Lower-bound Theory Research Articles

Uncertainty Quantification of Transient‐Based Leakage Identification: A Frequency Domain Framework

AbstractThe leak detection methods in the frequency domain have presently become widely established among transient‐based techniques. This study quantifies the uncertainty of the frequency domain multiple leak detection subjected to uncertain Gaussian, independent, and identically distributed measurements. The lower limit of the variances and covariances of the estimated leak locations and sizes are derived via the Cramer‐Rao Lower Bound theory, whose derivation consistency with the Taylor expansion is also revealed. A systematic methodology based on the Monte Carlo simulations and probing waves is carried out to examine and verify the proposed formulations and corresponding outcomes, leading to an in‐depth analysis of interactions between probing waves and leaks. The results justify the formation of points of minimum error in the pipeline corresponding to each specific harmonic wave, which has direct application in detectability using signals of limited bandwidth in actual practice. The findings resolve the reason behind localization failure for some test cases and success for others despite the same noise levels, as illustrated via numerical and experimental test cases. A workflow for efficient leak detection and the approach to estimating the corresponding localization uncertainty is proposed. According to the findings of this study, one may recommend the predicted error‐location results to accompany the wave‐based leak detection outputs to render a measure of accuracy in the identification process.

A General Non-Lipschitz Infimal Convolution Regularized Model: Lower Bound Theory and Algorithm

The convex infimal convolution model proposed in Chambolle and Lions [Numer. Math., 76 (1997), pp. 167--188] is a fundamental model to extract two useful components from a single input image and has various low level vision applications. In many of them, one target component has an (approximately) piecewise constant structure and the other is a smoothly varying function or repeated texture pattern. In this paper, we propose and study a general non-Lipschitz infimal convolution (GnLIC) regularization model, which covers most existing applications in this type. Therein the non-Lipschitz regularization enforces the piecewise constant property of the first component. For this GnLIC model, we prove a lower bound theory for its local minimizers and a local version for its stationary points. Motivated by these, we naturally extend previous works to design an inexact iterative support shrinking algorithm with proximal linearization for our GnLIC model (InISSAPL-GnLIC). Moreover, we establish the sequence convergence property and a sequence lower bound theory for InISSAPL-GnLIC, provided that an inexact subgradient condition generated by a subsolver holds. The subsolver is constructed by efficient ADMM and a specially designed feasibilization operation. We finally give numerical experiments in two low level vision applications: Retinex and cartoon-texture decomposition. These tests demonstrate that our non-Lipschitz regularization based method can indeed extract the piecewise constant component better than existing approaches, which is consistent with the established lower bound theory.

Damage-controlled allowable maximum stress of the stirrup for HSRC beam members based on lower-bound theory

The main purpose of this work is to investigate the damage-controlled allowable maximum stress of the stirrup of a high-strength reinforced concrete (HSRC) beam member in the general design using the lower-bound theory. Since several works have noted that the allowable stress of the stirrup of an RC beam in the general design is limited by the shear crack width, the relationship between shear crack width and applied shear force is obtained herein using available experimental data for HSRC beam members. Moreover, based on the fact that the allowable maximum shear crack widths that have been suggested elsewhere are related to the damage state, formulas for estimating the allowable maximum stress of the stirrup of an HSRC beam member with various specified compressive strengths of concrete and stirrup ratios are provided herein. Restated, the allowable maximum stress of the stirrup can be determined from the specified compressive strength of the concrete and the stirrup ratio.

Comparison of an improved self-consistent lower bound theory with Lehmann\u2019s method for low-lying eigenvalues

Ritz eigenvalues only provide upper bounds for the energy levels, while obtaining lower bounds requires at least the calculation of the variances associated with these eigenvalues. The well-known Weinstein and Temple lower bounds based on the eigenvalues and variances converge very slowly and their quality is considerably worse than that of the Ritz upper bounds. Lehmann presented a method that in principle optimizes Temple’s lower bounds with significantly improved results. We have recently formulated a Self-Consistent Lower Bound Theory (SCLBT), which improves upon Temple’s results. In this paper, we further improve the SCLBT and compare its quality with Lehmann’s theory. The Lánczos algorithm for constructing the Hamiltonian matrix simplifies Lehmann’s theory and is essential for the SCLBT method. Using two lattice Hamiltonians, we compared the improved SCLBT (iSCLBT) with its previous implementation as well as with Lehmann’s lower bound theory. The novel iSCLBT exhibits a significant improvement over the previous version. Both Lehmann’s theory and the SCLBT variants provide significantly better lower bounds than those obtained from Weinstein’s and Temple’s methods. Compared to each other, the Lehmann and iSCLBT theories exhibit similar performance in terms of the quality and convergence of the lower bounds. By increasing the number of states included in the calculations, the lower bounds are tighter and their quality becomes comparable with that of the Ritz upper bounds. Both methods are suitable for providing lower bounds for low-lying excited states as well. Compared to Lehmann’s theory, one of the advantages of the iSCLBT method is that it does not necessarily require the Weinstein lower bound for its initial input, but Ritz eigenvalue estimates can also be used. Especially owing to this property the iSCLBT method sometimes exhibits improved convergence compared to that of Lehmann’s lower bounds

Optimising sampling patterns for bi-exponentially decaying signals

A recently reported method, based on the Cramér-Rao Lower Bound theory, for optimising sampling patterns for a wide range of nuclear magnetic resonance (NMR) experiments is applied to the problem of optimising sampling patterns for bi-exponentially decaying signals. Sampling patterns are optimised by minimizing the percentage error in estimating the most difficult to estimate parameter of the bi-exponential model, termed the objective function. The predictions of the method are demonstrated in application to pulsed field gradient NMR data recorded for the two-component diffusion of a binary mixture of methane/ethane in a zeolite. It is shown that the proposed method identifies an optimal sampling pattern with the predicted objective function being within 10% of that calculated from the experiment dataset. The method is used to advise on the number of sampled points and the noise level needed to resolve two-component systems characterised by a range of ratios of populations and diffusion coefficients. It is subsequently illustrated how the method can be used to reduce the experiment acquisition time while still being able to resolve a given two-component system.

A general yield criterion for reinforced concrete slab elements

Synopsis The methods of plasticity theory are used to develop, from simple assumptions about the properties of the constituents, a numerical method of finding points on the yield criterion for elements of a reinforced concrete slab under general combinations of moments and membrane forces. The general method is modified to predict the results of tests on bent ‘push-off’ specimens, designed to simulate conditions in the loaded flange of a T beam. The experimental results agree reasonably well with the theoretical predictions, and also with a simpler lower-bound theory developed to deal with the test situation directly.