Crate uncertain_numerics 

Probabilistic numerical methods with explicit uncertainty over computational quantities.

uncertain-numerics treats numerical computation as an inference problem. Instead of returning only a point estimate of an integral or of the solution of a linear system, each method returns a validated Gaussian posterior: its mean is the estimate and its variance states how much the computation still does not know. Every statistical and numerical assumption behind that posterior is explicit, documented, and tested.

What is implemented

Area	Entry points
One-dimensional Bayesian quadrature	BayesianQuadrature, RbfKernel, GaussianMeasure, ScalarNormalPosterior
Active Bayesian quadrature	ActiveBayesianQuadrature, VarianceReductionAcquisition
Probabilistic linear solvers	SpdLinearSystem, GaussianLinearBelief, ResidualProjectionSolver, AConjugateProjectionSolver, CovarianceGreedyProjectionSolver
Building blocks	GaussianConditioner, KernelMean, KernelIntegral, ScalarKernel, ContinuousProbabilityMeasure

Example

Infer the integral of cos(x) against a standard Gaussian measure from seven function evaluations, together with its posterior uncertainty:

use uncertain_numerics::{BayesianQuadrature, GaussianMeasure, RbfKernel};

let kernel = RbfKernel::new(1.0, 1.0)?; // signal variance, length scale
let measure = GaussianMeasure::new(0.0, 1.0)?; // p(x) = N(0, 1)
let quadrature = BayesianQuadrature::new(kernel, measure, 1.0e-10);

let nodes = [-3.0, -2.0, -1.0, 0.0, 1.0, 2.0, 3.0];
let values: Vec<f64> = nodes.iter().copied().map(f64::cos).collect();
let posterior = quadrature.posterior(&nodes, &values)?;

// Exactly, E[cos X] = exp(-1/2) for X ~ N(0, 1).
let exact = (-0.5_f64).exp();
println!("{} ± {}", posterior.mean(), posterior.standard_deviation());
assert!((posterior.mean() - exact).abs() < 3.0 * posterior.standard_deviation());

Numerical policy

• Inputs are validated at the API boundary. Invalid values produce typed errors that implement std::error::Error; they never propagate as NaN.
• Linear systems are solved from a reusable Cholesky factorization. Explicit inverses are never formed.
• Jitter is a caller-supplied constant added to the diagonal. Because it changes the posterior, it is never increased automatically to make a factorization succeed.
• Posterior variances that are negative by more than machine roundoff are reported as errors rather than clamped.

Interpreting the uncertainty

Posterior uncertainty is conditional on the numerical model. Under the assumed prior the reported intervals are empirically calibrated, which the integration tests verify. Under misspecification, for example an RBF length scale that is far too smooth for the integrand, the posterior can be confidently wrong. The docs/ directory of the repository records the known failure modes and the calibration status of each linear-solver policy.

Minimum supported Rust version

Rust 1.85. Raising the MSRV is treated as at least a minor version bump.

Structs

AConjugateLinearSolveResult

Result of an A-conjugate probabilistic linear solve.

AConjugateLinearSolveStep

One A-conjugate projection-conditioning step.

AConjugateProjectionSolver

Probabilistic linear solver using residual directions orthogonalized in the A inner product.

ActiveBayesianQuadrature

Sequential active Bayesian quadrature over a finite candidate set.

ActiveDesignResult

Result of a sequential active Bayesian-quadrature run.

ActiveDesignStep

One function evaluation selected by the active design.

BayesianQuadrature

Bayesian quadrature with an RBF covariance kernel and Gaussian integration measure.

CovarianceGreedyProjectionSolver

Sequential probabilistic linear solver with data-independent covariance-greedy directions.

CovarianceGreedySolveResult

Result of a covariance-greedy probabilistic linear solve.

CovarianceGreedyStep

One covariance-greedy projection step.

CovarianceTraceAcquisition

Covariance-trace acquisition for exact linear-system projection observations.

GaussianConditioner

Reusable factorization of a symmetric positive-definite linear system.

GaussianLinearBelief

Gaussian belief over the unknown solution vector of a linear system.

GaussianMeasure

Gaussian probability measure (N(\mu, \sigma^2)).

LinearSolveStep

One residual-projection conditioning step.

ProbabilisticLinearSolveResult

Result of an iterative probabilistic linear solve.

RbfKernel

Squared-exponential (radial-basis-function) covariance kernel.

ResidualProjectionSolver

Residual-driven probabilistic solver for dense SPD systems.

ScalarNormalPosterior

Gaussian posterior for a scalar computational quantity.

SelectedCandidate

Candidate selected by posterior integral-variance reduction.

SelectedLinearDirection

Candidate projection chosen to maximize posterior covariance-trace reduction.

SpdLinearSystem

Dense symmetric positive-definite linear system A x = b.

VarianceReductionAcquisition

Expected reduction in posterior integral variance from evaluating one candidate node.

Enums

ActiveDesignError

Errors raised while running sequential active Bayesian quadrature.

ActiveSelectionError

Errors raised while selecting active Bayesian-quadrature candidates.

ActiveTermination

Reason a sequential active Bayesian-quadrature run terminated.

BayesianQuadratureError

Errors raised while constructing a Bayesian quadrature posterior.

ConditioningError

Errors raised while constructing or using a Gaussian conditioning system.

CovarianceGreedyTermination

Reason a covariance-greedy probabilistic solve terminated.

KernelError

Errors raised while constructing covariance kernels.

LinearSolveTermination

Reason a residual-projection probabilistic linear solve terminated.

LinearSolverError

Errors raised by probabilistic linear-system primitives.

MeasureError

Errors raised while constructing probability measures.

PosteriorError

Errors raised while constructing probabilistic numerical results.

Traits

ContinuousProbabilityMeasure

Contract for a normalized continuous probability measure on the real line.

KernelIntegral

Analytic double integral of a scalar kernel under a probability measure.

KernelMean

Analytic mean embedding of a scalar kernel under a probability measure.

ScalarKernel

Contract for a scalar covariance kernel.