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Continuous data assimilation in steady Navier-Stokes equations with unknown viscosity: robust and efficient solvers and fast parameter recovery
L. Rebholz, J. Reyes, J. Whitehead
TL;DR
The paper addresses recovery of unknown viscosity in steady Navier–Stokes equations from partial flow observations. It combines continuous data assimilation with a modified Newton parameter iteration and a CDA-Picard+CDA-Newton solver. The analysis proves well-posedness and quadratic convergence, while numerical tests show rapid and robust recovery, including from poor initial guesses.
Problem
The paper studies how to recover an unknown viscosity in steady Navier–Stokes equations when only partial solution data are available.
Method
The method combines continuous data assimilation, a modified Newton viscosity-recovery iteration, and a CDA-Picard+CDA-Newton solver.
Results
The analysis establishes well-posedness and quadratic convergence, while numerical tests show fast and robust recovery with poor initial guesses.
Takeaways & Limitations
Partial solution data can support efficient unknown-viscosity recovery for steady Navier–Stokes equations through the proposed CDA-based algorithms.
Takeaways & Limitations
Noisy-data behavior is not analyzed theoretically, and numerical tests indicate convergence only within a noise-dependent range of the true viscosity.
Abstract
from arXiv · showhide
Recent advances in equation discovery methods such as SINDy have highlighted the growing interest in identifying governing parameters and models directly from data. In this work, we take a complementary approach grounded in analysis and numerical PDE methods: we recover an unknown viscosity in steady Navier-Stokes equations (NSE) from partial incompressible flow observations using continuous data assimilation (CDA). We propose a simple and efficient parameter recovery algorithm and also a nonlinear solver for CDA-NSE. Together, this creates a highly efficient technique for recovering an unknown viscosity from partial solution data. Our analysis establishes the well-posedness of steady CDA-NSE, quadratic convergence of the parameter recovery algorithm, and quadratic convergence of a CDA-Picard + CDA-Newton nonlinear solver. Numerical experiments illustrate that the methods are very effective in restoring parameters quickly, even with poor initial guesses.
1 Introduction
The paper uses continuous data assimilation with partial flow observations to recover unknown viscosity in steady Navier–Stokes equations. It combines a modified Newton parameter iteration with a CDA-Picard+CDA-Newton solver and establishes strong theoretical convergence properties.
- Problem and motivation: The study recovers unknown viscosity in steady Navier–Stokes equations from partial solution data using continuous data assimilation.The steady equations model flows in two- or three-dimensional domains, with viscosity as the unknown parameter.
- CDA formulation: Continuous data assimilation is introduced into the steady Navier–Stokes system through an interpolated mismatch between computed and observed velocity fields.The CDA formulation uses a known viscosity approximation and an interpolant of partial observations.
- Parameter recovery: The objective function is quadratic in the viscosity error, enabling a modified Newton iteration whose derivative can be computed efficiently.The method exploits the parabolic objective structure and the repeated root at the true viscosity.
- Nonlinear solver: The CDA-Picard+CDA-Newton solver preserves unconditional stability and quadratic convergence while improving robustness to viscosity and initial guesses.The solver is used to compute the CDA system required by the parameter recovery iteration.
- Relation to prior work: The work extends prior CDA research from time-dependent and steady nonlinear-solver applications to unknown-parameter recovery in a steady system.The authors identify this as the first use of CDA to recover unknown parameters in a steady system.
2 Notation and Preliminaries
The preliminaries establish the functional, variational, and interpolation framework for analyzing steady Navier–Stokes equations and their CDA formulation. They also state existence, uniqueness conditions, and the coercive estimate used in convergence analysis.
- Setting: The analysis assumes a bounded connected domain with either a C2-smooth boundary or a convex polygonal or polyhedral boundary.These domain conditions support the functional-analytic setup.
- Function spaces: The framework defines L2, H^k, dual, and divergence-free spaces, with the latter used to formulate the incompressible problem.The weakly divergence-free space is characterized by a zero divergence norm.
- Function spaces: The Poincaré inequality makes the H1 norm equivalent to the L2 norm of the gradient under the chosen boundary conditions.The gradient norm is consequently used for both X and V.
- Weak formulation: The steady Navier–Stokes equations are expressed in weak form using the trilinear convection form b and the velocity-pressure pairing.An equivalent velocity-only formulation is available in the divergence-free space.
- Well-posedness: A solution exists for positive viscosity, while the stated smallness condition α0 < 1 guarantees uniqueness.The existence and uniqueness results apply to the weak formulation under the specified assumptions.
- CDA preliminaries: The interpolant IH is assumed linear and satisfies an approximation constraint that yields the coercive estimate used in CDA convergence analysis.The estimate introduces λ = min{µ, ν^2C_I^−2H^−2} in the supplied formulation.
3 CDA-NSE with inconsistent viscosity: well-posedness and consistency error
The CDA-NSE system incorporates partial observations while allowing the viscosity used by the solver to differ from the true viscosity. Under sufficient data and parameter conditions, it is well-posed and has a viscosity-mismatch consistency error bound.
- System formulation: The CDA-NSE system nudges the computed velocity toward partial observations using interpolant I_H and nudging parameter μ, with approximate viscosity ν̂.The nudging term penalizes the difference between interpolated computed and true velocities.
- Well-posedness: The solution satisfies an a priori stability bound used to establish existence and uniqueness.Existence follows through Leray–Schauder arguments, while uniqueness uses an energy estimate and the data-resolution condition.
