Linear Inversion

Linear-inversion is a point-estimate method for Kerr models whose remaining free parameters are complex amplitudes. It is designed for fast amplitude extraction and for cross-checking nested-sampler or minimization results.

The repository also contains a method note: Linear_inversion.pdf.

When To Use It

Use linear inversion when all of the following hold:

  • Template: the waveform template is Kerr.

  • Free parameters: the only unknowns are amplitude/phase pairs.

Do not use it when any free parameter is nonlinear:

  • damped-sinusoid frequencies;

  • damped-sinusoid damping times;

  • TEOB calibration coefficients;

  • KerrBinary phase parameters;

  • tail exponents.

Example

Run:

bayRing --config-file \
  config_files/config_SXS_0305_Kerr_220_linear_inversion.ini

The example selects:

[Model]
template  = Kerr
QNM-modes = 220,221,222,223

[Inference]
method = Linear-inversion
t-start = 0.0
t-end = 80.0
linear-inversion-eigenvalue-tol = 1e-10

With no fixed amplitude/phase pairs in [Priors], the solver estimates all listed complex amplitudes.

Linearisation Output

The default tolerance is:

[Inference]
linear-inversion-eigenvalue-tol = 1e-10

The solver prints:

  • Solved amplitudes: the number of complex amplitudes solved.

  • Cost: the weighted least-squares cost.

  • Raw Fisher spectrum: the eigenvalue range before regularization.

  • Tolerance: the eigenvalue floor used in the solve.

  • Conditioning: the condition number after regularization.

Interpretation:

  • a large condition number means the selected basis is poorly conditioned;

  • a result dominated by the eigenvalue floor means the data and error vector do not constrain the selected amplitudes cleanly.

Validation Checklist

After a linear-inversion run:

  • Point estimates: inspect Algorithm/point_estimates.dat.

  • Waveform plots: inspect waveform and residual reconstruction plots.

  • Method comparison: compare against method = Minimization for the same fixed/free parameter set.

  • Posterior check: run nested sampling if posterior structure or evidence matters.

  • Eigenvalue tolerance: vary linear-inversion-eigenvalue-tol only as a numerical diagnostic, not as a fitting knob.