[Paper Review] Inner product quadratures

Yu Chen|arXiv (Cornell University)|May 3, 2012

Mathematical Analysis and Transform Methods1 references3 citations

TL;DR

This paper introduces a novel n-term quadrature method for integrating inner products of n functions, termed the inner product quadrature, which generalizes classical Gaussian quadrature by directly targeting inner product integration rather than function integration. It establishes a duality between orthogonal columns of basis functions and quadrature nodes, enabling exact integration of Gramian matrices through eigenvalue decomposition of quotient matrices, with applications in imaging, integral equations, and high-dimensional quadrature design via deflation techniques.

ABSTRACT

We introduce a n-term quadrature to integrate inner products of n functions, as opposed to a Gaussian quadrature to integrate 2n functions. We will characterize and provide computational tools to construct the inner product quadrature, and establish its connection to the Gaussian quadrature.

Motivation & Objective

• To develop a quadrature method that directly integrates inner products of n functions, rather than integrating 2n functions as in classical Gaussian quadrature.
• To establish a duality between orthogonal column structures in function basis matrices and the existence of n-term quadrature rules with positive weights.
• To extend the framework to handle non-positive definite weights and construct quadratures for integral equations and imaging problems.
• To provide a constructive algorithm for generating such quadratures using eigenvalue decomposition of quotient matrices derived from Gramians.
• To generalize the method to higher dimensions and construct quadrature rules for bivariate and trivariate polynomials using deflation techniques to recover missing nodes.

Proposed method

• Proposes a Type-2 quadrature that integrates all inner products in an n-dimensional function space T(n,x) by ensuring the n columns of the orthogonal basis matrix Q(n, {x_j}) are orthogonal.
• Uses the Gramian matrices B(n,n) = T(n,x) · T(x,n) and A(n,n) = T(n,x) · μ(x)T(x,n) to define the inner product and moment conditions, respectively.
• Establishes that quadrature nodes x_j are eigenvalues of the matrix quotient AB⁻¹ when both Gramians are exactly integrated.
• Applies the duality condition: if the columns of Q(n, {x_j}) are orthogonal with norms 1/√w_j, then the quadrature rule {x_j, w_j} exactly integrates the identity Gramian.
• Introduces deflation techniques to iteratively remove known nodes and their contributions from Gramians A and B, enabling recovery of additional nodes via eigenvalue decomposition of deflated quotient matrices.
• Extends the method to higher dimensions by using multiple moment matrices (e.g., A_x, A_y) and applying deflation to recover nodes in bivariate and trivariate polynomial spaces.

Experimental results

Research questions

• RQ1 Can a quadrature rule with n nodes exactly integrate all inner products of n functions in a given function space, and under what conditions does such a rule exist?
• RQ2 How can the quadrature nodes and weights be computed when the classical Gaussian quadrature framework does not apply due to non-positive definite weights?
• RQ3 What is the relationship between the orthogonality of basis function columns and the existence of a valid n-term quadrature rule?
• RQ4 How can the method be extended to higher-dimensional integration problems where the number of required nodes exceeds the dimension of the function space?
• RQ5 Can deflation techniques be used to iteratively recover additional quadrature nodes beyond those provided by eigenvalue decomposition of quotient matrices?

Key findings

• A necessary and sufficient condition for the existence of an n-term inner product quadrature is that the columns of the orthogonal basis matrix Q(n, {x_j}) are mutually orthogonal.
• The quadrature nodes x_j are eigenvalues of the matrix AB⁻¹, where A and B are Gramians of the function space under the weight u and a simple function μ, respectively.
• For polynomial bases, the method recovers the classical Gaussian quadrature nodes as eigenvalues of AB⁻¹ when μ(x) = x.
• In bivariate polynomial spaces of degree less than n, the method can generate up to n(n+1)/2 nodes via eigenvalue decomposition, with additional nodes recovered via deflation.
• Deflation allows recovery of up to n(n−1)/6 additional nodes in 2D, and the method is effective for constructing Gauss-Radau and Gauss-Lobatto type rules in higher dimensions.
• Theoretical convergence of the deflation method is supported by the fast decay of Coulomb-like potentials, minimizing error propagation from initial node guesses.