<p>Conventional sensitivity analysis of long time-averaged functionals yields unbounded sensitivities when the simulation is chaotic or turbulent. Well-known methods for computing sensitivities in the presence of chaotic dynamical systems involve the use of the shadow trajectory. The least squares shadowing (LSS) is a popular approach to computing an approximation of the shadowing direction. While past literature has established the existence of the shadowing and the adjoint shadowing trajectories for a uniformly hyperbolic dynamical system, we note that the existence and uniqueness of the solution to the LSS equations are essential to ensure that the approximate shadowing direction can be computed. The existence and uniqueness of the solution to the LSS also ensure that the LSS equation can be discretized independently of the primal equation and that the true LSS solution is recovered as the time step is refined. The current paper proves the existence and uniqueness of the solution to the adjoint of the LSS equations for large integration times. The LSS operator is shown to be coercive and the condition number is bounded as <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_3075_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">$T\rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, in contrast to the existing literature showing that the LSS condition number increases at <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_3075_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {O}(T^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>T</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. The current work also derives a relation between the conditioning of the LSS and the time dilation factor, which explains the trend of deterioration of LSS conditioning for significantly small and large weights on time dilation factor, as observed in numerical simulations in previous LSS literature. Furthermore, having shown the boundedness of the inverse of the LSS operator as <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_3075_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">$T\rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, we provide an alternate proof to [<CitationRef CitationID="CR16">16</CitationRef>] of the convergence of the LSS sensitivity to the true sensitivity at the rate of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_3075_Article_IEq4.gif" Format="GIF" Height="33" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {O}\left( \frac{1}{\sqrt{T}}\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mfenced close=")" open="("> <mfrac> <mn>1</mn> <msqrt> <mi>T</mi> </msqrt> </mfrac> </mfenced> </mrow> </math></EquationSource> </InlineEquation>, which holds regardless of the boundary conditions imposed on the adjoint LSS, as long as the adjoint boundary conditions are bounded.</p>

Adjoint of Least Squares Shadowing: Existence, Uniqueness and Coarse Domain Discretization

期刊论文
发表日期：

2025年9月26日

查看全文

Pranshul Thakur,
Siva Nadarajah

摘要

Conventional sensitivity analysis of long time-averaged functionals yields unbounded sensitivities when the simulation is chaotic or turbulent. Well-known methods for computing sensitivities in the presence of chaotic dynamical systems involve the use of the shadow trajectory. The least squares shadowing (LSS) is a popular approach to computing an approximation of the shadowing direction. While past literature has established the existence of the shadowing and the adjoint shadowing trajectories for a uniformly hyperbolic dynamical system, we note that the existence and uniqueness of the solution to the LSS equations are essential to ensure that the approximate shadowing direction can be computed. The existence and uniqueness of the solution to the LSS also ensure that the LSS equation can be discretized independently of the primal equation and that the true LSS solution is recovered as the time step is refined. The current paper proves the existence and uniqueness of the solution to the adjoint of the LSS equations for large integration times. The LSS operator is shown to be coercive and the condition number is bounded as $T\rightarrow \infty $ $T \to \infty$ , in contrast to the existing literature showing that the LSS condition number increases at $\mathcal {O}(T^2)$ $O (T^{2})$ . The current work also derives a relation between the conditioning of the LSS and the time dilation factor, which explains the trend of deterioration of LSS conditioning for significantly small and large weights on time dilation factor, as observed in numerical simulations in previous LSS literature. Furthermore, having shown the boundedness of the inverse of the LSS operator as $T\rightarrow \infty $ $T \to \infty$ , we provide an alternate proof to [16] of the convergence of the LSS sensitivity to the true sensitivity at the rate of $\mathcal {O}\left( \frac{1}{\sqrt{T}}\right) $ $O (\frac{1}{\sqrt{T}})$ , which holds regardless of the boundary conditions imposed on the adjoint LSS, as long as the adjoint boundary conditions are bounded.