DEEMI — the improved variant¶

DEEMI (imported as from DEEM.DEEMI import DEEM) is an extended implementation of the same algorithm. It keeps the published behaviour by default where it matters, and adds opt-in improvements aimed at expensive, rugged objectives such as constitutive-model calibration, where a single evaluation can take many minutes and the cost function is discontinuous.

All additions are backward compatible: setting method_reset="density" and adapt=False reproduces the original algorithm.

What DEEMI adds¶

Hybrid restart — method_reset¶

In addition to the original density restart ("density"), DEEMI offers a hybrid restart ("hybrid", the default):

• the global best is kept exactly (true elitism across the restart);
• a diversity-controlled fraction is re-seeded around the most distinct archive elites using a shrinkage covariance estimated from the basin members, so the re-seeding respects parameter correlations rather than treating each dimension independently;
• the remaining candidates are placed by a scrambled low-discrepancy (Sobol) sequence for unbiased exploration of the whole box.

This targets the paper's own observation that model parameters are interdependent, so a per-dimension restart can break correlated parameter sets.

Success-history adaptation — adapt¶

With adapt=True (default), the crossover rate \(CR\) and the blend factor \(\phi\) are adapted from a success history weighted by the achieved improvement \(\Delta f\) (the SHADE idea), instead of the previous absolute-fitness weighting. The exploration/exploitation balance then responds to what is actually working.

Optional surrogate pre-screening — surrogate¶

For expensive objectives, an optional surrogate model can rank the trial vectors each iteration so that only the most promising fraction (plus an exploration quota) is evaluated on the real objective. Two surrogate managers are bundled in DEEM.surrogate, both with a feasibility classifier that biases evaluations away from regions that previously returned the penalty value (useful when the model is numerically unstable for some parameter sets):

• SurrogateManager — a regularised radial-basis-function (RBF) interpolant. Fast and a good general default; ranks candidates by predicted value plus a nearest-neighbour novelty term.
• GPSurrogateManager — a Gaussian-process (GP) surrogate with a squared-exponential kernel. Unlike the RBF, it returns a predictive variance, so it selects with a Lower-Confidence-Bound rule, \(\mu(x) - \kappa\,\sigma(x)\), evaluating candidates that are either promising (low predicted cost) or uncertain (poorly covered by past evaluations). This is the classic surrogate-assisted / Bayesian-optimisation acquisition and tends to explore more deliberately on expensive, multi-modal problems.

Both share the same interface, so they are interchangeable:

from DEEM.DEEMI import DEEM
from DEEM.surrogate import SurrogateManager, GPSurrogateManager
import numpy as np

LB = np.array([0.0] * 8); UB = np.array([1.0] * 8)

# RBF (default choice):
sm = SurrogateManager(LB, UB, eval_frac=0.5, explore_frac=0.3)

# or a Gaussian process with uncertainty-aware selection:
sm = GPSurrogateManager(LB, UB, eval_frac=0.5, explore_frac=0.3, kappa=1.5)

opt = DEEM(my_expensive_objective, LB, UB,
           nparticles_max=50, nparticles_min=50,
           npop_max=10, npop_min=4, maxiter=60,
           surrogate=sm)
opt.update()

Both are dependency-light (numpy + scipy only). On cheap analytic functions surrogate pre-screening is not worthwhile — its purpose is the expensive-objective regime (≈40 min per evaluation), where skipping evaluations dominates the cost.

Evaluation cache — cache_tol¶

When cache_tol is set, identical (within a relative tolerance) positions are answered from a cache instead of being re-simulated. It is off by default (None), which reproduces the original number of evaluations. On smooth landscapes exact revisits are rare, but on rugged / near-discrete calibration cost functions the same penalty-triggering points recur and the cache avoids paying for them twice.

Reproducibility and reporting — seed, return value¶

Passing seed seeds NumPy's global RNG for reproducible runs. DEEMI.update() additionally returns a structured result dictionary:

result = opt.update()
# {'x', 'f', 'nit', 'nfev', 'n_restarts', 'cache_size', 'time'}

Restart budget — restart_budget¶

restart_budget caps the number of global restarts (None = unlimited).

Parallel evaluation on Windows¶

DEEMI evaluates candidate solutions through a process pool when nworkers > 1. On Windows, worker processes are spawned and do not inherit the parent's runtime state. The bundled evaluation.py therefore starts the pool with an initializer that re-establishes the required state (e.g. the ACT.globals used by numgeo-ACT) in every worker. On Linux the state is inherited through fork. See the calibration example.

Summary of new constructor arguments¶

See the DEEMI API reference for the full signature.