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Layer 4 V10 — Smart warm-start + PTC (research)

V4 (line search), V5 (LM), V8 (κ-homotopy), and V9 (V_F0- homotopy) all failed to solve the κ=20 stiff sinusoidal rectifier from x = 0. V10 was scoped to ship pseudo-transient continuation (PTC) as the robust fallback.

Empirical finding during V10: pure PTC is fundamentally unsuitable for Pulsim MNA systems with voltage-source constraints. The actual fix is much simpler — a STRUCTURAL warm-start helper that places source values onto the corresponding source nodes. Combined with plain Newton + line search, this finally cracks the κ=20 stiff rectifier that's been deferred for 6 successive OpenSpecs.

API

Primary: smart warm-start

#include "pulsim/pwl/initial_guess.hpp"

[[nodiscard]] Vector make_diode_aware_initial_guess(
    const Graph& graph,
    const DevicePool& pool,
    const Vector& b_extra);

Walks the graph's branches. For each branch stored in the DevicePool as VoltageSource, computes the effective voltage V_eff = pool.V − b_extra[source_branch_var] and writes it onto the source's "from" node in the returned x_init.

For canonical source → diode → load circuits, this places Newton inside the correct basin of attraction: - Diode anode at V_source, cathode at 0 → v_diode = V_source. - If V_source > V_F0, Newton pulls the cathode up toward V_source − V_F0 (forward conducting). - If V_source < V_F0, Newton finds v_cathode ≈ 0 (diode off).

Research-grade: pseudo-transient continuation

#include "pulsim/pwl/pseudo_transient.hpp"

[[nodiscard]] Vector pseudo_transient_solve(
    const PwlSegment& seg,
    const NonlinearRefreshFn& refresh,
    const Graph& graph,
    const DevicePool& pool,
    const Vector& x_init,
    const Vector& b_extra,
    Real dt_init = 1.0,
    Real dt_max  = 1e10,
    Size max_iters = 500,
    Real tol_res = 1e-7);

Converts F(x) = 0 into dx/dt = -F(x), integrates with implicit Euler + trust-region dt adaptation. Ships as research-grade for circuits with positive-real-part J eigenvalues. NOT recommended for canonical Pulsim MNA circuits (see "PTC limitation" below).

Recipe: cracking the κ=20 stiff sinusoidal rectifier from x=0

for (Size k = 0; k < n_steps; ++k) {
    const Real t = k * dt_sim;
    const Vector b_extra = make_sinusoidal_b_extra(t);

    // STEP 1: smart warm-start.
    const Vector x_init =
        make_diode_aware_initial_guess(graph, pool, b_extra);

    // STEP 2: plain Newton + line search.
    Vector x = solve_with_newton_b_extra(
        seg, &refresh_smooth_diodes, graph, pool,
        x_init, b_extra,
        /*max_iters=*/100,
        /*tol_dx=*/1e-7, /*tol_res=*/1e-5,
        /*enable_line_search=*/true);

    // Use x ...
}

Verified across 2 cycles at dt=100µs (334 steps): every step converges, > 95 % pos-half + neg-half tracking, mean power within 15 % of analytical (V_amp − V_F0)² / (4·R_load).

PTC limitation (why it doesn't work for MNA)

PTC requires the artificial dynamics dx/dt = -F(x) to be STABLE near the solution. This holds when J = ∂F/∂x has positive-real-part eigenvalues.

Pulsim's MNA Jacobian for a voltage-source-driven circuit has a constraint row like: 

J[constraint_row] = [..., 1 (on v_node), ..., 0 (on i_src), ...]
The constraint coupling creates eigenvalue pairs whose real parts can be EITHER sign depending on topology.

Concrete example — linear V_dc → R → GND: - State: [v_n0, i_src] - J = [[1, 1], [1, 0]] (MNA stamp) - Eigenvalues = (1 ± √5) / 2 ≈ 1.618 and −0.618

The negative eigenvalue means PTC's -J has eigenvalue +0.618 — UNSTABLE. The iterate is repelled from the solution along that eigenvector. NO dt schedule recovers.

Multiple PTC variants were attempted (SER, trust-region, exponential growth, rollback-on-rebound) — all failed. Documented as a research note in pwl/pseudo_transient.hpp.

Test coverage

Three tests in tests/v2/layer5_v4/test_pseudo_transient_rectifier.cpp:

  1. Helper writes source value onto from-node — pool.V = 3.5, b_extra = 0 → x_init[n0] = 3.5.
  2. Helper folds b_extra into the effective voltage — pool.V = 0, b_extra[src_var] = −7.5 → x_init[n0] = +7.5.
  3. κ=20 stiff sinusoidal rectifier from auto warm-start + plain Newton + line search — the deferred test from V4 → V9, finally cracked.

What V10 deliberately does NOT do

  • Diode load-line correction beyond v_to = 0. The V0 helper sets only source nodes; diode cathodes are left at 0 (a "diode-off" assumption that works for the canonical rectifier topology). Future work could compute a proper load-line correction for multi-diode networks.
  • Cap/inductor warm-start. Trap-companion history is handled separately by HistoryState; the warm-start helper doesn't touch those branches.
  • Make PTC work for MNA. The PTC primitive ships as research-grade; a future OpenSpec could explore sign-corrected PTC, hybrid PTC+warm-start, or affine-invariant Newton as alternatives.

Files

  • NEW core/include/pulsim/pwl/initial_guess.hpp
  • NEW core/include/pulsim/pwl/pseudo_transient.hpp (research-grade)
  • NEW core/tests/v2/layer5_v4/test_pseudo_transient_rectifier.cpp
  • NEW openspec/changes/pulsim-v2-pseudo-transient/
  • MODIFIED core/CMakeLists.txt (test wired)