arXiv subjects

Danilo Aballay

Publications and source records attributed to Danilo Aballay.

5 recordsLinked to original sources

arXiv2026

Robust Hierarchical Matrix Compression of Acoustic Volume and Boundary Integral Operators

Discretizing integral formulations of the Helmholtz equation yields dense linear systems. Hence, simulating acoustic models at larger scales or higher frequencies is typically constrained by memory capacity. Fast algorithms, such as hierarchical matrix compression, reduce the memory footprint substantially while controlling the approximation error in matrix-vector multiplications. However, the commonly used Adaptive Cross Approximation suffers from early-convergence problems, where the iterative construction of low-rank decompositions stops before reaching the targeted error tolerance. This failure arises when the error estimator does not capture significant components of the matrix structure under partial pivoting. This manuscript proposes a new diagonal convergence criterion, additional matrix elements for the pivoting strategy, an extended admissibility condition, and a sustained convergence check to improve the robustness of hierarchical matrix compression. These modifications improve compression reliability without increasing memory. We tested our compression strategy on various discretized volume and boundary integral operators. The computational results show that our approach successfully compresses all benchmark matrices within predefined tolerances, thereby resolving the early-convergence issues encountered in standard algorithms. This robust matrix compression was achieved at the same memory footprint as alternative compression strategies. Furthermore, a complexity analysis shows log-linear memory scaling with mesh refinement at constant frequency. Finally, we successfully applied our robust matrix compression algorithm to a coupled system of volume and boundary integral operators that models transcranial ultrasound propagation. This confirms the feasibility of our robust algorithm to accelerate large-scale simulations with high-resolution meshes in a biomedical application.

math.NA↗

arXiv2026

Nested Volume-Surface Integral Equations for Acoustics

The simulation of high-frequency acoustic wave propagation in unbounded domains with local heterogeneous materials and high-contrast interfaces poses significant challenges to numerical methods. The volume-surface integral equation (VSIE) method is an attractive approach as it automatically satisfies the radiation condition at infinity via Green's functions, handles heterogeneous materials via Newton potentials, and models scattering at high-contrast interfaces via surface integral operators. However, its effectiveness in practical simulations has been limited by high computational costs, sensitivity to sharp interfaces, and insufficient computational verification. This study extends the applicability of VSIE by deriving integral formulations for nested heterogeneous materials with parameter jumps at interfaces. We also develop extensive benchmarks against coupled finite-element and boundary-element methods to verify the VSIE's accuracy and mesh convergence. The various benchmarks using open-source software demonstrate the effectiveness of VSIE for large-scale acoustic simulations.

math.NA↗

arXiv2026

Improved global stability bounds for two-dimensional plane Poiseuille flow

This work provides new lower bounds on the global (nonlinear) stability limit of pressure-driven two-dimensional plane Poiseuille flow, improving on the energy stability limit, $Re_E$, originally computed by Orr in 1907. Using a computer we carefully construct quartic Lyapunov functionals of the velocity perturbations about the laminar profile, which certify the nonlinear stability of the flow to arbitrary perturbations. The formulation combines a decomposition of the velocity into finitely many energy eigenmodes, referred to as a 'mode set', and an infinite-dimensional 'tail', together with explicit bounds that recast the Lyapunov inequality conditions as semidefinite programs, whose feasibility is tested. Over the streamwise lengths considered, the certified stability limit exceeds the classical energy bound. In particular, at the critical energy-stable streamwise length, where $Re_E\approx 87.59$, the flow is found to be globally stable up to $Re \approx 106.8$ (representing a $22\%$ improvement). Various modestly-sized mode sets, capable of capturing sufficient features of the nonlinear dynamics of energy growth and subsequent decay, are proposed and found to be successful in producing improved bounds, with the simplest one involving only five modes.

physics.flu-dyn↗

arXiv2025

Full-Wave Modeling of Transcranial Ultrasound using Volume-Surface Integral Equations and CT-Derived Heterogeneous Skull Data

Transcranial ultrasound therapy uses focused acoustic energy to induce therapeutic bioeffects in the brain. Ultrasound must be transmitted through the skull, which is highly attenuating and heterogeneous, causing beam distortion, reducing focal pressure, and shifting the target location. Computational models are frequently used to predict beam aberration, assess cranial heating, and correct the phase of ultrasound transducers. These models often rely on computed tomography (CT) images to build patient-specific geometries and estimate skull acoustic properties. However, the coarse voxel resolution of CT limits accuracy for differential equation solvers at ultrasound frequencies. This paper presents an efficient numerical method based on volume-surface integral equations to model full-wave acoustic propagation through heterogeneous skull bone. We show that our approach effectively simulates transcranial ultrasound, even when using the original CT voxels as the computational mesh, where the 0.5 mm voxel length is relatively coarse compared to the shortest wavelength of 3 mm. The method is validated against a high-resolution boundary element model using an averaged skull representation. Simulations using a CT-based skull model and a bowl transducer reveal significant beam distortion of 7.8 mm attributed to the skull's heterogeneous acoustical properties.

physics.med-ph↗

arXiv2025

An $r$-adaptive finite element method using neural networks for parametric self-adjoint elliptic problem

This work proposes an $r$-adaptive finite element method (FEM) using neural networks (NNs). The method employs the Ritz energy functional as the loss function, currently limiting its applicability to symmetric and coercive problems, such as those arising from self-adjoint elliptic problems. The objective of the NN optimization is to determine the mesh node locations. For simplicity in two-dimensional problems, these locations are assumed to form a tensor product structure. The method is designed to solve parametric partial differential equations (PDEs). For each PDE parameter instance, the optimal $r$-adapted mesh generated by the NN is then solved with a standard FEM. The construction of FEM matrices and load vectors is implemented such that their derivatives with respect to mesh node locations, required for NN training, can be efficiently computed using automatic differentiation. However, the linear equation solver does not need to be differentiable, enabling the use of efficient, readily available `out-of-the-box' solvers. Consequently, the proposed approach retains the robustness and reliability guarantees of the FEM for each parameter instance, while the NN optimization adaptively adjusts the mesh node locations. The method's performance is demonstrated on parametric Poisson problems using one- and two-dimensional tensor product meshes.

math.NA↗