arXiv subjects

Yasuhiro Matsumoto

Publications and source records attributed to Yasuhiro Matsumoto.

9 recordsLinked to original sources

arXiv2026

Proxy-surface-based fast direct solver for TE-mode scattering problems on distributed memory systems

This paper describes an MPI/OpenMP hybrid parallelized fast direct solver for the scattering problem of transverse electric (TE)-mode electromagnetic waves. Because TE-mode scattering can be reduced to the two-dimensional Helmholtz equation, solvers based on the hierarchically semiseparable (HSS) representation are highly attractive due to their high parallel efficiency. However, as the HSS representation applies low-rank approximations to all off-diagonal blocks, it exhibits poor compatibility with high-order discretization methods. We developed a fast direct solver with $O(h^3)$ convergence for Helmholtz transmission problems, whereas conventional HSS solvers typically yield only $O(h)$ convergence (where $h$ represents intervals between the quadrature nodes). It is based on the weakly singular Burton-Miller boundary integral equation and the Nyström method with a one-point correction. Furthermore, recognizing that matrix component calculation, rather than matrix factorization, dominates the total computational time of HSS-type boundary integral solvers, we introduced a load-balancing method to maximize parallel efficiency. Numerical results demonstrate that the direct solver achieves high-accuracy convergence and nearly ideal strong and weak scalabilities.

math.NA↗

arXiv2026

Well-posedness of the weakly singular Burton-Miller equation for Helmholtz transmission problems

Although various boundary integral formulations are available for the Helmholtz transmission problem, the weakly singular Burton-Miller (BM) equation is promising because it is well-suited for the Nyström discretization. Moreover, unlike other formulations such as the PMCHWT or Müller equations, its fictitious eigenvalues do not coincide with eigenvalues of a different transmission problem. This paper rigorously shows that the weakly singular BM equation is well-posed.

math.AP↗

arXiv2026

A fast direct solver for two-dimensional transmission problems of elastic waves

This paper describes a fast direct boundary element method for elastodynamic transmission problems in two dimensions, which can be used for analyzing elastic wave scattering by an inclusion. We develop an efficient solver based on a discretization method that is broadly applicable regardless of the inclusion shape. From the smoothness of the solutions of the Navier--Cauchy equation, it is reasonable that the displacement is approximated by the piecewise linear bases and the traction is approximated by the piecewise constant bases. However, in this mixed bases strategy, Calderón preconditioning, that is, an analytical preconditioning with excellent performance, cannot be applied. To circumvent this issue, we developed a fast direct solver formulated using both Burton--Miller and Poggio--Miller--Chang--Harrington--Wu--Tsai (PMCHWT) boundary integral equations. Our method uses a technique based on the proxy method for low-rank approximation of the coefficient matrix's off-diagonal blocks. To handle transmission problems, the proposed fast direct solver uses separate binary tree partitions for nodes and elements. Numerical examples demonstrate that our solver achieves linear computational complexity at fixed low frequencies and can efficiently handle problems with multiple right-hand sides. Notably, the solver based on the Burton--Miller formulation is approximately 20\% faster than the one using the PMCHWT formulation. Our new method provides a versatile, fast solver, whose performance is relatively independent of the shape of inclusions and computational parameters, such as density, for elastodynamic transmission problems.

math.NA↗

arXiv2026

An accelerated direct solver for scalar wave scattering by multiple transmissive inclusions in two dimensions

This paper discusses a fast direct solver using boundary integral equations for Helmholtz transmission problems involving multiple inclusions in two dimensions. Efficiently addressing scattering problems in the presence of numerous inclusions remains a key challenge for various practical applications. For problems involving a large number of scatterers, the number of iterations in Krylov subspace methods is known to increase significantly. This occurs even when using second-kind boundary integral equations, which are typically recognized for their rapid convergence. We consider a fast direct solver as an alternative, an approach that has been less commonly explored for transmission problems with disjoint multiple inclusions. The low-rank approximation based on the proxy method achieve speedup by calculating interactions between disjoint scatterers without the terms derived from the internal integral representation. Notably, this advantage applies to the Poggio--Miller--Chang--Harrington--Wu--Tsai (PMCHWT) formulation but breaks down in the Burton--Miller case. Numerical examples demonstrate that the proposed solver can compress the system of linear algebraic equations to a size of $O(ωD)$, where $ω$ is the frequency of the incident wave and $D$ is the diameter of the (smallest) bounding box enclosing the multiple inclusions. The total computational cost scales as $O(N^{1.5})$ $(= O(\sqrt{N}^3))$ at most for a fixed $ω$ when the inclusions are arranged on a grid. Moreover, the PMCHWT formulation, that omits the interior term in the proxy method, is approximately six times faster than the Burton--Miller formulation when treating each inclusion as a cell. Furthermore, in the same setting, the former can compress the size of the system of linear algebraic equations by half compared to the latter.

