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Data assimilation in 2D nonlinear coupled sound and heat flow, using a stabilized explicit finite difference scheme marched backward in time

Alfred S. Carasso

TL;DR

Addresses ill-posed data assimilation for time-reversed 2D coupled sound and heat flow. A stabilized backward explicit finite-difference scheme with a compensating smoothing operator is developed and analyzed for linear selfadjoint operators, with extensions to nonlinear problems on non-rectangular domains via FFT-based Laplacian smoothing. Theoretical stability and error-bounds are derived, and computations on 512×512 images demonstrate both successful reconstructions and limitations due to data uncertainty and the choice of $T_{max}$. The results provide a fast, non-iterative baseline suitable for initialization and validation of ML-based refinements in backward-in-time data assimilation.

Abstract

This paper considers the ill-posed data assimilation problem associated with hyperbolic/parabolic systems describing 2D coupled sound and heat flow. Given hypothetical data at time T > 0, that may not correspond to an actual solution of the dissipative system at time T, initial data at time t = 0 are sought that can evolve, through the dissipative system, into a useful approximation to the desired data at time T. That may not always be possible. A stabilized explicit finite difference scheme, marching backward in time, is developed and applied to nonlinear examples in non rectangular regions. Stabilization is achieved by applying a compensating smoothing operator at each time step, to quench the instability. Analysis of convergence is restricted to the transparent case of linear, autonomous, selfadjoint spatial differential operators. However, the actual computational scheme can be applied to more general problems. Data assimilation is illustrated using 512x512 pixel images. Such images are associated with highly irregular non smooth intensity data that severely challenge ill-posed reconstruction procedures. Successful and unsuccessful examples are presented.

Data assimilation in 2D nonlinear coupled sound and heat flow, using a stabilized explicit finite difference scheme marched backward in time

TL;DR

Addresses ill-posed data assimilation for time-reversed 2D coupled sound and heat flow. A stabilized backward explicit finite-difference scheme with a compensating smoothing operator is developed and analyzed for linear selfadjoint operators, with extensions to nonlinear problems on non-rectangular domains via FFT-based Laplacian smoothing. Theoretical stability and error-bounds are derived, and computations on 512×512 images demonstrate both successful reconstructions and limitations due to data uncertainty and the choice of . The results provide a fast, non-iterative baseline suitable for initialization and validation of ML-based refinements in backward-in-time data assimilation.

Abstract

This paper considers the ill-posed data assimilation problem associated with hyperbolic/parabolic systems describing 2D coupled sound and heat flow. Given hypothetical data at time T > 0, that may not correspond to an actual solution of the dissipative system at time T, initial data at time t = 0 are sought that can evolve, through the dissipative system, into a useful approximation to the desired data at time T. That may not always be possible. A stabilized explicit finite difference scheme, marching backward in time, is developed and applied to nonlinear examples in non rectangular regions. Stabilization is achieved by applying a compensating smoothing operator at each time step, to quench the instability. Analysis of convergence is restricted to the transparent case of linear, autonomous, selfadjoint spatial differential operators. However, the actual computational scheme can be applied to more general problems. Data assimilation is illustrated using 512x512 pixel images. Such images are associated with highly irregular non smooth intensity data that severely challenge ill-posed reconstruction procedures. Successful and unsuccessful examples are presented.
Paper Structure (6 sections, 5 theorems, 22 equations, 3 figures)

This paper contains 6 sections, 5 theorems, 22 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A stabilized explicit scheme for linear selfadjoint time-reversed coupled wave and diffusion equations
  3. Application to data assimilation
  4. Using the Laplacian for smoothing when $L$ has variable coefficients
  5. Data assimilation in nonlinear coupled sound and heat flow in non-rectangular region, using FFT Laplacian smoothing
  6. Concluding Remarks

Key Result

Lemma 1

With $p >1$, and $~\zeta_m, ~q_m,$ as in Eq. (eq:9.0005a), fix a positive integer $J,$ and choose $\omega \geq (\zeta_J)^{1-p}$. Then,

Figures (3)

  • Figure 1.1: FIGURE 1. Non smooth intensity data plot, associated with Abraham Lincoln image, is typical of many natural images.
  • Figure 3.1: FIGURE 2. Successful data assimilation experiment. See summary in Table 2. Above nonlinear coupled sound and heat flow experiment lies outside scope of linear theory developed in Section 2. As explained in in the discussion following Eq. (\ref{['eq:309.0001']}), enclosing quarter circle region $\Omega$ in unit square $\Psi$, allows use of FFT Laplacian smoothing operator $Q_{\Delta}$, in backward reconstruction with scheme in Eq. (\ref{['eq:9.0006a']}).
  • Figure 3.2: FIGURE 3. Failure of data assimilation with significantly larger value for $T_{max}$. See summary in Table 3. As previously explained, enclosing quarter circle region $\Omega$ in unit square $\Psi$, allows use of FFT Laplacian smoothing operator $Q_{\Delta}$, in backward reconstruction with scheme in Eq. (\ref{['eq:9.0006a']}).

Theorems & Definitions (5)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 1
  • Theorem 2