A Variational Data Assimilation Procedure for the Incompressible Navier-Stokes Equations in Hemodynamics

We propose a data assimilation (DA) technique for including noisy measurements of the velocity field into the simulation of the Navier-Stokes equations (NSE) driven by hemodynamics applications. The technique is formulated as an inverse problem where we use a Discretize-then-Optimize approach to minimize the misfit between the recovered velocity field and the data, subject to the incompressible NSE. The DA procedure for this nonlinear problem is a combination of two approaches: the Newton method for the NSE and the DA procedure we designed and tested for the linearized problem. We discuss conditions on the location of velocity measurements that guarantee the well-posedness of the minimization process for the linearized problem. Numerical results, with both noise-free and noisy data, certify the theoretical analysis. Moreover, we consider 2D non-trivial geometries and 3D axisymmetric geometries. Also, we study the impact of noise on non-primitive variables of medical interest.

[1]  M. Benzi,et al.  INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids (2010) Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/fld.2267 Modified augmented Lagrangian preconditioners for the incompressible Navier , 2022 .

[2]  P. Hansen.  Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion , 1987 .

[3]  Alfio Quarteroni,et al.  Cardiovascular mathematics : modeling and simulation of the circulatory system , 2009 .

[4]  C. Taylor,et al.  Predictive medicine: computational techniques in therapeutic decision-making. , 1999, Computer aided surgery : official journal of the International Society for Computer Aided Surgery.

[5]  J L Lions,et al.  On the controllability of distributed systems. , 1997, Proceedings of the National Academy of Sciences of the United States of America.

[6]  Thomas A. Manteuffel,et al.  Weighted least-squares finite elements based on particle imaging velocimetry data , 2010, J. Comput. Phys..

[7]  A. Quarteroni,et al.  Numerical Approximation of Partial Differential Equations , 2008 .

[8]  B Wang,et al.  Data assimilation and its applications. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[9]  Robert Atlas,et al.  Ooce Note Series on Global Modeling and Data Assimilation Estimation Theory and Foundations of Atmospheric Data Assimilation , 2022 .

[10]  A. Veneziani,et al.  A DATA ASSIMILATION TECHNIQUE FOR INCLUDING NOISY MEASUREMENTS OF THE VELOCITY FIELD INTO NAVIER-STOKES SIMULATIONS , 2010 .

[11]  L. Hou,et al.  Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with Dirichlet controls , 1991 .

[12]  Max Gunzburger,et al.  Perspectives in flow control and optimization , 1987 .

[13]  Max Gunzburger,et al.  SENSITIVITIES, ADJOINTS AND FLOW OPTIMIZATION , 1999 .

[14]  Alessandro Veneziani,et al.  Methods for assimilating blood velocity measures in hemodynamics simulations: Preliminary results , 2010, ICCS.

[15]  Rolf Rannacher,et al.  Hemodynamical Flows: Modeling, Analysis and Simulation (Oberwolfach Seminars) , 2007 .

[16]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[17]  K. Kunisch,et al.  Second order methods for boundary control of the instationary Navier‐Stokes system , 2004 .

[18]  Trevor Q. Robbie,et al.  Hemodynamic predictors of aortic dilatation in bicuspid aortic valve by velocity-encoded cardiovascular magnetic resonance , 2010, Journal of cardiovascular magnetic resonance : official journal of the Society for Cardiovascular Magnetic Resonance.

[19]  L. Hou,et al.  Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with distributed and Neumann controls , 1991 .

[20]  Ionel M. Navon,et al.  Data Assimilation for Geophysical Fluids , 2009 .

[21]  M. Hinze.  Optimal and instantaneous control of the instationary Navier-Stokes equations , 2002 .

[22]  Giovanni Paolo Galdi,et al.  Hemodynamical Flows: Modeling, Analysis and Simulation , 2008 .

[23]  H. Elman,et al.  Efficient preconditioning of the linearized Navier-Stokes , 1999 .

[24]  Rolf Rannacher,et al.  ARTIFICIAL BOUNDARIES AND FLUX AND PRESSURE CONDITIONS FOR THE INCOMPRESSIBLE NAVIER–STOKES EQUATIONS , 1996 .

[25]  Alfio Quarteroni,et al.  Complex Systems in Biomedicine , 2006 .

[26]  Ricardo Todling.  Estimation Theory and Atmospheric Data Assimilation , 2013 .

[27]  R. Dwight.  Bayesian inference for data assimilation using least-squares finite element methods , 2010 .

[28]  Christian Vergara,et al.  An approximate method for solving incompressible Navier–Stokes problems with flow rate conditions , 2007 .

[29]  M. Heinkenschloss,et al.  Optimal control of unsteady compressible viscous flows , 2002 .

[30]  Andrew J. Wathen,et al.  A Preconditioner for the Steady-State Navier-Stokes Equations , 2002, SIAM J. Sci. Comput..

[31]  T. J. Rivlin.  An Introduction to the Approximation of Functions , 2003 .