[Paper Review] Interval-based parameter identification for structural static problems

Naijia Xiao, Francesco Fedele+1 more|arXiv (Cornell University)|Aug 14, 2014

Probabilistic and Robust Engineering Design33 references3 citations

TL;DR

This paper proposes an interval-based parameter identification method for structural static problems using Interval Finite Element Method (IFEM) and adjoint-based optimization. The two-step algorithm first applies a deterministic solver, then switches to interval extension with a novel containment-stopping criterion, guaranteeing verified bounds on uncertain parameters—demonstrated through numerical examples to provide rigorous parameter enclosures.

ABSTRACT

We present an interval-based approach for parameter identification in structural static inverse problems. The proposed inverse formulation exploits the Interval Finite Element Method (IFEM) combined with adjoint-based optimization. The inversion consists of a two-step algorithm: first, an estimate of the parameters is obtained by means of a deterministic iterative solver. Then, the algorithm switches to the interval extension of the previous solver, using the deterministic estimate of the parameters as an initial guess. The iterations are terminated based on a new containment-stopping criterion, which is intrinsic to intervals. Various numerical examples show that the proposed method provides guaranteed interval enclosures of the parameters.

Motivation & Objective

• Address uncertainty in structural parameter identification by providing guaranteed bounds rather than point estimates.
• Overcome limitations of deterministic methods in handling bounded uncertainties in material or geometric parameters.
• Develop a robust inverse analysis framework that ensures all possible parameter values within intervals are captured.
• Introduce a stopping criterion intrinsic to interval analysis to improve convergence reliability.
• Ensure verified and computationally efficient parameter identification under bounded uncertainty.

Proposed method

• Formulate the inverse problem using Interval Finite Element Method (IFEM) to represent uncertain parameters as intervals.
• Apply a deterministic iterative solver in the first step to obtain an initial parameter estimate.
• Switch to the interval extension of the solver, using the deterministic result as an initial guess for interval iterations.
• Implement a new containment-stopping criterion based on interval width and convergence behavior to terminate iterations reliably.
• Use adjoint-based optimization to efficiently compute sensitivity information required for parameter updates in the interval framework.
• Ensure all computations are performed with interval arithmetic to maintain verified bounds throughout the solution process.

Experimental results

Research questions

• RQ1 Can interval-based optimization provide verified bounds for structural parameters under bounded uncertainty?
• RQ2 How does the proposed two-step algorithm improve upon purely deterministic or interval-only approaches in parameter identification?
• RQ3 What is the effectiveness of the proposed containment-stopping criterion in ensuring convergence within guaranteed intervals?
• RQ4 How does the integration of adjoint-based optimization enhance efficiency in interval parameter identification?
• RQ5 Can the method reliably enclose true parameter values within computed intervals across diverse structural problems?

Key findings

• The proposed method successfully provides guaranteed interval enclosures for structural parameters, ensuring all possible true values are contained within the computed bounds.
• The two-step algorithm—starting with deterministic optimization and transitioning to interval extension—improves convergence and robustness.
• The novel containment-stopping criterion effectively terminates iterations when interval widths reach a desired tolerance, ensuring verified convergence.
• Numerical examples demonstrate the method's ability to handle various uncertainties in material and geometric properties with verified results.
• The integration of adjoint-based optimization reduces computational cost while maintaining interval validity.
• The method outperforms standard deterministic approaches by providing not just a point estimate, but a verified range of feasible parameters.