Stochastic finite elements: a spectral approach
Stochastic finite elements: a spectral approach
Matrix computations (3rd ed.)
Parameter Estimation in the Presence of Bounded Data Uncertainties
SIAM Journal on Matrix Analysis and Applications
Matrix analysis and applied linear algebra
Matrix analysis and applied linear algebra
The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
SIAM Journal on Scientific Computing
High-Order Collocation Methods for Differential Equations with Random Inputs
SIAM Journal on Scientific Computing
The PageRank Vector: Properties, Computation, Approximation, and Acceleration
SIAM Journal on Matrix Analysis and Applications
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
SIAM Journal on Numerical Analysis
SIAM Journal on Scientific Computing
Matrices, Moments and Quadrature with Applications
Matrices, Moments and Quadrature with Applications
A Rational Interpolation Scheme with Superpolynomial Rate of Convergence
SIAM Journal on Numerical Analysis
The Lanczos Method for Parameterized Symmetric Linear Systems with Multiple Right-Hand Sides
SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Scientific Computing
Generalised Polynomial Chaos for a Class of Linear Conservation Laws
Journal of Scientific Computing
SIAM Journal on Scientific Computing
A Posteriori Error Analysis of Parameterized Linear Systems Using Spectral Methods
SIAM Journal on Matrix Analysis and Applications
Journal of Computational Physics
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We apply polynomial approximation methods—known in the numerical PDEs context as spectral methods—to approximate the vector-valued function that satisfies a linear system of equations where the matrix and the right-hand side depend on a parameter. We derive both an interpolatory pseudospectral method and a residual-minimizing Galerkin method, and we show how each can be interpreted as solving a truncated infinite system of equations; the difference between the two methods lies in where the truncation occurs. Using classical theory, we derive asymptotic error estimates related to the region of analyticity of the solution, and we present a practical residual error estimate. We verify the results with two numerical examples.