# Past Research

The LBNL Mathematics Group has made fundamental contributions to some of the key mathematical and algorithmic technologies in common practice and everyday use worldwide. These projects are lead by one of our four principle investigators: Alexandre Chorin, James A. Sethian, Jon Wilkening and Per-Olof Persson.

## Low Mach Number Models and Algorithms

Low Mach number models and algorithms solve the problem of vastly different wave speeds in a host of fluid calculations in which pressure waves move much faster than advection. Through a formal analysis based on a low Mach number expansion in the Navier-Stokes equations, the resulting methods solve a mathematically consistent set of equations for a spatially constant but time-varying pressure term coupled to the time-evolution of the fluid flow. They are the backbone of combustion calculations,… Read More »

## Level Set Methods for Propagating Interfaces

Level set methods are computational methods for tracking moving interfaces. They rely on a central embedding of a moving front as the zero level set of an implicitly defined function, which results in an initial value Hamilton-Jacobi equation, and is accurately and efficiently solved by building on numerical schemes originally designed for hyperbolic conservation laws. They handle topological change, sharp corners and cusps, and naturally lend themselves to computational problems in which… Read More »

## Fast Marching and Ordered Upwind Methods for Wave Propagation and Control

Fast Marching Methods, and the more general class of Ordered Upwind Methods, are fast methods to solve Eikonal and Hamilton-Jacobi equations which arise in control and anisotropic front propagation. They exploit a fundamental ordering in Dijkstra’s method for the cheapest path on network graphs, and couple this to upwind entropy-satisfying weak viscosity solutions to the gradient and derivative operators. The resulting methods are used in path planning, fast simulations in phololithography,… Read More »

## PDE-based Image Segmentation Techniques

PDE-based image segmentation techniques couple level set methods and Fast Marching Methods to quickly and accurately extract boundaries from image data. They do not require prior knowledge about the number or topology of objects in the image data. They are in common use throughout medical imaging, such as in cardiac scanners, as well as in use to analyze porous materials from light source data and in robotic devices on assembly… Read More »

## Voronoi Implicit Interface Methods for Multiphase Physics

Voronoi Implicit Interface Methods, known as “VIIM”, provide the first robust, accurate, and reliable mathematical and computational methodology to track multiple interacting and evolving regions (“phases”) whose motion is determined by complex physics (fluids, mechanics, elasticity, etc.). This approach automatically handles multiple junctions, triple points/lines and quadruple points, and topological changes in the system occur naturally, with no surgery required. These methods are… Read More »

## Vortex Methods for Turbulent Flow

Vortex methods are meshless, particle-based methods which solve high Reynolds fluid flow in vorticity form. They accurately compute far beyond the flow regimes of meshed finite difference/finite element methods, relying on the mathematical properties of random walks to model viscosity and employing a natural adaptivity. They are used to compute flows past bluff bodies, turbines, liquid jets, etc., and have been used extensively to model optimal engine design and wake effects in ship… Read More »

## Projection Methods for Incompressible Fluid Dynamics

Projection methods are the core mathematical and computational fluid methodology to accurately account for incompressibility in the Navier-Stokes equations. By continually projecting the evolving solution onto the subspace of divergence-free vectors fields through the Hodge decomposition, these methods provide a robust way to compute a wide host of pratical engineering problems. They are ubiquitous in computational fluid mechanics, and are at the core of current day aircraft simulations,… Read More »

## Optimal Prediction and Dimensional Reduction

Optimal prediction is an adaptation of a formalism from irreversible statistical mechanics to the reduction of the dimension of large systems of time-dependent differential equations, and to the optimal estimation of solutions of differential equations when the data are noise or incomplete. It reduces such problems to the solution of generalized Langevin equations, where the memory and noise terms are found numerically. It is used in climate modeling and in problems of condensed matter physics. Read More »

## Multiple and Escape Arrival Techniques for Geophysical Imaging and Inversion

Escape arrivals methods solve problem of computing multiple arrivals, based on high frequency approximations to the wave equation. These non-viscosity solutions, which contain swallowtails and overlaps, are required to accurately perform seismic inversion, since first arrivals may not contain the most energy. The methods rely on recasting the time-dependent Lagrangian characteristic equations in phase space as time-dependent Eulerian Liouville equations. These are then converted into coupled… Read More »

## Compact Discontinuous Galerkin Methods for Fluids

Finite element methods, while low order, naturally work on unstructured meshes, and can compute physics inside and around complex, moving boundaries. Discontinuous Galerkin methods, which use high order polynomials within mesh elements, are significantly more accurate, but suffer because the complexity of linking the influence of the elements results in very slow methods. Compact DG Methods, which change the numerical fluxes in LDG schemes to achieve stencils that are compact (only connects… Read More »