Executive Summary

SimV2 introduces the adaptive multi-fidelity framework that combines low-fidelity (LF) and high-fidelity (HF) simulations. We hope to achieve computational speedups of 8-12× while maintaining prediction accuracy within RMSE < 2.5 K. This module implements co-kriging surrogates, error estimators, and threshold-based fidelity switching logic to optimize the accuracy-cost tradeoff in real-time photovoltaic thermal simulations.

Multi-Fidelity Hierarchy

Fidelity Level Definitions

SimV2 orchestrates between three computational tiers:

Fidelity	Grid Resolution	Physics Model	Cost [s]	Accuracy [K RMSE]
Low (LF)	$25 \times 25 \times 5$	Poisson + Coarse CFD	1.2	3.8
Medium (MF)	$50 \times 50 \times 5$	Diffusion BTE + S₄ DOM	8.5	1.7
High (HF)	$100 \times 100 \times 10$	MC-BTE + Fine LES	97.3	0.5

The fundamental question: When should we promote from LF to HF?

Co-Kriging Surrogate Model

Mathematical Foundation

Co-kriging extends Gaussian Process (GP) regression to leverage correlations between low- and high-fidelity data. Let $y_L(\mathbf{x})$ and $y_H(\mathbf{x})$ denote LF and HF responses at parameter vector $\mathbf{x} = [T_\infty, G, u_\infty, \dots]^T$.

The autoregressive form assumes:

$$ y_H(\mathbf{x}) = \rho \cdot y_L(\mathbf{x}) + \delta(\mathbf{x}) $$

where:

• $\rho$ is the correlation coefficient (trained via maximum likelihood)
• $\delta(\mathbf{x}) \sim \mathcal{GP}(0, k_\delta)$ is the discrepancy GP with covariance kernel $k_\delta$

Prediction Equations

Given training data $\mathcal{D}_L = \{(\mathbf{x}_i^L, y_i^L)\}_{i=1}^{N_L}$ and $\mathcal{D}_H = \{(\mathbf{x}_j^H, y_j^H)\}_{j=1}^{N_H}$, the co-kriging predictor at test point $\mathbf{x}^*$ is:

$$ \begin{aligned} \mu_{CK}(\mathbf{x}^*) &= \rho \mu_L(\mathbf{x}^*) + \mathbf{k}_\delta^T(\mathbf{x}^*) \left[\mathbf{K}_\delta + \sigma_n^2 \mathbf{I}\right]^{-1} (\mathbf{y}_H - \rho \mathbf{y}_L) \\[10pt] \sigma_{CK}^2(\mathbf{x}^*) &= k_\delta(\mathbf{x}^*, \mathbf{x}^*) - \mathbf{k}_\delta^T(\mathbf{x}^*) \left[\mathbf{K}_\delta + \sigma_n^2 \mathbf{I}\right]^{-1} \mathbf{k}_\delta(\mathbf{x}^*) \end{aligned} $$

Squared Exponential Kernel

SimV2 will employ an anisotropic SE kernel with automatic relevance determination (ARD):

$$ k_\delta(\mathbf{x}, \mathbf{x}') = \sigma_f^2 \exp\left(-\frac{1}{2} \sum_{d=1}^{D} \frac{(x_d - x_d')^2}{\ell_d^2}\right) $$

Typical learned values:

• $\ell_{T_\infty} \approx 2.3$ K (ambient temperature)
• $\ell_G \approx 150$ W m$^{-2}$ (irradiance)
• $\ell_{u_\infty} \approx 0.8$ m s$^{-1}$ (wind speed)

Error Estimation Framework

Residual-Based Indicators

SimV2 will compute a posteriori error indicators at each timestep to decide fidelity level. Define the energy balance residual:

$$ \mathcal{R}_E(\mathbf{T}) = \left| \frac{\int_\Omega \left(Q^* - Q_H - Q_E - Q_G\right) dA}{\int_\Omega Q^* \, dA} \right| $$

Criterion 1: If $\mathcal{R}_E > \tau_E = 0.03$ (3% imbalance), promote to HF.

