray line indicates chaotic evolution of the flow, influenced by the presence of unstable fixed points, indicated in red.

9 RPO: Flow exhibits a steady relative periodic orbit at Re = I I 1 I Projection onto the imaginary parts of the three dominant Fourier modes. Gray line indicates the evolution of an RPO.

10 General idea: Project the scalar fields to the space of persistence diagrams Study the dynamics in the space of persistence diagrams Q: Why persistent homology? A: Invariance under coordinate transformations.

11 Step. Generate the data. REQ: Choose initial solutions from chaotic trajectory and use Newton s method to solve for fixed points. Get 67 fixed points. RPO: Numerically integrate the equation and sample the trajectory at equally-spaced time points. Get 5 samples.

12 Step 1. Project each scalar field (image) to a vector of persistence diagrams PD 3 2 PD ω ω Death ω = 1.5 ω = ω =.75 ω = ω Birth ω Death ω = 1.5 ω = ω =.75 ω = ω Birth (a) (b) (c) Figure: (a) Vorticity plot and persistence diagrams for (b) H and (c) H 1. (Diagram for H 2 not shown.) Video.

13 Step 2. Choose a metric (e.g. d B, d Wp ) and generate the distance matrix D ij corresponding to the projected set of scalar fields in the space of persistence diagrams. Bottleneck Distance d B (PD, PD ) = max k inf γ : PD k PD k sup p γ(p), p PD k where (a, b ) (a 1, b 1 ) := max{ a a 1, b b 1 } and γ ranges over all bijections between persistence points. Degree-p Wasserstein Distance d W p(pd, PD ) = k inf γ : PD k PD k p γ(p) p p PD k 1/p.

14 ω (a) (b) (c) PD PD Death Death 2 (a) (b) (c) Birth 2 (a) (b) (c) Birth

15 PD PD Death Death 2 (a) (b) (c) Birth 2 (a) (b) (c) Birth d B d W 2 d W 1 (PD a, PD b ) (PD a, PD c ) Table: Distances between selected persistence diagrams (rounded to 3 decimal places).

16 Solution Solution (a) Sample point Sample point Figure: Distance matrices using d B for (a) fixed-point solutions and (b) samples taken from stable RPO. (b)

17 Step 3. Generate a filtration of Vietoris-Rips complexes from D ij. Vietoris-Rips Complex Given a point cloud X = {x,..., x N } in a metric space with distance function d, the Vietoris-Rips complex at scale θ, denoted R(X, θ), is the simplicial complex defined by the collection of simplicies {x n,..., x nk } R(X, θ) if and only if d(x ni, x nj ) 2θ, for all i, j {, 1, 2,..., k}. Definition only relies on the distance matrix! Filtration sweeps through the scales θ. Video.

18 A detailed look at our computational framework so far... Pick a metric! Mathematical object: Scalar field Point cloud Input data: Bitmap image Distance matrix Complex structure: Cubical complex Vietoris-Rips complex Filtration: Sublevelset filtration Distance filtration Output data: Persistence diagrams Persistence diagrams Collection of persistence diagram vectors!

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20 Step 4. Compute the persistence diagrams corresponding to the Vietoris-Rips filtration of the point cloud in the space of persistence diagrams. Step 5. Analyze the results.

21 REQ: Clustering symmetry-related equilibria PD Solution Death.2.1 Multiplicity = Solution Birth Existing method using Fourier modes confirms seven clusters of solutions after identifying symmetry-related solutions.

22 REQ: Clustering symmetry-related equilibria. Solution Death PD Multiplicity = Solution Birth Using Fourier method confirms seven distinct clusters of solutions after identifying symmetry-related solutions.

23 RPO: Studying a relative periodic orbit. Video..15 PD.15 PD 1 Death.1 Death Birth Birth

24 For a closer look, visit arxiv.org

25 What about more complicated dynamics?

26 Rayleigh-Bénard Convection: Almost-periodic orbit. Video. 3 PD 3 PD 1 Death 2 1 Death Birth Birth

27 Rayleigh-Bénard Convection: Spiral-defect chaos.

28 Many thanks to... My collaborators: Rutgers University Miroslav Kramár Konstantin Mischaikow Georgia Institute of Technology Jeffrey Tithof Balachandra Suri Michael F. Schatz Virginia Tech Mu Xu Mark Paul The software creators: Vidit Nanda (Perseus) Shaun Harker (Subsampling/Cluster-delegator) Miro Kramár (Diffusion Map projection) Jonathan Reeve (who taught me all about scripting) Funding: NSF, AFOSR, and DARPA.