math.NA↗

arXiv2025

Efficient LU factorization exploiting direct-indirect Burton-Miller equation for Helmholtz transmission problems

This paper proposes a direct-indirect mixed Burton-Miller boundary integral equation for solving Helmholtz scattering problems with transmissive scatterers. The proposed formulation has three unknowns, one more than the number of unknowns for the ordinary formulation. However, we can construct efficient numerical solvers based on LU factorization by exploiting the sparse alignment of the boundary integral operators of the proposed formulation. Numerical examples demonstrate that the direct solver based on the proposed formulation is approximately 40% faster than the ordinary formulation when the LU-factorization-based solver is used. In addition, the proposed formulation is applied to a fast direct solver employing LU factorization in its algorithm. In the application to the fast direct solver, the proxy method with a weak admissibility low-rank approximation is developed. The speedup achieved using the proposed formulation is also shown to be effective in finding nonlinear eigenvalues, which are related to the uniqueness of the solution, in boundary value problems. Furthermore, the well-posedness of the proposed boundary integral equation is established for scatterers with boundaries of class $C^2$, using the mapping property of boundary integral operators in Hölder space.

math.NA↗

arXiv2025

Injectivity of boundary integral operator in direct-indirect mixed Burton-Miller equation for wave scattering problems with transmissive circular inclusion

This study proves that the injectivity condition for the integral operator of the direct-indirect mixed Burton-Miller (BM) boundary integral equation (BIE) for Helmholtz transmission problems is identical to that for the ordinary BM BIE for Helmholtz transmission problems with a transmissive circular inclusion. Although some numerical methods based on the direct-indirect mixed BM BIE can be computed faster than the ordinary BM BIE, its well-posedness has been unclear. This study resolves a part of the well-posedness, namely the injectivity of the integral operator with a transmissive circular inclusion.

math.AP↗

arXiv2025

Automatically well-conditioned collocation boundary element method for transmission problems based on the Burton--Miller formulation

This paper proposes a collocation boundary element method based on the Burton--Miller method for solving transmission problems, which is rapidly convergent within the Krylov subspace solver framework. Our study enhances Burton--Miller-type boundary integral equations tailored for transmission problems by exploiting the Calderon formula. In cases where a single material exists in an unbounded host medium, we demonstrate the formulation of the boundary integral equation such that the underlying integral operator ${\cal A}$ is spectrally well-conditioned. Specifically, ${\cal A}$ can be designed such that ${\cal A}^2$ has only a single eigenvalue accumulation point. Furthermore, we extend this to the multi-material case, proving that the square of the proposed operator has only a few eigenvalues clustering points. When the collocation method is used to discretise the proposed boundary integral equations, the good spectral properties of the integral operator are naturally inherited to the coefficient matrix $\mathsf{A}$ similarly to the Nystöm methods; Almost all eigenvalues of $\mathsf{A}^2$ cluster at a few points in the complex plane ensuring the small condition number for $\mathsf{A}$. Through numerical examples of several benchmark problems, we illustrate that our formulation reduces the iteration number required by iterative linear solvers, even in the presence of material junction points.

math.NA↗

arXiv2024

Linearly scalable fast direct solver based on proxy surface method for two-dimensional elastic wave scattering by cavity

This paper proposes an $O(N)$ fast direct solver for two-dimensional elastic wave scattering problems. The proxy surface method is extended to elastodynamics to obtain shared coefficients for low-rank approximations from discretized integral operators. The proposed method is a variant of the Martinsson-Rokhlin-type fast direct solver. Our variant avoids the explicit computation of the inverse of the coefficient matrix, thereby reducing the required number of matrix-matrix multiplications. Numerical experiments demonstrate that the proposed solver has a complexity of $O(N)$ in the low-frequency range and has a highly parallel computation efficiency with a strong scaling efficiency of 70\%. Furthermore, multiple right-hand sides can be solved efficiently; specifically, when solving problems with 180 right-hand side vectors, the processing time per vector from the second vector onward was approximately 28,900 times faster than that for the first vector. This is a key advantage of fast direct methods.

math.NA↗

arXiv2024

Fast wavefield evaluation method based on modified proxy-surface-accelerated interpolative decomposition for two-dimensional scattering problems

This paper presents a fast wavefield evaluation method for two-dimensional wave scattering problems. The proposed method is based on a modified version of proxy-surface-accelerated interpolative decomposition, making it effective even if the evaluation points are near the boundary. The commonly known fast multipole method requires the use of direct evaluations near the boundaries of scatterers because the analytical expansion of kernel functions does not converge. On the one hand, the proposed method does not require the analytical expansion of kernel functions. The validity and effectiveness of the proposed method are demonstrated using numerical examples.

math.NA↗