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Inverse Conduction
Published in M. Necati Özisik, Helcio R. B. Orlande, Inverse Heat Transfer, 2018
M. Necati Özisik, Helcio R. B. Orlande
Figure 3.1.3 shows the variation of the determinant of FI* with time. An analysis of this figure reveals that, for three sensors located at (0, 0.9, 0.9), (0.9, 0, 0.9) and (0.9, 0.9, 0), the duration of the experiment should be taken as tf = 0.22 where such determinant is maximum, so that the confidence region of the estimated parameters is minimized. A similar analysis involving three sensors located at (0, 0.5, 0.5), (0.5, 0, 0.5) and (0.5, 0.5, 0) yields a maximum determinant of 7 × 10− 11 for tf = 0.3. Such a value for the determinant is about three orders of magnitude smaller than the maximum determinant of figure 3.1.3. Similarly to the analysis of the sensitivity coefficients, this gives also an indication that the measurements of sensors located at (0, 0.9, 0.9), (0.9, 0, 0.9) and (0.9, 0.9, 0) provide more accurate estimates than the measurements of sensors located at (0, 0.5, 0.5), (0.5, 0, 0.5) and (0.5, 0.5, 0).
Parameter Estimation
Published in M. Necati Özisik, Helcio R. B. Orlande, Inverse Heat Transfer, 2021
M. Necati Özisik, Helcio R. B. Orlande
Figure 2.8 presents the timewise variation of the determinant of FI* for different heating times, for sensors located at (0,0.9,0.9), (0.9,0,0.9) and (0.9,0.9,0) in Test case 1. Such locations for the sensors are the ones that resulted on the largest values for the determinant of FI* for this test case. We note in Figure 2.8 that, as observed by Taktak et al. [89] for a one-dimensional property estimation problem, the use of heating times smaller than the final experimental time yielded larger values for the determinant of FI* for Test case 1. Such is the case because any tendency of the sensitivity coefficients to become linearly dependent is reversed after the heating is stopped, that is, when a new transient period is created. This can be clearly noticed in Figures 2.9a-c for the reduced sensitivity coefficients at each of the measurement locations, respectively, for Test case 1, with th = 0.03 and tf = 0.05, which are the heating and final times, respectively, resulting on the maximum determinant of FI*.
Application of Eigenvalues and Eigenvectors
Published in Timothy Bower, ®, 2023
The maximum determinant of a Markov matrix is one, which is the case for the identity matrix. All other Markov matrices have a determinant less than one. Therefore, the product of the eigenvalues is ≤1.
Dual-Orthogonal Arrays for Order-of-Addition Two-Level Factorial Experiments
Published in Technometrics, 2023
The proof of Theorem 1 is presented in the supplementary materials. Because many statistical software and libraries provide built-in functions to compute the determinant of a matrix, the D-optimality criterion will be used later to search for dual-orthogonal arrays. Specifically, a design is said to be D-optimal if its moment matrix has maximum determinant among all competing designs.
Exploiting aggregate sparsity in second-order cone relaxations for quadratic constrained quadratic programming problems
Published in Optimization Methods and Software, 2022
Heejune Sheen, Makoto Yamashita
The matrix has the maximum determinant among all possible matrix completion of , i.e.
Response surface optimization for a nonlinearly constrained irregular experimental design space
Published in Engineering Optimization, 2019
There have been several different design criteria for optimal design experiments. First, Smith (1918) stated the basic concept of optimal designs. Then, Wald (1943) extended Smith's optimal design framework by investigating the quality of the parameter estimates. Wald (1943) also proposed a design criterion for maximizing the determinant of the information matrix. This criterion was termed D-optimality by Kiefer and Wolfowitz (1959). Fedorov (1972) further developed the research in optimal designs to solve numerical optimal designs using the exchange algorithm. John and Draper (1975) further developed the study of D-optimality for regression designs by creating the procedures used to obtain D-optimal experimental designs. Along the same lines, Cook and Nachtsheim (1980) provided an empirical comparison of various algorithms to generate D-optimal experimental designs. Similarly, Nguyen and Miller (1992) and Miller and Nguyen (1994) developed discrete D-optimal designs. DuMouchel and Jones (1994) reported that increasing the determinant of the information matrix could decrease the error variance of the regression coefficients. Another important property of experimental designs is their orthogonality. Thus, de Aguiar et al. (1995) observed that a maximum determinant could be useful for providing a near orthogonality for optimal experimental designs. In addition to these studies, Cook and Fedorov (1995) discussed several approaches to optimal experimental designs with consideration of the costs involved in conducting the experiment and the inclusion of supporting design points and auxiliary objective functions. Productive techniques in optimal designs may yield new insights; therefore, Poston, Wegman, and Solka (1998) discussed a productive method of finding D-optimal experimental designs to determine the robust estimator with the minimum volume of an ellipsoid. Model fitting is another significant research area for a number of practical situations. Thus, Fedorov, Gagnon, and Leonov (2002) proposed iterated estimators for optimal design algorithms based on the convex design theory. Jin, Chen, and Sudjianto (2005) proposed a stochastic evolutionary algorithm and investigated the execution time in terms of constructing optimal designs. Pronzato (2003) and Harman and Pronzato (2007) studied various D-optimal experimental design search algorithms on the removal of non-optimal support points from the experimental design space. In addition, Myers, Montgomery, and Anderson-Cook (2009) surveyed potential applications of optimal experimental designs. Other optimal design articles by Sahraian and Kodiyalam (2000), Wang, Dong, and Aitchison (2001), Bandaru and Deb (2011), Aspenberg, Jergeus, and Nilsson (2013) and Bruggi and Mariani (2013) illustrated the applications of D-optimal design in practical engineering optimization. Finally, Ozdemir (2017) investigated various engineering optimization problems with special types of response surface designs and D-optimal designs.