Parallel in time numerical method for turbulent fluid flow
Dr. Mohamed Kamel Riahi
- The increasing demand on numerical simulations has pushed toward inventing new methods that could benefit from the huge capacity that high performance computing technology offers nowadays. Parallelism across the time direction has been for a long time a difficult task. As the sequential nature of the propagation stands against the time decomposition. Indeed, with a lack of knowledge of the present the future could not be predicted anyways. The parareal in time algorithm, with its apparition twenty years ago broke this tradition and proposed to solve the sequentially natural problems on high performance computers. The algorithm iteratively corrects and predicts intermediate solutions until convergence. With all its success in many applications, the plain version of the parareal algorithms needs some adjustment in many cases in order to deal with the advection-dominated problems. One of the main challenges in applied mathematics is the turbulence (usually occurs with high Reynold’s number in Navier-Stokes equations). We aim with this project at proposing a parallel in time algorithm that helps in simulating turbulence in time-parallelized fashion using newly developed numerical techniques.
Robust parallel in space and in time optimal control algorithm
Dr. Mohamed Kamel Riahi and Dr. Mohamed al Khaleel
- The optimal control of partial differential equations is an old field with many direct applications in engineering and finance. With this project we aim at new parallel numerical approaches in order to accelerate the decision and or the predictions for the handled problem using sophisticated numerical methods. Traditionally, optimal control problems are solved using iterative optimization techniques based on forward and backward resolutions of the treated model. This classical approach is memory consuming and time burdening. With this research we are working on developing robust numerical schemes based on parareal-type methods and also on Waveform techniques. A newly proposed technique based on the control decomposition “virtual controls” is also of utmost importance in our study to propose efficient optimal control tools.