Karhunen-Loeve Expansion

The Karhunen-Loeve expansion represents a random field as a finite sum of orthogonal modes weighted by uncorrelated random coefficients.

The expansion

A random field g(x, omega) with mean m(x) and covariance C(x, x') can be written as g(x, omega) = m(x) + sum_k sqrt(lambda_k) phi_k(x) xi_k(omega), where (lambda_k, phi_k) are the eigenvalues and eigenfunctions of the covariance operator and the xi_k are uncorrelated, zero-mean, unit-variance random variables. This is the optimal linear representation in mean-square error for a given number of terms.

Solving the eigenproblem

The eigenpairs solve the Fredholm integral equation: integral of C(x, x') phi_k(x') dx' = lambda_k phi_k(x). In practice the field is discretized on a grid and the integral becomes a matrix eigenvalue problem, closely related to principal component analysis of the covariance matrix. Eigenvalues decay fastest for smooth, strongly correlated fields.

Truncation and dimension reduction

Because lambda_k typically decays quickly, keeping the first few modes captures most of the field's variance. This turns an infinite-dimensional random field into a handful of scalar random variables xi_k, the key enabler for propagating spatial uncertainty through expensive models. The number of retained modes is chosen so the cumulative eigenvalue ratio exceeds a threshold such as 0.99.

Gaussian and non-Gaussian fields

For a Gaussian field the xi_k are independent standard normals, which makes downstream sampling trivial. For non-Gaussian fields the coefficients are uncorrelated but not independent, and additional transformation or a polynomial-chaos representation of the xi_k is needed to sample correctly.

Use in UQ

KL expansion is the standard front end for propagating uncertain material properties, boundary conditions, or spatially varying loads through simulations. In fusion modeling it can represent an uncertain profile, such as a plasma density or wall-load distribution, with a few random coefficients, which are then fed to stochastic collocation or polynomial chaos for efficient propagation.