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Real-Time Operation of River-Reservoir Systems
Published in Larry W. Mays, Optimal Control of Hydrosystems, 1997
The original problem, GOM, can be converted into a reduced problem as suggested by the implicit function theorem (Luenberger, 1984). The implicit function theorem states that if some of the problem variables can be solved in terms of the remaining variables, then a reduced problem can be devised which can be manipulated more easily. The approach is applied to the problem given by Eqs. (5.2.1)–(5.2.6) in such a way that the hydraulic constraints [Eq. (5.2.2)] are handled separately by the simulator and the other constraints by the optimizer. The simulation model computes the values of the state variables, h and Q, for given values of the control variables r and the optimization model seeks the optimal values of r that will minimize the objective function. The implicit function theorem states that h(r) and Q(r) exist if and only if the basic matrix [the Jacobian of the system of equations given by Eq. (5.2.2)] is nonsingular. This condition is always satisfied when a solution is possible, as the simulator (DWOPER) uses the same matrix for the finite difference unsteady flow computations.
Vector analysis
Published in John P. D’Angelo, Linear and Complex Analysis for Applications, 2017
Consider an open set Ω in ℝn × ℝk and a continuously differentiable function f : Ω → ℝk. We write the points in Ω as pairs (x, y), where x ∈ ℝn and y ∈ ℝk. Suppose that f(x0, y0) = 0. The implicit function theorem provides circumstances in which we can solve the equation f(x, y) = 0 for y as a function of x.
Engineering Optimization
Published in Ganesh M. Kakandikar, Vilas M. Nandedkar, Sheet Metal Forming Optimization, 2017
Ganesh M. Kakandikar, Vilas M. Nandedkar
Optimization problems are classified based on various criteria:The nature of the variable sets: a variable may be continuous (e.g., a geometrical dimension), discrete (e.g., cross sections of beams are often available by discrete steps in catalogues), or integer (e.g., the number of layers in a composite material). There are often mixed variables in engineering problems.The nature of the constraints and the objective functions: they may be linear, quadratic, nonlinear, or even nondifferentiable. For instance, gradient-based algorithms, based on the computation of the sensitivities, require the functions to be differentiable in order to compute their first-order (and sometimes also their second-order) derivatives.The analytical properties of the functions, for example, linearity in linear programming. Convexity or monotonicity can also be successfully exploited to converge to a global optimal solution.The presence (or absence) of constraints. Equality constraints are usually tackled by converting them into inequality constraints.The size of the problem: to remain applicable even when the number of variables is very large (more than about 10,000 for continuous problems), optimization algorithms have to be adapted, because of limited memory or computational time.Implicit or explicit functions: in shape optimization, for instance, when finite element models are needed to compute the stresses and displacements, the objective function (generally the mass) is almost always an implicit function of the variables. Therefore, the objective(s) and constraints are approximated to a linear, quadratic, or other (cubic, posynomial, etc.) model. Neural networks may also be used to construct an approximation of the functions.Local or global optimization: in single-objective optimization, local optimization is used commonly with smooth functions in order to find a local optimum.Single-objective or multi-objective: though the first studies in structural optimization used only one objective (most of the time minimizing the mass), an increasing number of studies deal with multiple criteria (mass, cost, specific performances, etc.). Indeed, in the industrial context, optimal solutions must be good compromises between the different (and often contradictory) criteria.To solve optimization problems, a large number of methods have been proposed in the literature. These are briefly summarized in the following section.
Implicit modelling and dynamic update of tunnel unfavourable geology based on multi-source data fusion using support vector machine
Published in Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, 2023
Binru Yang, Yulin Ding, Qing Zhu, Liguo Zhang, Haoyu Wu, Yongxin Guo, Mingwei Liu, Wei Wang
For modelling problems, the existing modelling methods are divided into explicit and implicit modelling. Most design institutes still use the explicit modelling method, which means that the geological interface is established manually according to the borehole data, leading to serious topology inconsistencies at the interface (Guo et al. 2020). Implicit modelling can fit the surface of the model by establishing an implicit function (Carr et al. 2001). It only needs the 3D coordinates of discrete data points, so it is widely used in medical image restoration (Carr, Richard Fright, and Beatson 1997). However, the implicit function is established by taking the weighted sum of the radial basis kernel function of each discrete point, so when the data change, the corresponding implicit function needs to be solved again and cannot be updated locally. The initially built models are often revised and updated during tunnel excavation due to new advanced prediction data and additional expert knowledge, so a dynamic modelling method is needed to keep the model consistent with the real-time changing environment.
The impact of procedural and conceptual teaching on students' mathematical performance over time
Published in International Journal of Mathematical Education in Science and Technology, 2021
Vahid Borji, Farzad Radmehr, Vicenç Font
Finally, the lecturer used the following constructions of implicit differentiation: actions of formally substituting the symbolic expression f(x) for y in given equations of the form K(x,y) = c; then differentiating both sides with respect to x; and finally substituting the symbol y for the symbol f(x), for , and so on3. Further coordination with other schema will allow the algebraic manipulation needed to solve for . Repetition and reflection on this action eventually allows the students to omit the symbolic substitution of f(x) for y. The construction of implicit differentiation requires the coordination of derivative rules, chain rule, and implicit function.
Boundary simulation – a hierarchical approach for multiple categories
Published in Applied Earth Science, 2021
Flavio Azevedo Neves Amarante, Roberto Mentzingen Rolo, João Felipe Coimbra Leite Costa
Another family of cell-based methods is the implicit modelling. Implicit models are created from scattered sample data, usually categorical and structural. An implicit function or implicit model provides a continuous mathematical representation of the attribute throughout the volume (a scalar field). Implicit models contain an infinite number of isosurfaces, and to visualise a geological model an isosurface must be extracted from the model, usually the zero isosurface. Different volume functions can be used in geological modelling.