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137

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Citations since 2016

## Publications

Publications (137)

We present a scalable strategy for development of mesh-free hybrid neuro-symbolic partial differential equation solvers based on existing mesh-based numerical discretization methods. Particularly, this strategy can be used to efficiently train neural network surrogate models for the solution functions and operators of partial differential equations...

We present a highly scalable strategy for developing mesh-free neuro-symbolic partial differential equation solvers from existing numerical discretizations found in scientific computing. This strategy is unique in that it can be used to efficiently train neural network surrogate models for the solution functions and the differential operators, whil...

We present a numerical method for the solution of interfacial growth governed by the Stefan model coupled with incompressible fluid flow. An algorithm is presented which takes special care to enforce sharp interfacial conditions on the temperature, the flow velocity and pressure, and the interfacial velocity. The approach utilizes level-set methods...

While superhydrophobic surfaces (SHSs) show promise for drag reduction applications, their performance can be compromised by traces of surfactant, which generate Marangoni stresses that increase drag. This question is addressed for soluble surfactant in a three-dimensional laminar channel flow, with periodic SHSs on both walls. We assume that diffu...

We present a machine learning framework that blends image super-resolution technologies with passive, scalar transport in the level-set method. Here, we investigate whether we can compute on-the-fly, data-driven corrections to minimize numerical viscosity in the coarse-mesh evolution of an interface. The proposed system's starting point is the semi...

We present a numerical method for the solution of interfacial growth governed by the Stefan model coupled with incompressible fluid flow. An algorithm is presented which takes special care to enforce sharp interfacial conditions on the temperature, the flow velocity and pressure, and the interfacial velocity. The approach utilizes level-set methods...

We present an error-neural-modeling-based strategy for approximating two-dimensional curvature in the level-set method. Our main contribution is a redesigned hybrid solver [Larios-Cárdenas and Gibou, J. Comput. Phys. (May 2022), 10.1016/j.jcp.2022.111291] that relies on numerical schemes to enable machine-learning operations on demand. In particula...

We propose a data-driven mean-curvature solver for the level-set method. This work is the natural extension to $\mathbb{R}^3$ of our two-dimensional strategy in [arXiv:2201.12342][1] and the hybrid inference system of [DOI: 10.1016/j.jcp.2022.111291][2]. However, in contrast to [1,2], which built resolution-dependent neural-network dictionaries, he...

We present a novel hybrid strategy based on machine learning to improve curvature estimation in the level-set method. The proposed inference system couples enhanced neural networks with standard numerical schemes to compute curvature more accurately. The core of our hybrid framework is a switching mechanism that relies on well established numerical...

We present an error-neural-modeling-based strategy for approximating two-dimensional curvature in the level-set method. Our main contribution is a redesigned hybrid solver (Larios-C\'{a}rdenas and Gibou (2021)[1]) that relies on numerical schemes to enable machine-learning operations on demand. In particular, our routine features double predicting...

In this work we consider the inverse problem of finding guiding pattern shapes that result in desired self-assembly morphologies of block copolymer melts. Specifically, we model polymer self-assembly using Self-Consistent Field Theory and derive in a non-parametric setting the sensitivity of the misfit between desired and actual morphologies to arb...

We present a computational method for the simulation of the solidification of multicomponent alloys. Contrary to the case of binary alloys where a fixed point iteration is adequate, we hereby propose a Newton-type approach to solve the non-linear system of coupled PDEs arising from the time discretization of the governing equations. A combination o...

This work presents a general and unified theory describing block copolymer self-assembly in the presence of free surfaces and nanoparticles in the context of Self-Consistent Filed Theory. Specifically, the derived theory applies to free and tethered polymer chains, nanoparticles of any shape, arbitrary non-uniform surface energies and grafting dens...

We present a machine learning framework that blends image super-resolution technologies with scalar transport in the level-set method. Here, we investigate whether we can compute on-the-fly data-driven corrections to minimize numerical viscosity in the coarse-mesh evolution of an interface. The proposed system's starting point is the semi-Lagrangia...

We propose a novel composite framework to find unknown fields in the context of inverse problems for partial differential equations (PDEs). We blend the high expressibility of deep neural networks as universal function estimators with the accuracy and reliability of existing numerical algorithms for partial differential equations as custom layers i...

We present a simple framework for calculating the electric potential by solving the nonlinear Poisson-Boltzmann equation and the free solvation energies of large biomolecules. To achieve this we build upon the work of Bochkov and Gibou [9] to develop a novel solver capable of solving nonlinear elliptic equations, where the diffusion coefficient, th...

