On the Physical Interpretation of Adjoint Methods for Sensitivity Analysis, Part II: Non-Self-Adjoint Linear Systems

Published

May 28, 2026

Author(s)

Vijay Srinivasan

Abstract

Adjoint methods have gained prominence in science and engineering as the preferred approach for sensitivity analysis of mechanistic models involving a large number of parameters. Part I of this series explored adjoint methods for self-adjoint (symmetric) linear systems, which possess a mathematically symmetric structure. In such systems, the adjoint vector has a direct physical interpretation as a system response (e.g., a displacement field), and the mathematical symmetry is a manifestation of the physical concept of reciprocity. This direct interpretation is lost in non-self-adjoint (non-symmetric) linear systems, where the matrix operator (A) and its transpose (A^T) are distinct. Such systems, common in engineering, are characterized by directional transport in space and directional evolution in time. This report, the second in the series, explores the physical meaning of the adjoint vector in these non-self-adjoint systems by analyzing two fundamental examples: (1) a one-dimensional convection-diffusion problem, which exhibits spatial asymmetry, and (2) a one-dimensional damped structural dynamics problem, which exhibits temporal asymmetry. We demonstrate that in both cases, the adjoint operator represents a reversed physical system. The adjoint equation for convection reverses the flow velocity, while the adjoint equation for dynamics reverses the direction of time (solving backward from a final condition). Consequently, the adjoint vector v is interpreted not as a physical response of the original system, but as a general influence map representing receptivity. It quantifies how a local perturbation in space or time affects the final quantity of interest, providing a unified principle for sensitivity analysis and uncertainty quantification.

Citation

Advanced Manufacturing Series (NIST AMS) - 100-79

Report Number

100-79

NIST Pub Series

Advanced Manufacturing Series (NIST AMS)

Pub Type

NIST Pubs

Download Paper

https://doi.org/10.6028/NIST.AMS.100-79

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Keywords

Adjoint Method, Calibration, Digital Twin, Influence Map, Non-Self-Adjoint Systems, Optimization, Receptivity, Reversed System, Sensitivity Analysis, Verification, Validation, and Uncertainty Quantification (VVUQ).

Citation

Srinivasan, V. (2026), On the Physical Interpretation of Adjoint Methods for Sensitivity Analysis, Part II: Non-Self-Adjoint Linear Systems, Advanced Manufacturing Series (NIST AMS), National Institute of Standards and Technology, Gaithersburg, MD, [online], https://doi.org/10.6028/NIST.AMS.100-79 , https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=962290 (Accessed May 29, 2026)

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