[Paper Review] Approximating semigroups by using pseudospectra

E. B. Davies|ArXiv.org|Mar 19, 2003

Advanced Optimization Algorithms Research14 references3 citations

TL;DR

This paper introduces a pseudospectral method for approximating semigroups generated by highly non-self-adjoint operators, using approximate eigenvalues and eigenvectors to construct a stable, accurate expansion. The method achieves superior accuracy over standard spectral methods, especially for convection-diffusion operators, by leveraging dense sets of approximate eigenvalues even when true eigenvectors are poorly conditioned or non-basis-forming.

ABSTRACT

We study evolution equations with non-self-adjoint generators, for example the convection-diffusion equation. Spectral expansions are not a reliable method of solving such equations, because they are so ill-conditioned. We introduce a new method using pseusospectra to produce an approximated spectral expansion, and explain its theoretical status. We also give some simple numerical examples to show that it is much better conditioned.

Motivation & Objective

• To address the instability and inaccuracy of standard spectral methods when applied to non-self-adjoint operators with poor spectral basis properties.
• To develop a numerical method for computing semigroups $ T_t = e^{At} $ that remains accurate even when the spectrum is ill-conditioned or the spectral projections grow rapidly.
• To demonstrate that a large number of approximate eigenvalues—often ignored in classical spectral theory—can be used as a computational resource for accurate semigroup approximation.
• To provide a stable, efficient method for solving initial value problems for evolution equations in $ L^2 $, particularly for high-dimensional or convection-diffusion operators.

Proposed method

• Define a pseudospectral transform $ ilde{f G} $ mapping $ L^2(S) $ to the Hilbert space $ ilde{\cal H} $, using a family of unit vectors $ u_s $ and complex numbers $ \lambda_s $ satisfying $ \|Au_s - \lambda_s u_s\| < \varepsilon $.
• Construct the operator $ B = \tilde{\bf G}^* \tilde{\bf G} $, a Hilbert-Schmidt operator on $ L^2(S) $, and assume it is invertible to enable stable reconstruction.
• Use the inverse $ B^{-1} $ to compute the optimal $ \phi = B^{-1} \tilde{\bf G}^* f $ that minimizes $ \| \tilde{\bf G} \phi - f \| $, enabling accurate approximation of $ f \in \tilde{\cal H} $.
• Apply the method to the evolution equation $ f'(t) = A f(t) $ by evolving the coefficients $ \phi_t $ via $ f_t = \tilde{\bf G} \phi_t $, with $ \phi_t $ computed using the pseudospectral expansion.
• Use a finite set $ S $ of approximate eigenpairs $ (\lambda_s, u_s) $, with $ \lambda_s $ in the $ 2\varepsilon $-pseudospectrum of $ A $, to form a low-dimensional subspace for efficient computation.
• Validate the method numerically on convection-diffusion operators, showing high accuracy and stability even for non-smooth initial data and large $ b $.

Experimental results

Research questions

• RQ1 Can a large number of approximate eigenvalues of a non-self-adjoint operator be used effectively to construct a stable and accurate approximation of the semigroup $ e^{At} $?
• RQ2 How does the pseudospectral method compare in accuracy to classical spectral expansion when the true eigenvectors do not form a basis or when spectral projections grow rapidly?
• RQ3 To what extent can the pseudospectral method handle initial data that do not satisfy boundary conditions, such as discontinuous or non-smooth functions?
• RQ4 Does the method remain robust and accurate for different values of the convection and diffusion parameters, especially when the spectrum is highly non-normal?

Key findings

• For $ N = 71 $, the approximation error $ \|f - P_N f\| $ dropped to $ 1.3 \times 10^{-14} $, demonstrating exponential convergence in the number of approximate eigenpairs.
• The pseudospectral method produced a non-negative solution $ f_t $ with minimum value $ \geq -3 \times 10^{-4} $, indicating numerical stability and physical consistency.
• The maximum of the solution $ m(t) $ matched the theoretical asymptotic decay $ m_\infty = (1 + 4t/b)^{-1/2} $ up to $ t = 10 $, confirming accuracy in long-time behavior.
• For the initial function $ g(x) = 1 $, the method produced a smoothed approximation of the characteristic function of $ [0, a - t] $, with no Gibbs phenomenon due to diffusion.
• The method remained accurate across a wide range of $ b $ from 5 to 100, showing robustness to parameter variation.
• The real parts of the approximate eigenvalues $ \mu_s $ ranged from $ -0.488 $ to $ -0.729 $, and their dense distribution near zero was key to the method’s accuracy.