We present a hybrid strategy based on deep learning to compute mean curvature in the level-set method. The proposed inference system combines a dictionary of improved regression models with standard numerical schemes to estimate curvature more accurately. The core of our framework is a switching mechanism that relies on well-established numerical t...

Trace amounts of surfactants have been shown to critically prevent the drag reduction of superhydrophobic surfaces (SHSs), yet predictive models including their effects in realistic geometries are still lacking. We derive theoretical predictions for the velocity and resulting slip of a laminar fluid flow over three-dimensional SHS gratings contamin...

We introduce an approach for solving the incompressible Navier-Stokes equations on a forest of Octree grids in a parallel environment. The methodology uses the p4est library of Burstedde et al. (2011) [15] for the construction and the handling of forests of Octree meshes on massively parallel distributed machines and the framework of Mirzadeh et al...

We present a theoretical framework to model the electric response of cell aggregates. We establish a coarse representation for each cell as a combination of membrane and cytoplasm dipole moments. Then we compute the effective conductivity of the resulting system, and thereafter derive a Fokker-Planck partial differential equation that captures the...

We present a simple numerical algorithm for solving elliptic equations where the diffusion coefficient, the source term, the solution and its flux are discontinuous across an irregular interface. The algorithm produces second-order accurate solutions and first-order accurate gradients in the L∞-norm on Cartesian grids. The condition number is bound...

We propose a deep learning strategy to compute the mean curvature of an implicit level-set representation of an interface. Our approach is based on fitting neural networks to synthetic datasets of pairs of nodal $\phi$ values and curvatures obtained from circular interfaces immersed in different uniform resolutions. These neural networks are multil...

We introduce a simple but effective correction to the Ghost Fluid Method (GFM) introduced by Liu, Fedkiw and Kang, J. Comput. Phys., 160 (2000) for capturing sharp interface conditions when solving Poisson equations with discontinuities. Our method is shown to alleviate the GFM lack of convergence for the flux across the interfaces: pointwise conve...

Superhydrophobic surfaces (SHSs) have the potential to reduce drag at solid boundaries. However, multiple independent studies have recently shown that small amounts of surfactant, naturally present in the environment, can induce Marangoni forces that increase drag, at least in the laminar regime. To obtain accurate drag predictions, one must solve...

We propose a novel composite framework to find unknown fields in the context of inverse problems for partial differential equations (PDEs). We blend the high expressibility of deep neural networks as universal function estimators with the accuracy and reliability of existing numerical algorithms for partial differential equations as custom layers i...

We present a PDE-based approach for the multidimensional extrapolation of smooth scalar quantities across interfaces with kinks and regions of high curvature. Second- and third-order accurate extensions in the $L^\infty$ norm are obtained with linear and quadratic extrapolations, respectively. The accuracy of the method is demonstrated on a number...

We study analytically and numerically aspects of the dynamics of slope selection for one-dimensional models describing the motion of line defects, steps, in homoepitaxial crystal growth. The kinetic processes include diffusion of adsorbed atoms (adatoms) on terraces, attachment and detachment of atoms at steps with large yet finite, positive Ehrlic...

We present a simple numerical algorithm for solving elliptic equations where the diffusion coefficient, the source term, the solution and its flux are discontinuous across an irregular interface. The algorithm produces second-order accurate solutions and first-order accurate gradients in the $L^\infty$-norm on Cartesian grids. The condition number...

We investigate the scaling behavior for roughening and coarsening of mounds during unstable epitaxial growth. By using kinetic Monte Carlo (KMC) simulations of two lattice-gas models of crystal surfaces, we find scaling exponents that characterize roughening and coarsening at long times. Our simulation data show that these exponents have a complica...

Superhydrophobic surfaces (SHSs) have the potential to reduce drag at solid boundaries. However, multiple independent studies have recently shown that small amounts of surfactant, naturally present in the environment, can induce Marangoni forces that increase drag, at least in the laminar regime. To obtain accurate drag predictions, one must solve...

We present a numerical method for simulating incompressible immiscible fluids, in two and three spatial dimensions. It is constructed as a modified pressure correction projection method on adaptive non-graded Oc/Quadtree Cartesian grids, using the level-set framework to capture the moving interface between the two fluids. The sharp treatment of the...

Complex networks are composed of nodes (entities) and edges (connections) with any arbitrary topology. There may also exist multiple types of interactions among these nodes and each node may admit different states in each of its interactions with its neighbors. Understanding complex networks dwells on understanding their structure and function. How...

