Research

21 pages, 5043 KiB

Open AccessArticle

Reliability-Based Design Optimization of Structures Considering Uncertainties of Earthquakes Based on Efficient Gaussian Process Regression Metamodeling

by Yanjie Xiao, Feng Yue, Xinwei Wang and Xun’an Zhang

Axioms 2022, 11(2), 81; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms11020081 - 20 Feb 2022

Cited by 5 | Viewed by 2076

Abstract

The complexity of earthquakes and the nonlinearity of structures tend to increase the calculation cost of reliability-based design optimization (RBDO). To reduce computational burden and to effectively consider the uncertainties of ground motions and structural parameters, an efficient RBDO method for structures under [...] Read more.

The complexity of earthquakes and the nonlinearity of structures tend to increase the calculation cost of reliability-based design optimization (RBDO). To reduce computational burden and to effectively consider the uncertainties of ground motions and structural parameters, an efficient RBDO method for structures under stochastic earthquakes based on adaptive Gaussian process regression (GPR) metamodeling is proposed in this study. In this method, the uncertainties of ground motions are described by the record-to-record variation and the randomness of intensity measure (IM). A GPR model is constructed to obtain the approximations of the engineering demand parameter (EDP), and an active learning (AL) strategy is presented to adaptively update the design of experiments (DoE) of this metamodel. Based on the reliability of design variables calculated by Monte Carlo simulation (MCS), an optimal solution can be obtained by an efficient global optimization (EGO) algorithm. To validate the effectiveness and efficiency of the developed method, it is applied to the optimization problems of a steel frame and a reinforced concrete frame and compared with the existing methods. The results show that this method can provide accurate reliability information for seismic design and can deal with the problems of minimizing costs under the probabilistic constraint and problems of improving the seismic reliability under limited costs. Full article

(This article belongs to the Special Issue Computing Methods in Mathematics and Engineering)

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13 pages, 1708 KiB

Open AccessArticle

Zero-Aware Low-Precision RNS Scaling Scheme

by Amir Sabbagh Molahosseini

Axioms 2022, 11(1), 5; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms11010005 - 23 Dec 2021

Cited by 1 | Viewed by 2110

Abstract

Scaling is one of the complex operations in the Residue Number System (RNS). This operation is necessary for RNS-based implementations of deep neural networks (DNNs) to prevent overflow. However, the state-of-the-art RNS scalers for special moduli sets consider the 2^k modulo as [...] Read more.

Scaling is one of the complex operations in the Residue Number System (RNS). This operation is necessary for RNS-based implementations of deep neural networks (DNNs) to prevent overflow. However, the state-of-the-art RNS scalers for special moduli sets consider the 2^k modulo as the scaling factor, which results in a high-precision output with a high area and delay. Therefore, low-precision scaling based on multi-moduli scaling factors should be used to improve performance. However, low-precision scaling for numbers less than the scale factor results in zero output, which makes the subsequent operation result faulty. This paper first presents the formulation and hardware architecture of low-precision RNS scaling for four-moduli sets using new Chinese remainder theorem 2 (New CRT-II) based on a two-moduli scaling factor. Next, the low-precision scaler circuits are reused to achieve a high-precision scaler with the minimum overhead. Therefore, the proposed scaler can detect the zero output after low-precision scaling and then transform low-precision scaled residues to high precision to prevent zero output when the input number is not zero. Full article

(This article belongs to the Special Issue Computing Methods in Mathematics and Engineering)

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11 pages, 762 KiB

Open AccessArticle

The Representation of D-Invariant Polynomial Subspaces Based on Symmetric Cartesian Tensors

by Xue Jiang and Kai Cui

Axioms 2021, 10(3), 193; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms10030193 - 19 Aug 2021

Cited by 1 | Viewed by 1212

Abstract

Multivariate polynomial interpolation plays a crucial role both in scientific computation and engineering application. Exploring the structure of the D-invariant (closed under differentiation) polynomial subspaces has significant meaning for multivariate Hermite-type interpolation (especially ideal interpolation). We analyze the structure of a D [...] Read more.

Multivariate polynomial interpolation plays a crucial role both in scientific computation and engineering application. Exploring the structure of the D-invariant (closed under differentiation) polynomial subspaces has significant meaning for multivariate Hermite-type interpolation (especially ideal interpolation). We analyze the structure of a D-invariant polynomial subspace

P_{n}

in terms of Cartesian tensors, where

P_{n}

is a subspace with a maximal total degree equal to

n, n \geq 1

. For an arbitrary homogeneous polynomial

p^{(k)}

of total degree k in

P_{n}

,

p^{(k)}

can be rewritten as the inner products of a kth order symmetric Cartesian tensor and k column vectors of indeterminates. We show that

p^{(k)}

can be determined by all polynomials of a total degree one in

P_{n}

. Namely, if we treat all linear polynomials on the basis of

P_{n}

as a column vector, then this vector can be written as a product of a coefficient matrix

A^{(1)}

and a column vector of indeterminates; our main result shows that the kth order symmetric Cartesian tensor corresponds to

p^{(k)}

is a product of some so-called relational matrices and

A^{(1)}

. Full article

(This article belongs to the Special Issue Computing Methods in Mathematics and Engineering)

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