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ACMD Seminar: Computing Ill-Posed Time-Reversed 2D Navier-Stokes Equations, Using a Stabilized Explicit Finite Difference Scheme Marching Backward in Time

Alfred S. Carasso
Applied and Computational Mathematics Division, NIST

Tuesday, August 27, 2019, 3:00-4:00
Building 101, Lecture Room D

Tuesday, August 27, 2019, 1:00-2:00
Building 1, Room 3304

This talk will be broadcast on-line using BlueJeans.  Contact acmdseminar [at] (acmdseminar[at]nist[dot]gov) for details.

Abstract: This talk discusses an unconditionally stable explicit finite difference scheme, marching backward in time, that can solve an interesting but limited class of ill-posed, time-reversed, 2D incompressible Navier-Stokes initial value problems. Stability is achieved by applying a compensating smoothing operator at each time step to quench the instability. This leads to a distortion away from the true solution. However, in many interesting cases, the cumulative error is sufficiently small to allow for useful results. Effective smoothing operators based on (−∆) p , with real p > 2, can be efficiently synthesized using FFT algorithms. Similar stabilizing techniques were successfully applied in other ill-posed evolution equations. The analysis of numerical stabilty is restricted to a related linear problem. However, extensive numerical experiments indicate that such linear stability results remain valid when the explicit scheme is applied to a significant class of time-reversed nonlinear 2D Navier-Stokes initial value problems. Several reconstruction examples are included, based on the stream function-vorticity formulation, and focusing on 256 × 256 pixel images of recognizable objects. Such images, associated with non-smooth underlying intensity data, are used to create severely distorted data at time T > 0. Successful backward recovery is shown to be possible at parameter values far exceeding expectations.

Bio: Alfred S. Carasso received the Ph.D degree in mathematics at the University of Wisconsin in 1968. He was a professor of mathematics at the University of New Mexico, and a visiting staff member at the Los Alamos National Laboratory, prior to joining the National Institute of Standards and Technology in 1982. His major research interests lie in the theoretical and computational analyses of ill-posed continuation problems in partial differential equations, together with their application in system identification, non-destructive evaluation, inverse heat transfer, and image reconstruction and computer vision. He is the author of original theoretical papers, is a patentee in the field of image processing, and is an active speaker at national and international conferences in applied mathematics.

Note: Visitors from outside NIST must contact Cathy Graham; (301) 975-3800; at least 24 hours in advance.

Created July 1, 2019, Updated June 2, 2021