# Highlighted Research

These computational and mathematical methodologies drive a variety of applications.

## Implicit Sampling

The difficulty in sampling many-dimensional probability densities on a computer is a major obstacle to progress in many fields of science, for example in economics and finance, weather forecasting, quantum field theory, and statistics. There are typically too many states to sample, most of which have very small probability, while the location of the important states is not known in advance. The Mathematics Group at LBNL has developed a methodology, implicit sampling, which finds unbiased… Read More »

## High-Order Methods for Fluid-Structure Interaction with Applications to Vertical Axis Wind Turbine Simulations

The Mathematics Group has developed new numerical schemes for high-order accurate simulations of fluids and solids. This has led to efficient solvers that scale well on the new generation of multi-core computer architectures. The research focuses on applying these methods to relevant applications, such as wind turbine simulations. The studies have been used to determine optimal operating conditions and to maximize the power outputs of vertical-axis wind turbines, or VAWTs. Zoom-in of a… Read More »

## Mathematical and Algorithmic Methodologies for Computing Multi-phase Multiphysics: Industrial Foams, Materials, Manufacturing, and Cell Mechanics

Many problems involve the physics of multiply-connected moving interfaces. Examples include liquid foams (e.g. soap bubbles, polyurethane and colloidal mixtures), and solid foams, such as wood and bone. Manufactured solid foams lead to lightweight cellular engineering materials, including crash absorbent aluminum foams, and controlling foams is critical in chemical processing. Diagram showing the components of the Navier-Stokes equation, which includes velocity u, and surface tension. (Select… Read More »

## Long-time Dynamics and Optimization of Nonlinear PDE

The Math Group is developing new computational tools to study dynamic problems with time-periodic forcing or symmetry. A key step in this research is to devise adjoint-based optimization techniques suitable for large-scale systems governed by nonlinear evolutionary partial differential equations. Because the solutions of the PDEs are time-periodic, Floquet theory can be used to characterize their stability. The framework developed to compute time-periodic solutions is easily adapted to… Read More »