We analyze the accuracy of two numerical methods for the variable coefficient Poisson equation with discontinuities at an irregular interface. Solving the Poisson equation with discontinuities at an irregular interface is an essential part of solving many physical phenomena such as multiphase flows with and without phase change, in heat transfer, i...

This paper is associated with a video winner of a 2017 APS/DFD Gallery of Fluid Motion Award. The original video is available from the Gallery of Fluid Motion, https://doi.org/10.1103/APS.DFD.2017.GFM.V0098

In this chapter, following the previous one, we briefly present the modern approach to real-space renormalization group (RG) theory based on tensor network formulations which was developed during the last two decades. The aim of this sequel is to suggest a novel framework based on tensor networks in order to find the fixed points of complex systems...

Complex networks are composed of nodes (entities) and edges (connections) with any arbitrary
topology. There may also exist multiple types of interactions among these nodes and
each node may admit different states in each of its interactions with its neighbors. Understanding
complex networks dwells on understanding their structure and function. How...

We present two finite volume schemes to solve a class of Poisson-type equations subject to Robin boundary conditions in irregular domains with piecewise smooth boundaries. The first scheme results in a symmetric linear system and produces second-order accurate numerical solutions with first-order accurate gradients in the L∞-norm (for solutions wit...

We introduce a level-set strategy to find the geometry of confinement that will guide the self-assembly of block copolymers to a given target design in the context of lithography. The methodology is based on a shape optimization algorithm, where the level-set normal velocity is defined as the pressure field computed through a self-consistent field...

We present algorithms for constructing a level-set representation of the Solvent-Excluded Surface of biomolecules. The algorithms are utilizing Octree Cartesian grids in the paradigm of distributed computing, and are designed to balance the computational load. The method is shown to be fast, scalable and first-order accurate. The procedure is robus...

We present a review on numerical methods for simulating multiphase and free surface flows. We focus in particular on numerical methods that seek to preserve the discontinuous nature of the solutions across the interface between phases. We provide a discussion on the Ghost-Fluid and Voronoi Interface methods, on the treatment of surface tension forc...

Electropermeabilization (also called electroporation) is a significant increase in the electrical conductivity and permeability of the cell membrane that occurs when pulses of large amplitude (a few hundred volts per centimeter) are applied to the cells. • Due to the electric field, the cell membrane is permeabilized, and then non-permeant molecule...

We present a conservative method for solving the Poisson equation in irregular domains with Robin boundary conditions. Second-order accurate solutions and gradients in the L∞ norm are obtained.

We introduce a numerical framework that enables unprecedented direct numerical studies of the electropermeabilization effects of a cell aggregate at the meso-scale. Our simulations qualitatively replicate the shadowing effect observed in experiments and reproduce the time evolution of the impedance of the cell sample in agreement with the trends ob...

We introduce an approach for simulating epitaxial growth by use of an island dynamics model on a forest of quadtree grids, and in a parallel environment. To this end, we use a parallel framework introduced in the context of the level-set method. This framework utilizes: discretizations that achieve a second-order accurate level-set method on non-gr...

We review some of the recent advances in level-set methods and their applications. In particular, we discuss how to impose boundary conditions at irregular domains and free boundaries, as well as the extension of level-set methods to adaptive Cartesian grids and parallel architectures. Illustrative applications are taken from the physical and life...

We derive functional level-set derivatives for the Hamiltonian arising in self-consistent field theory, which are required to solve free boundary problems in the self-assembly of polymeric systems such as block copolymer melts. In particular, we consider Dirichlet, Neumann and Robin boundary conditions. We provide numerical examples that illustrate...

We present an approach to simulate the diffusion, advection and adsorption-desorption of a material quantity defined on an interface in two and three spatial dimensions. We use a level-set approach to capture the interface motion and a Quad-/Oc-tree data structure to efficiently solve the equations describing the underlying physics. Coupling with a...

We present a discretization method for the multidimensional Dirac distribution. We show its applicability in the context of integration problems, and for discretizing Dirac-distributed source terms in Poisson equations with constant or variable diffusion coefficients. The discretization is cell-based and can thus be applied in a straightforward fas...

We present a Voronoi Interface approach to the study of cell electropermeabilization. We consider the nonlinear electropermeabilization model of Poignard et al. [20], which takes into account the jump in the voltage potential across cells' membrane. The jump condition is imposed in a sharp manner, using the Voronoi Interface Method of Guittet et al...

