Target Head Well (String) Control points

[Huite]

Related to one of the suggestions noted in #60 (area of control points for TargetHeadWellString), here are some thoughts.

A potentially user-friendly approach is to let users define a set of control points inside their excavation / building pit footprint. This could reduce some of the current trial-and-error that might be involved with picking points and target heads.

There are (at least) two reasonable formulations:

1) Least-squares head objective (equality / soft targets)

Use a weighted least-squares objective that tries to match a target head z at the control points:

$$ \min_q \ \tfrac12|h(q)-z|_R^2 + \tfrac12 q^\top W q $$

In confined linear AEM, head is linear in discharge, so this reduces to a single linear system. This objective is symmetric: it penalizes both “too wet” and “too dry”. That may actually be desirable, since “too dry” typically implies over-pumping. (As with the current target-head elements, without explicitly enforcing q >= 0 the solution could in principle include injection.)

2) Dryness constraints (inequality / “keep the pit dry”)

More directly, enforce “dryness” at each control point as an inequality:

$$ h(q) \le z $$

This is naturally a convex quadratic programming exercise once we add an objective to pick a unique solution, e.g. a smooth “least pumping” objective:

$$ \min_q \ \tfrac12 q^\top W q \quad \text{s.t.}\quad h(q) \le z \quad (\text{optionally } q\ge0) $$

Because constraints become active/inactive depending on whether a control point is still wet, this can't be solved with a single linear solve, so an active-set / KKT approach is appropriate.

Fortunately, I think this can be implemented efficiently because if the expensive part is evaluating influence functions, not the dense linear solve:

• Precompute/cache head influence rows at the control points (and known-head contributions).
• Solve the KKT system for the current active set (dense solve; active set is typically small).
• Evaluate heads at control points (dot products using cached rows), update active set, repeat.

A good warm-start is to first do the least-squares solve so the discharges are already near the “just dry enough” regime before enforcing the inequalities.

(The inequality formulation guarantees feasibility (“dry everywhere at the chosen points”), while least-squares provides a simple one-shot approximation that may leave small wet spots unless weights/targets are adjusted.)