Error Budgets

An error budget identifies which approximations and uncertainties limit a reported observable. Its practical purpose is to direct refinement toward the dominant uncertainty. Tightening an eigensolver by six orders of magnitude is unhelpful when finite-box bias already exceeds the desired accuracy.

Required background. Convergence studies explains refinement studies; numerical error and stability explains what residuals can bound.

A telescoping decomposition

For a target $Q_{\mathrm{phys}}$, distinguish a chosen physical model, its infinite-resolution solution, its finite discretization, and the computed output. A schematic decomposition is

$$Q_{\mathrm{num}}-Q_{\mathrm{phys}} ={}(Q_{\mathrm{num}}-Q_{\mathrm{disc}}) +(Q_{\mathrm{disc}}-Q_{\mathrm{model}}) +(Q_{\mathrm{model}}-Q_{\mathrm{phys}}).$$

The first term includes solver and arithmetic error, the second discretization and finite-domain bias, and the third model inadequacy. Further insert intermediate quantities to separate box, basis, time-step, and sampling effects. Such a decomposition is exact once the intermediate problems are defined; estimating its terms may be difficult.

Varying one parameter gives a sensitivity estimate, not necessarily a strict bound on its error. Coupled approximations can cancel or interact. Record whether a budget entry is a proven bound, an asymptotic estimate, an empirical change, or a statistical standard error.

Deterministic bounds and statistical uncertainties

If $|e_i|\le b_i$ are valid deterministic bounds, the triangle inequality gives $|\sum_i e_i|\le\sum_i b_i$. Adding them in quadrature has no general justification. For random errors with a specified joint distribution,

$$\operatorname{Var}\left(\sum_i e_i\right) =\sum_i\operatorname{Var}(e_i) +2\sum_{i<j}\operatorname{Cov}(e_i,e_j).$$

Quadrature follows only when the covariance terms vanish. A time-step bias is not converted into an independent random error by calling it an “uncertainty.” Report deterministic estimates and statistical intervals separately unless a probabilistic model supports their combination.

For parameters $\theta$ with covariance $\Sigma$, a locally smooth observable has approximate propagated variance $\nabla Q^T\Sigma\nabla Q$. This linear approximation can fail near thresholds, small denominators, or phase transitions. Parameter sensitivity is also distinct from the numerical error of solving the model at fixed $\theta$.

A bound-state budget

Suppose the target is a dimensionless ground energy with desired absolute numerical error below $10^{-5}$. A refinement study estimates spatial bias at $3\times10^{-6}$ and box bias below $2\times10^{-6}$; the Hermitian eigenpair residual bounds distance to a matrix eigenvalue by $10^{-9}$, with the eigenvalue identified independently.

If the first two entries are only empirical estimates, their sum suggests that the target is met, but does not certify it. If all three are established bounds, the conservative sum is $5.001\times10^{-6}$ and meets the target. In either case, reducing the residual from $10^{-9}$ to $10^{-13}$ barely improves the budget. Refining the grid or enlarging the box is more consequential.

For a tunneling splitting, budgeting the two absolute energies separately may be too pessimistic or insufficiently informative. Study the splitting directly under refinement and investigate correlations in the energy errors. Do not assume cancellation merely because both energies came from the same solver.

Allocating effort

A rough cost model can expose inefficient precision choices. Suppose errors $e_i$ are positive adjustable estimates and independent computational costs behave as $C_i(e_i)=a_i e_i^{-\alpha_i}$, with $a_i,\alpha_i>0$. Minimizing $\sum_iC_i$ subject to $\sum_i e_i=\varepsilon$ yields

$$\alpha_i a_i e_i^{-\alpha_i-1}=\mu$$

for all interior allocations. The marginal cost of reducing error is equalized, not necessarily the errors themselves. The model is only a planning aid: actual solver costs may be discontinuous, and numerical errors may not decrease monotonically.

Usually a simpler policy suffices: make cheap algebraic errors much smaller than the expensive discretization error, then balance independent physical resolution parameters through measured refinement. Reserve precision increases for cases where a precision study indicates that roundoff limits the result.

Limits of the budget

An error budget cannot include an unknown coding mistake as a trustworthy small number. Independent benchmarks, symmetry tests, and review are needed to discover such defects. Nor does a numerical budget establish the adequacy of an effective Hamiltonian, Markov approximation, exchange-correlation functional, or experimental calibration.

State the result at a precision supported by the least well controlled relevant error. Keep additional internal digits for later computation, but avoid interpreting them as physical accuracy. Convergence reports provides a compact way to present the result and its limitations.

Exercises

Two independent Monte Carlo estimates each have standard error $0.01$. What is the standard error of their sum? How does the answer change if their correlation coefficient is $0.8$?

Solution

For independent estimates it is $\sqrt{2}\,0.01\approx0.0141$. With correlation $0.8$, it is $0.01\sqrt{2+2(0.8)}\approx0.0190$. Neither calculation applies to two deterministic bias bounds.

References

• Higham, N. J., Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, 2002 — algebraic error and conditioning.
• Joint Committee for Guides in Metrology, Evaluation of Measurement Data—Guide to the Expression of Uncertainty in Measurement, JCGM 100:2008, official publications — covariance and uncertainty propagation, applied here with numerical bias explicitly distinguished.
• LeVeque, R. J., Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM, 2007 — discretization error and refinement.