arXiv subjects

Masado Ishii

Publications and source records attributed to Masado Ishii.

8 recordsLinked to original sources

arXiv2026

Towards Fully Dynamic Omnitrees: Moment-Conserving Anisotropic Compression With Wavelets

Recently, omnitrees were introduced as a flexible space partitioning tree that improves upon the benefits of both octrees and k-d trees: Omnitrees' efficient encoding of anisotropic refinements holds particular interest for applications with anisotropic features and high dimensionality. These include, but are not limited to, computer graphics, databases, machine learning, and physics simulations. The present paper defines new operations on the omnitree encoding that extend its capabilities from the existing refinement to also include coarsening and therefore fully adaptive compression. It demonstrates natural integration of omnitrees with wavelets, which conserves moments of the stored function by design. For omnitrees, the wavelet coefficients can be interpreted as local refinement priorities, which can be used to guide the adaptation process. We derive algorithms for coarsening and downsplit that are guided by wavelet coefficients, and show their application to a large dataset of 3D shapes, as well as the continuous-valued density field of a cloud. The comparison to OpenVDB, a widely-used data structure for sparse volumetric data in computer graphics, enables a demonstration of the practical benefits of omnitrees even for moderately anisotropic three-dimensional data. Compared to OpenVDB, objects can be stored using up to 28x less space, and asymptotically show savings that exceed theoretical expectations. Using lossy compression, the cloud dataset can be compressed by $\approx5\times$ compared to OpenVDB, with negligible loss of visual quality. This demonstrates the potential of omnitrees for efficient storage and processing, and motivates further research into their applications in various domains.

cs.DS↗

arXiv2026

The Beauty of Anisotropic Mesh Refinement: Omnitrees for Efficient Dyadic Discretizations

Structured adaptive mesh refinement (AMR), commonly implemented via quadtrees and octrees, underpins a wide range of applications including databases, computer graphics, physics simulations, and machine learning. However, octrees enforce isotropic refinement in regions of interest, which can be especially inefficient for problems that are intrinsically anisotropic--much resolution is spent where little information is gained. This paper presents omnitrees as an anisotropic generalization of octrees and related data structures. Omnitrees allow to refine only the locally most important dimensions, providing tree structures that are less deep than bintrees and less wide than octrees. As a result, the convergence of the AMR schemes can be increased by up to a factor of the dimensionality d for very anisotropic problems, quickly offsetting their modest increase in storage overhead. We validate this finding on the problem of binary shape representation across 4,166 three-dimensional objects: Omnitrees increase the mean convergence rate by 1.5x, require less storage to achieve equivalent error bounds, and maximize the information density of the stored function faster than octrees. These advantages are projected to be even stronger for higher-dimensional problems. We provide a first validation by introducing a time-dependent rotation to create four-dimensional representations, and discuss the properties of their 4-d octree and omnitree approximations. Overall, omnitree discretizations can make existing AMR approaches more efficient, and open up new possibilities for high-dimensional applications.

cs.DS↗

arXiv2026

Field conserving adaptive mesh refinement (AMR) scheme on massively parallel adaptive octree meshes

Adaptive mesh refinement (AMR) is widely used to efficiently resolve localized features in time-dependent partial differential equations (PDEs) by selectively refining and coarsening the mesh. However, in long-horizon simulations, repeated intergrid interpolations can introduce systematic drift in conserved quantities, especially for variational discretizations with continuous basis functions. While interpolation from parent-to-child during refinement in continuous Galerkin (CG) discretizations is naturally conservative, the standard injection-based child-to-parent coarsening interpolation is generally not. We propose a simple, scalable field-conserving coarsening operator for parallel, octree-based AMR. The method enforces discrete global conservation during coarsening by first computing field conserving coarse-element values at quadrature points and then recovering coarse nodal degrees of freedom via an $L^2$ projection (mass-matrix solve), which simultaneously controls the $L_2$ error. We evaluate the approach on mass-conserving phase-field models, including the Cahn--Hilliard and Cahn--Hilliard--Navier--Stokes systems, and compare against injection in terms of conservation error, solution quality, and computational cost.

math.NA↗

arXiv2023

Scalable adaptive algorithms for next-generation multiphase flow simulations

High-fidelity flow simulations are indispensable when analyzing systems exhibiting multiphase flow phenomena. The accuracy of multiphase flow simulations is strongly contingent upon the finest mesh resolution used to represent the fluid-fluid interfaces. However, the increased resolution comes at a higher computational cost. In this work, we propose algorithmic advances that aim to reduce the computational cost without compromising on the physics by selectively detecting key regions of interest (droplets/filaments) that require significantly higher resolution. The framework uses an adaptive octree-based meshing framework that is integrated with PETSc's linear algebra solvers. We demonstrate scaling of the framework up to 114,688 processes on TACC's Frontera. Finally, we deploy the framework to simulate one of the most resolved simulations of primary jet atomization. This simulation -- equivalent to 35 trillion grid points on a uniform grid -- is 64 times larger than the current state-of-the-art simulations and provides unprecedented insights into an important flow physics problem with a diverse array of engineering applications.

