[Paper Review] Total wave based fast direct solver for volume scattering problems

Yu Chen|arXiv (Cornell University)|Feb 8, 2013

Electromagnetic Scattering and Analysis6 references3 citations

TL;DR

This paper presents a fast direct solver for volume scattering problems governed by the Helmholtz equation using a total wave solution space (TWSS) approach. By constructing the null space of the homogeneous PDEs with variable coefficients first, then incorporating boundary conditions and incident waves, the method enables efficient, direct solution with simplified discretization and improved accuracy over traditional finite difference, finite element, or boundary integral methods.

ABSTRACT

We present a fast direct solver for the volume scattering problem of the Helmholtz equation. The algorithm is faster than existing methods. Moreover, discretization for our method is much simpler and more accurate than that for finite difference, finite elements, and integral equations.

Motivation & Objective

• To develop a fast direct solver for volume scattering problems (VSP) governed by the Helmholtz equation with inhomogeneous media.
• To overcome the challenges of ill-conditioning and complex discretization in existing methods such as finite elements, finite differences, and boundary integral equations.
• To simplify solution representation and discretization by avoiding iterative solvers and direct solution of boundary value problems.
• To enable efficient handling of global multiple scattering effects through a hierarchical, recursive construction of the total wave solution space (TWSS).
• To extend the approach to general elliptic PDEs and Maxwell's equations by first solving the homogeneous PDEs without boundary conditions.

Proposed method

• The method constructs the total wave solution space (TWSS) as the null space of the homogeneous Helmholtz equation with variable coefficients, without imposing boundary conditions.
• It uses a hierarchical domain decomposition (e.g., quadtree in 2D), building TWSS recursively from bottom-level subdomains upward.
• Merging waves from child subdomains into parent subdomains is achieved via a recursive algorithm that preserves the solution space structure.
• The approach employs Green’s third identity and layer potential operators (S, K, K′, T) to represent Dirichlet and Neumann data on interfaces.
• It applies jump relations and projector operators P₊ and P₋ to decompose waves into incoming and outgoing components, enabling efficient merging.
• Discretization is performed via collocation and weak formulations on the TWSS, ensuring high accuracy with minimal degrees of freedom.

Experimental results

Research questions

• RQ1 Can a fast direct solver be constructed for volume scattering problems without solving ill-posed or well-posed boundary value problems directly?
• RQ2 How can global multiple scattering effects be captured efficiently in inhomogeneous media using a solution space approach?
• RQ3 Can the discretization complexity of boundary integral equations be significantly reduced by working with the null space of the homogeneous PDE?
• RQ4 What is the role of the total wave solution space (TWSS) in enabling a direct, non-iterative solver for elliptic PDEs with variable coefficients?
• RQ5 How can the hierarchical merging of solution spaces across subdomains be formulated to maintain accuracy and computational efficiency?

Key findings

• The proposed method achieves a fast direct solver for volume scattering problems with significantly simpler and more accurate discretization than finite difference, finite element, or boundary integral methods.
• The total wave solution space (TWSS) is constructed as the null space of the homogeneous Helmholtz equation with variable coefficients, enabling global multiple scattering effects to be captured inherently.
• The method avoids the need for iterative solvers and instead constructs the solution directly through recursive merging of TWSS across a hierarchical domain decomposition.
• The use of layer potential operators and jump relations (e.g., P₊, P₋, S, K, K′, T) enables precise decomposition of wave components and accurate interface treatment.
• The approach is generalizable to other elliptic PDEs and Maxwell’s equations, as demonstrated by the formulation of first-order systems with symmetry-breaking.
• The method maintains high accuracy even in complex geometries and near singularities, such as corners and edges, where traditional quadrature-based methods fail.