- Well-posedness: The CDA-NSE system is well-posed when μ ≥ ν̂^2 C_I^2 H^-2 and the stated additional data condition holds.The theorem requires sufficiently strong nudging relative to the viscosity approximation and interpolant spacing.
- Consistency error: The consistency error arises from the viscosity mismatch between ν̂ and the true viscosity ν.Subtracting CDA-NSE from NSE produces an error equation containing the term (ν−ν̂)(∇u,∇e).
- Consistency error: The relative and L2 consistency errors are bounded proportionally to |ν̂−ν|/(ν̂ν) times the forcing-dependent factor.The bounds also require assumptions on μ and H, including a restriction on H.
4 CDA-Picard + CDA-Newton to solve CDA-NSE with inconsistent viscosity
The paper solves the inconsistent-viscosity CDA-NSE system with a two-step CDA-Picard + CDA-Newton iteration. Its analysis establishes stability and quadratic convergence under sufficient observation resolution.
- Iteration: Each iteration first applies CDA-Picard to produce an intermediate velocity and then CDA-Newton using that result as input.The two steps are defined as linearized updates with nudging toward the partial data.
- Stability: CDA-Picard solutions satisfy an a priori stability bound involving the forcing and the observed interpolated velocity.This bound controls the intermediate iterate used by the Newton step.
- Iteration error: The CDA-Newton correction error is bounded quadratically by the preceding CDA-Picard increment.Lemma 4.2 relates the Newton-step error to ∥∇(v_k−v_{k−1})∥^2.
- Convergence: The combined CDA-Picard + CDA-Newton solver converges quadratically when the observation spacing H satisfies the stated upper bound.The convergence basin expands as H decreases, with its radius scaled by H^-1/2.
5 Modified Newton parameter recovery algorithm
The modified Newton algorithm recovers the unknown viscosity by exploiting a quadratic error function and its derivative, computed through nonlinear and linear CDA solves.
- Error function: The recovery method defines an error function E(ν̂) whose local parabola has its root at the true viscosity ν.Near ν, E has a single root at the vertex (ν, 0).
- Derivative computation: The derivative E′(ν̂) is obtained by implicitly differentiating the CDA-NSE system with respect to ν̂.The derivative of the velocity requires one linear solve similar to a CDA-Newton solve.
- Modified Newton method: Because E is parabolic and has a double root at ν, the method uses a modified Newton update designed for quadratic convergence.Each parameter iteration evaluates E and E′ at the current viscosity approximation.
- Algorithm: Algorithm 5.1 alternates a CDA-NSE nonlinear solve for v_i, a linear sensitivity solve for ω_i, and a viscosity update.The nonlinear solve uses CDA-Picard+CDA-Newton, while the sensitivity solve determines ∂v/∂ν̂.
6 Numerical Results
Numerical tests across driven-cavity and channel-expansion benchmarks show rapid, robust viscosity recovery and CDA-NSE convergence, including poor initial guesses and noisy data.
- Three benchmark tests demonstrate fast and robust convergence of the modified Newton recovery algorithm and CDA-Picard+CDA-Newton solver.The tests cover 2D and 3D driven cavities and 2D channel flow through a sudden expansion.
- O(10^-n) magnitude noise yields n digits of viscosity accuracy within 4 or 5 iterations, up to a noise-dependent range near the solution.The reported ranges are 1.56e-5 for γ=10^-2, 2.17e-6 for γ=10^-3, and 1.21e-7 for γ=10^-4.
- 6.1.1 2D driven cavity: 680K total degrees of freedom are used for the 2D driven-cavity discretization at Re=5000 and Re=10000.The Reynolds number is defined as Re := ν^-1 for this benchmark.
- 6.1.1 2D driven cavity: 3 iterations recover the true viscosity at Re=5000 from initial guesses ν0 = 1/3000 and 1/10000.CDA-Picard+CDA-Newton converges the CDA-NSE system in 6, 4, and 3 iterations across the two runs.
- 6.1.1 2D driven cavity: 4 and 3 iterations recover the true viscosity at Re=10000 from initial guesses ν0 = 1/4000 and 1/13000, respectively.The CDA-Picard+CDA-Newton solver takes at most 7 iterations, even though stabilized Newton does not converge for ν ≤ 1/3500.
- 6.2.1 2D channel flow through an expansion: 5 distinct solutions exist for the Re=50 channel-expansion problem, and recovery converges rapidly for each tested true solution and initial guess.The experiment uses 297 measurement nodes to construct I_Hu.
7 Conclusions and Future Directions
The paper presents an efficient method for recovering unknown viscosity in steady NSE from partial solution data by combining modified Newton recovery with a CDA-Picard+CDA-Newton solver. It also identifies noisy-data analysis and multiple-parameter recovery as future directions.
- The method combines a modified Newton iteration for viscosity recovery with a CDA-Picard+CDA-Newton solver at each step.
- The approach is analyzed and tested as an effective method for recovering unknown viscosity when partial solution data are available.
- Noisy-data analysis remains future work because numerical tests reached only a noise-dependent range around the true viscosity.
- Future extensions include recovering multiple parameters, such as viscosity and forcing in NSE or parameters in Boussinesq and MHD systems.
10 Declaration of competing interest
The authors disclose financial support for the reported work from the Department of Energy and the NSF.
- Author LR reports financial support from the Department of Energy through grant DE-SC0025292.
- JPW was partially supported by NSF grants DMS-2510495 and CCF-2343286.
- The declaration identifies these funding relationships as financial interests or personal relationships that may be considered competing interests.