73 lines
2.6 KiB
ReStructuredText
73 lines
2.6 KiB
ReStructuredText
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McKean–Vlasov — propagation of chaos
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====================================
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Interacting-particle Euler scheme for $dX_t = θ(\bar X_t - X_t) dt + σ dW_t$ (`mean_reverting_mckean_vlasov`). The empirical mean is preserved; the empirical variance approaches the diffusion-only equilibrium.
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.. note:: Companion executed notebook: `14_mckean_vlasov.ipynb <../../examples/notebooks/14_mckean_vlasov.ipynb>`_
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14 — McKean–Vlasov mean-reverting dynamics
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==========================================
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.. code-block:: python
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import numpy as np
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import matplotlib.pyplot as plt
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from optimizr import _core as opt
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plt.rcParams['figure.figsize'] = (7, 4)
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plt.rcParams['figure.dpi'] = 110
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.. code-block:: python
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init = np.linspace(-2.0, 2.0, 200).tolist()
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init_mean = float(np.mean(init))
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res = opt.mean_reverting_mckean_vlasov(
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initial=init, theta=1.0, sigma=0.1,
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n_steps=1000, t_horizon=1.0, seed=42,
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)
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n_t = res['n_steps']; n_p = res['n_particles']
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X = np.array(res['paths_flat']).reshape(n_t, n_p)
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tg = np.array(res['time_grid'])
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print('initial mean =', init_mean)
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print('final mean =', float(X[-1].mean()))
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print('final std =', float(X[-1].std()))
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.. code-block:: python
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fig, ax = plt.subplots()
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ax.plot(tg, X[:, ::20], color='tab:blue', alpha=0.2, lw=0.6)
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ax.plot(tg, X.mean(axis=1), color='red', lw=2, label='empirical mean')
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ax.axhline(init_mean, color='k', ls=':', label='initial mean')
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ax.set_xlabel('t'); ax.set_ylabel('X^i_t'); ax.legend(); ax.grid(alpha=0.3)
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ax.set_title('Mean-reverting McKean–Vlasov — 200 particles')
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fig.tight_layout(); plt.show()
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.. image:: ../_static/v2/mckean_vlasov/plot_01.png
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:align: center
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:width: 80%
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.. code-block:: python
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fig, ax = plt.subplots()
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ax.hist(X[0], bins=30, alpha=0.5, label='t = 0', density=True)
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ax.hist(X[-1], bins=30, alpha=0.5, label='t = T', density=True)
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ax.set_xlabel('x'); ax.set_ylabel('empirical density'); ax.legend(); ax.grid(alpha=0.3)
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ax.set_title('Marginal density at t = 0 and t = T')
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fig.tight_layout(); plt.show()
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.. image:: ../_static/v2/mckean_vlasov/plot_02.png
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:align: center
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:width: 80%
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**Verified:** empirical mean stays within `0.05` of the initial mean.
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API
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---
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.. code-block:: rust
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pub fn simulate_mckean_vlasov<B>(initial: &[f64], drift: B, cfg: &McKeanVlasovConfig) -> Result<McKeanVlasovResult>
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where B: Fn(f64, &[f64]) -> f64;
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pub struct McKeanVlasovConfig { pub n_particles: usize, pub n_steps: usize, pub t_horizon: f64, pub sigma: f64, pub seed: u64 }
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pub struct McKeanVlasovResult { pub paths: Array2<f64>, pub time_grid: Array1<f64> }
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