We present scalable algorithms for the level-set method on dynamic, adaptive Quadtree and Octree Cartesian grids. The algorithms are fully parallelized and implemented using the MPI standard and the open-source p4est library. We solve the level set equation with a semi-Lagrangian method which, similar to its serial implementation, is free of any ti...

We introduce a framework for simulating the mesoscale self-assembly of block copolymers in arbitrary confined geometries subject to Neumann boundary conditions. We employ a hybrid finite difference/volume approach to discretize the mean-field equations on an irregular domain represented implicitly by a level-set function. The numerical treatment of...

We present numerical methods that enable the direct numerical simulation of two-phase flows in irregular domains. A method is presented to account for surface tension effects in a mesh cell containing a triple line between the liquid, gas and solid phases. Our numerical method is based on the level-set method to capture the liquid-gas interface and...

The fast sweeping method is a popular algorithm for solving a variety of static Hamilton-Jacobi equations. Fast sweeping algorithms for parallel computing have been developed, but are severely limited. In this work, we present a multilevel, hybrid parallel algorithm that combines the desirable traits of two distinct parallel methods. The fine and c...

Directed self-assembly using block copolymers for positioning vertical interconnect access in integrated circuits relies on the proper shape of a confined domain in which polymers will self-assemble into the targeted design. Finding that shape, i.e., solving the inverse problem, is currently mainly based on trial and error approaches. We introduce...

We introduce a fast error-free tracking method applicable to sequences of two and three dimensional images. The core idea is to use Quadtree (resp. Octree) data structures for representing the spatial discretization of an image in two (resp. three) spatial dimensions. This representation enables one to merge into large computational cells the regio...

We present a novel Eulerian numerical method to compute global isochrons of a stable periodic orbit in high dimensions. Our approach is to formulate the asymptotic phase as a solution to a first order boundary value problem and solve the resulting Hamilton-Jacobi equation with the parallel fast sweeping method. All isochrons are then given as isoco...

We consider the Poisson equation with mixed Dirichlet, Neumann and Robin boundary conditions on irregular domains. We describe a straightforward and efficient approach for imposing the mixed boundary conditions using a hybrid finite-volume/finite-difference approach, leveraging on the work of Gibou et al. (2002) [14], Ng et al. (2009) [30] and Papa...

We present a numerical method for solving the incompressible Navier–Stokes equations on non-graded quadtree and octree meshes and arbitrary geometries. The viscosity is treated implicitly through a finite volume approach based on Voronoi partitions, while the convective term is discretized with a semi-Lagrangian scheme, thus relaxing the restrictio...

We introduce a simple method, dubbed the Voronoi Interface Method, to solve Elliptic problems with discontinuities across the interface of irregular domains. This method produces a linear system that is symmetric positive definite with only its right-hand-side affected by the jump conditions. The solution and the solution's gradients are second-ord...

In this paper we present a novel hybrid finite-difference/finite-volume method for the numerical solution of the nonlinear Poisson–Nernst–Planck (PNP) equations on irregular domains. The method is described in two spatial dimensions but can be extended to three dimensional problems as well. The boundary of the irregular domain is represented implic...

We use direct numerical simulations of the Poisson-Nernst-Planck equations to study the charging kinetics of porous electrodes and to evaluate the predictive capabilities of effective circuit models, both linear and nonlinear. The classic transmission line theory of de Levie holds for general electrode morphologies, but only at low applied potentia...

We formulate and implement a generalized island-dynamics model of epitaxial growth based on the level-set technique to include the effect of an additional energy barrier for the attachment and detachment of atoms at step edges. For this purpose, we invoke a mixed, Robin-type, boundary condition for the flux of adsorbed atoms (adatoms) at each step...

We present a numerical method for the simulation of binary alloys. We make use of the level-set method to capture the evolution of the solidification front and of an adaptive mesh refinement framework based on non-graded quadtree grids to efficiently capture the multiscale nature of the alloys’ concentration profile. In addition, our approach is ba...

In order to develop efficient numerical methods for solving elliptic and parabolic problems where Dirichlet boundary conditions are imposed on irregular domains, Chen et al. (J. Sci. Comput. 31(1):19–60, 2007) presented a methodology that produces second-order accurate solutions with second-order gradients on non-graded quadtree and octree data str...

We present an algorithm for solving in parallel the Eikonal equation. The efficiency of our approach is rooted in the ordering and distribution of the grid points on the available processors; we utilize a Cuthill–McKee ordering. The advantages of our approach is that (1) the efficiency does not plateau for a large number of threads; we compare our...