Research

The Sublime Lab develops computational electromagnetics and scientific computing methods for modeling electromagnetic interactions in complex biological and engineered systems. We focus on scalable algorithms for problems that are computationally challenging because of geometric complexity, multiscale structure, or the need for repeated large-scale simulation. Our methodological contributions include new integral equation formulations, fast direct and iterative solvers, matrix compression algorithms, uncertainty quantification frameworks, and multiphysics electromagnetic models. This work is organized around three research thrusts; each project below links to a page with its background, approach, results, and publications.

1. Fast solvers and uncertainty quantification for transcranial magnetic stimulation (TMS)

In brain stimulation the “dose” is an electric field inside the brain that cannot be measured and must be computed. Brain electric fields vary with anatomy, MRI segmentation errors, and coil placement, so reliable dosing, targeting, and coil design require uncertainty quantification and optimization over thousands to millions of simulations, far beyond what standard solvers allow. Our goal is accurate, reproducible, clinic-speed TMS dosing and targeting. An overview of this thrust appears in our invited IEEE Antennas and Propagation Magazine article (2026).

Low-rank compression across coil placements (the auxiliary dipole method and its extensions), giving brain electric fields for about one million coil placements from a handful of simulations.
Real-time electric field solvers that factorize the head model once and return the whole-brain electric field for any coil position in milliseconds, enabling electric-field-informed neuronavigation.
Uncertainty quantification of TMS dosimetry: stochastic models of MRI segmentation error and anatomical variability, and conditions for numerically accurate simulation.
Group-level computational dosimetry and coil placement optimization for clinical TMS.
Computational design of TMS coils with optimal trade-offs between focality, depth, and energy, including experimentally validated focal-deep coils and low-cost stimulation hardware for animal experiments.
Deep learning emulators and classifiers for transcranial electric stimulation dosimetry and for brain imaging and electroencephalography data.
Our solvers are released as open-source software (see Software) and are being integrated into neuronavigation systems.

2. Multiscale (bidomain) modeling of neurons in electric fields

Understanding how device-generated electric fields activate neurons is essential for explaining experimental observations and improving therapies. Conventional cable models ignore the neuron’s effect on its local electric field and miss phenomena such as transverse stimulation, ephaptic coupling, and weak-field entrainment, while volumetric bidomain models are intractable for realistic neuron morphologies.

BidomainBEM, the first integral-equation bidomain formulation, which captures full coupling among neurons, the extracellular medium, and device fields using surface meshes only.
Hierarchical-matrix fast direct solvers that factorize once and are reused over thousands of time steps, enabling simulation of morphologically realistic reconstructed neurons and groups of neurons.
Boundary element and cable hybrid formulations, realistic neuron mesh generation, and fast multipole acceleration toward network-scale simulations.
Long-term goal: cortical-column networks of thousands of realistic neurons with full electromagnetic coupling under external stimulation.

3. Scalable computational electromagnetics for electrically large industrial systems

Layered-media structures, chip packages, and lithography masks now approach a thousand wavelengths in size, and low-frequency breakdown and high material contrast break classical formulations. Through the Consortium for Electromagnetic Technologies, which Dr. Gomez co-directs with Professors Weng Cho Chew and Dan Jiao, we develop solvers that go beyond what general-purpose commercial tools can handle.

A hybrid discrete exterior calculus and surface integral equation framework for potential-based electromagnetic analysis of heterogeneous media.
Kernel-independent fast multipole methods for layered Green’s functions and scalable integral equation solvers for electrically large quasi-planar and layered systems, including hierarchical off-diagonal low-rank inversion of sparse matrices and interconnect solvers for chip packaging.
High-contrast-stable volume integral equations for broad frequency and material parameter ranges.
Full-wave-guided optical proximity correction for lithography mask optimization.
Long-term goal: full-wave simulation and mask optimization of circuits approaching a thousand wavelengths on a single compute node.

Additional projects

Magnetoacoustic tomography with magnetic induction for accurate, high-resolution electrical conductivity imaging, and the fast, broadband acoustic volume integral equation solvers it requires.
Tensor compression algorithms, fast direct volume integral equation solvers, and other optimal-complexity solvers.
Stochastic partial differential equation approaches for rough-surface and geometric uncertainty in electromagnetic scattering.

Software

We share the code behind our papers so that others can reproduce our results and build on them. Everything is on GitHub; questions and pull requests are welcome.

Auxiliary dipole method: fast coil placement optimization for TMS (Gomez, Dannhauer, and Peterchev, NeuroImage 2021). The method is also included in the SimNIBS platform.
PMD-TMS: probabilistic matrix decomposition solvers for group-level electric field dosimetry and coil placement (Hasan, Wang, and Gomez, Computers in Biology and Medicine 2023).
Real-Time-TMS: whole-brain TMS electric field computation in under 4 ms per coil placement (Hasan et al., Imaging Neuroscience 2025).
TMS_Efield_Solvers: finite element (first to third order), finite difference, and boundary element TMS electric field solvers in MATLAB, from our study of the conditions for numerically accurate TMS simulation (Gomez et al., Brain Stimulation 2020).
fdTMS: coil design companion that generates focal-deep TMS coil windings on triangular-mesh coil supports (Gomez, Goetz, and Peterchev, Journal of Neural Engineering 2018).
BidomainBEM: boundary element bidomain solver for neuronal responses to electromagnetic fields (Czerwonky, Aberra, and Gomez, Journal of Neural Engineering 2024).