physics.comp-ph↗

arXiv2022

A projection-based, semi-implicit time-stepping approach for the Cahn-Hilliard Navier-Stokes equations on adaptive octree meshes

The Cahn-Hilliard Navier-Stokes (CHNS) system provides a computationally tractable model that can be used to effectively capture interfacial dynamics in two-phase fluid flows. In this work, we present a semi-implicit, projection-based finite element framework for solving the CHNS system. We use a projection-based semi-implicit time discretization for the Navier-Stokes equation and a fully-implicit time discretization for the Cahn-Hilliard equation. We use a conforming continuous Galerkin (cG) finite element method in space equipped with a residual-based variational multiscale (RBVMS) formulation. Pressure is decoupled using a projection step, which results in two linear positive semi-definite systems for velocity and pressure, instead of the saddle point system of a pressure-stabilized method. All the linear systems are solved using an efficient and scalable algebraic multigrid (AMG) method. We deploy this approach on a massively parallel numerical implementation using parallel octree-based adaptive meshes. The overall approach allows the use of relatively large time steps with much faster time-to-solve than similar fully-implicit methods. We present comprehensive numerical experiments showing detailed comparisons with results from the literature for canonical cases, including the single bubble rise and Rayleigh-Taylor instability.

math.NA↗

arXiv2022

Breakup dynamics in primary jet atomization using mesh- and interface- refined Cahn-Hilliard Navier-Stokes

We present a technique to perform interface-resolved simulations of complex breakup dynamics in two-phase flows using the Cahn-Hilliard Navier-Stokes equations. The method dynamically decreases the interface thickness parameter in relevant regions and simultaneously increases local mesh resolution, preventing numerical artifacts. We perform a detailed numerical simulation of pulsed jet atomization that shows a complex cascade of break-up mechanisms involving sheet rupture and filament formation. To understand the effect of refinement on the breakup, we analyze the droplet size distribution. The proposed approach opens up resolved simulations for various multiphase flow phenomena.

physics.flu-dyn↗

arXiv2021

Case study of SARS-CoV-2 transmission risk assessment in indoor environments using cloud computing resources

Complex flow simulations are conventionally performed on HPC clusters. However, the limited availability of HPC resources and steep learning curve of executing on traditional supercomputer infrastructure has drawn attention towards deploying flow simulation software on the cloud. We showcase how a complex computational framework -- that can evaluate COVID-19 transmission risk in various indoor classroom scenarios -- can be abstracted and deployed on cloud services. The availability of such cloud-based personalized planning tools can enable educational institutions, medical institutions, public sector workers (courthouses, police stations, airports, etc.), and other entities to comprehensively evaluate various in-person interaction scenarios for transmission risk. We deploy the simulation framework on the Azure cloud framework, utilizing the Dendro-kT mesh generation tool and PETSc solvers. The cloud abstraction is provided by RocketML cloud infrastructure. We compare the performance of the cloud machines with state-of-the-art HPC machine TACC Frontera. Our results suggest that cloud-based HPC resources are a viable strategy for a diverse array of end-users to rapidly and efficiently deploy simulation software.

cs.DC↗

arXiv2021

Scalable adaptive PDE solvers in arbitrary domains

Efficiently and accurately simulating partial differential equations (PDEs) in and around arbitrarily defined geometries, especially with high levels of adaptivity, has significant implications for different application domains. A key bottleneck in the above process is the fast construction of a `good' adaptively-refined mesh. In this work, we present an efficient novel octree-based adaptive discretization approach capable of carving out arbitrarily shaped void regions from the parent domain: an essential requirement for fluid simulations around complex objects. Carving out objects produces an $\textit{incomplete}$ octree. We develop efficient top-down and bottom-up traversal methods to perform finite element computations on $\textit{incomplete}$ octrees. We validate the framework by (a) showing appropriate convergence analysis and (b) computing the drag coefficient for flow past a sphere for a wide range of Reynolds numbers ($\mathcal{O}(1-10^6)$) encompassing the drag crisis regime. Finally, we deploy the framework on a realistic geometry on a current project to evaluate COVID-19 transmission risk in classrooms.

math.NA↗