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Clarify finite precision positivity of log density

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Emily Boudreaux
2026-09-19 12:04:27 -04:00
parent 8fbfc4f191
commit 91334b5f69
1 changed files with 14 additions and 7 deletions
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experiments/LOG_DENSITY_ASSESSMENT_2026-09-19.md
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@@ -17,8 +17,10 @@ eventual EOS to a barotrope. The important choice is what gets exponentiated:
coordinate change on positive coefficient vectors. The measured negative coordinate change on positive coefficient vectors. The measured negative
density fields already have positive nodal coefficients. density fields already have positive nodal coefficients.
2. Evaluating `rho = rho_scale * exp(u_h(x))` pointwise guarantees strictly 2. Evaluating `rho = rho_scale * exp(u_h(x))` pointwise guarantees strictly
positive density wherever `u_h` is finite. This is a different approximation positive density in exact arithmetic wherever `u_h` is finite. Floating-point
family from polynomial density. Every density-dependent operator must use it. evaluation also needs checks for underflow and overflow. This is a different
approximation family from polynomial density. Every density-dependent
operator must use it.
3. A finite polynomial `u_h` cannot represent an exact zero-density vacuum 3. A finite polynomial `u_h` cannot represent an exact zero-density vacuum
surface with that raw exponential. A vanishing reference factor can encode surface with that raw exponential. A vanishing reference factor can encode
the surface, but its asymptotic behavior becomes part of the representation. the surface, but its asymptotic behavior becomes part of the representation.
@@ -256,11 +258,16 @@ corresponding numbers are `6.65377`, `13.7153`, and `74.9786`.
The first-order identity `delta rho/rho=delta u` makes a log step a relative The first-order identity `delta rho/rho=delta u` makes a log step a relative
density step. That can be useful across large density contrasts, but large log density step. That can be useful across large density contrasts, but large log
steps create exponential changes. Underflow, overflow and weak response near steps create exponential changes. Underflow can produce zero from finite log
vacuum need explicit finite-value checks and meaningful globalization; silently coefficients, so finite-value checks alone do not ensure positivity. A raw
clamping the exponential would change the equations. A vanishing factor removes exponential implementation must also detect or reject underflow and nonpositive
the singular log profile when it matches the asymptotics, but retains the reconstructed density in the material interior, including underflow when
density-weighted sensitivity near vacuum. multiplying by its reference scale or factor. Prescribed zero-density traces
from a vanishing reference factor remain intentional. Overflow and weak
response near vacuum also need explicit checks and meaningful globalization;
silently clamping the exponential would change the equations. A vanishing
factor removes the singular log profile when it matches the asymptotics, but
retains the density-weighted sensitivity near vacuum.
With the explicit constant test retained, a converged weak exponential closure With the explicit constant test retained, a converged weak exponential closure
preserves its target cell mass under the closure quadrature. Exponentiation preserves its target cell mass under the closure quadrature. Exponentiation