Thermal Gradient Monitoring

Large spatial gradients indicate regions requiring fine resolution:

$$ \mathcal{G}_T = \max_{\mathbf{x} \in \Omega} \|\nabla T(\mathbf{x})\|_2 $$

Criterion 2: If $\mathcal{G}_T > \tau_G = 50$ K m$^{-1}$, promote to HF.

Temporal Rate of Change

Rapid transients (e.g., cloud shadows) necessitate high temporal fidelity:

$$ \dot{T}_{\max} = \max_{\mathbf{x} \in \Omega} \left|\frac{dT(\mathbf{x}, t)}{dt}\right| $$

Criterion 3: If $\dot{T}_{\max} > \tau_{\dot{T}} = 0.5$ K s$^{-1}$, promote to HF.

Adaptive Refinement Strategy

Particle Swarm Optimization (PSO)

SimV2 will use PSO to tune the threshold parameters $\{\tau_E, \tau_G, \tau_{\dot{T}}\}$ offline to maximize computational efficiency subject to accuracy constraints.

Objective Function:

$$ J(\boldsymbol{\tau}) = \frac{1}{N_{\text{test}}} \sum_{i=1}^{N_{\text{test}}} \left[ w_1 \cdot \text{RMSE}_i(\boldsymbol{\tau}) + w_2 \cdot \frac{t_i(\boldsymbol{\tau})}{t_i^{\text{HF}}} \right] $$

where $w_1 = 0.7$, $w_2 = 0.3$ are penalty weights.

PSO Update Rules:

$$ \begin{aligned} \mathbf{v}_j^{(k+1)} &= \omega \mathbf{v}_j^{(k)} + c_1 r_1 (\mathbf{p}_j^{\text{best}} - \boldsymbol{\tau}_j^{(k)}) + c_2 r_2 (\mathbf{g}^{\text{best}} - \boldsymbol{\tau}_j^{(k)}) \\[8pt] \boldsymbol{\tau}_j^{(k+1)} &= \boldsymbol{\tau}_j^{(k)} + \mathbf{v}_j^{(k+1)} \end{aligned} $$

where $\omega = 0.7$ (inertia), $c_1 = c_2 = 1.5$ (cognitive/social factors), and $r_1, r_2 \sim \mathcal{U}(0,1)$.

Multi-Fidelity Workflow

Decision Tree Logic

FOR each timestep t:
    1. Solve LF model → T_LF, Q_LF
    2. Compute indicators: R_E, G_T, dT/dt
    3. Evaluate promotion criteria:
        IF (R_E > τ_E) OR (G_T > τ_G) OR (dT/dt > τ_dT):
            4a. Prolong T_LF to HF grid
            4b. Solve HF model → T_HF, Q_HF
            4c. Update co-kriging surrogate
        ELSE:
            4d. Accept T_LF as solution
            4e. Query co-kriging for uncertainty
        END IF
    5. Advance to t + Δt
END FOR

Hysteresis Prevention

To avoid chattering (rapid LF ↔ HF switching), SimV2 enforces a cooldown period:

$$ t_{\text{next\_HF}} \geq t_{\text{current}} + \Delta t_{\text{min}} = 5 \, \text{min} $$

References

1. Kennedy, M. C., & O'Hagan, A. (2000). Predicting the output from a complex computer code when fast approximations are available. Biometrika, 87(1), 1-13.
2. Forrester, A. I., Sóbester, A., & Keane, A. J. (2008). Engineering Design via Surrogate Modelling. Wiley.
3. Peherstorfer, B., Willcox, K., & Gunzburger, M. (2018). Survey of multifidelity methods in uncertainty propagation, inference, and optimization. SIAM Review, 60(3), 550-591.
4. Kennedy, J., & Eberhart, R. (1995). Particle swarm optimization. Proc. IEEE Int. Conf. Neural Networks, 4, 1942-1948.