Generalization of Lagrange's implicit function theorem to n-dimensions
Lagrange implicit functions generalization to n dimensions, developing inversion as power series
Feagin, T.↗
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Results for “Implicit functions”
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Local mapping relations and global implicit function theorems
Nonlinear equation solution based on local mapping relations and global implicit function theorems for Banach space problems
Rheinboldt, W. C.↗
Local mapping relations and global implicit function theorems
Local mapping relations and global existence theorems
Rheinboldt, W. C.↗
Implicit Function Method for the Liouville Equation
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Kheifets, S.↗
An advanced probabilistic structural analysis method for implicit performance functions
In probabilistic structural analysis, the performance or response functions usually are implicitly defined and must be solved by numerical analysis methods such as finite element methods. In such cases, the most commonly used probabilistic analysis tool is the mean-based, second-moment method which provides only the first two statistical moments. This paper presents a generalized advanced mean value (AMV) method which is capable of establishing the distributions to provide additional information for reliability design. The method requires slightly more computations than the second-moment method but is highly efficient relative to the other alternative methods. In particular, the examples show that the AMV method can be used to solve problems involving non-monotonic functions that result in truncated distributions.
Wu, Y.-T.↗
Geometric prior of multi-resolution yielding manifolds and the local closest point projection for nearly non-smooth plasticity
Elastoplasticity models often introduce a scalar-valued yield function to implicitly represent the boundary between elastic and plastic material states. Here this paper introduces a new alternative where the yield envelope is represented by a manifold of which the topology and the geometry are learned from a set of data points in a parametric space (e.g. principal stress space, π-plane). Here, deep geometric learning enables us to reconstruct a highly complex yield envelope by breaking it down into multiple coordinate charts. The global atlas that consists of these coordinate charts in return allows us to represent the yield surface via multiple overlapping patches, each with a specific local parametrization. This setup provides several advantages over the classical implicit function representation approach. For instance, the availability of coordinate charts enables us to introduce an alternative stress integration algorithm where the trial stress may project directly on a local patch and hence circumvent the issues related to non-smoothness and the lack of convexity of yield surfaces. Meanwhile, the local parametric approach also enables us to predict hardening/softening locally in the parametric space, even without complete knowledge of the yield surface. Comparisons between the classical yield function approach on the non-smooth plasticity and anisotropic cam-clay plasticity model are provided to demonstrate the capacity of the models for highly precise yield surface and the feasibility of the implementation of the learned model in the local stress integration algorithm.
42 ENGINEERING↗
Spillover, nonlinearity, and flexible structures
Many systems whose evolution in time is governed by Partial Differential Equations (PDEs) are linearized around a known equilibrium before Computer Aided Control Engineering (CACE) is considered. In this case, there are infinitely many independent vibrational modes, and it is intuitively evident on physical grounds that infinitely many actuators would be needed in order to control all modes. A more precise, general formulation of this grave difficulty (spillover problem) is due to A.V. Balakrishnan. A possible route to circumvention of this difficulty lies in leaving the PDE in its original nonlinear form, and adding the essentially finite dimensional control action prior to linearization. One possibly applicable technique is the Liapunov Schmidt rigorous reduction of singular infinite dimensional implicit function problems to finite dimensional implicit function problems. Omitting details of Banach space rigor, the formalities of this approach are given.
Bass, Robert W.↗
Differentiable McCormick relaxations
McCormick's classical relaxation technique constructs closed-form convex and concave relaxations of compositions of simple intrinsic functions. These relaxations have several properties which make them useful for lower bounding problems in global optimization: they can be evaluated automatically, accurately, and computationally inexpensively, and they converge rapidly to the relaxed function as the underlying domain is reduced in size. They may also be adapted to yield relaxations of certain implicit functions and differential equation solutions. However, McCormick's relaxations may be nonsmooth, and this nonsmoothness can create theoretical and computational obstacles when relaxations are to be deployed. This article presents a continuously differentiable variant of McCormick's original relaxations in the multivariate McCormick framework of Tsoukalas and Mitsos. Gradients of the new differentiable relaxations may be computed efficiently using the standard forward or reverse modes of automatic differentiation. Furthermore, extensions to differentiable relaxations of implicit functions and solutions of parametric ordinary differential equations are discussed. A C++ implementation based on the library MC++ is described and applied to a case study in nonsmooth nonconvex optimization.
97 MATHEMATICS AND COMPUTING↗
A Conservative Finite-Volume Based Interface-Tracking Algorithm Using the Signed Distance Function
Methods for tracking an interface between two fluid phases are developed to ensure desired fluid properties, conservation, and stability are preserved in a finitevolume (FV) discretization. Here, the interface is tracked using a level set method where the signed distance function implicitly defines the interface. Marching methods are used to evaluate the value of the signed distance function, including a novel initialization method to initialize any implicit function to the signed distance function around sharp corners in the level set. Global conservation and consistency with a set of governing equations is enforced by a compression coefficient that measures the volumetric compression or expansion due to inaccuracies in the level set evaluation. A redistribution method is integrated into the volume correction to eliminate the small-cell instability while maintaining global conservation. This suite of methods is implemented and tested using static uniform velocity, and potential flow cases with multiple interface geometries. Results show these methods achieve up to second order accuracy, and are conservative. The application for these methods is intended to track the interface of a 3D printing filament in a finite-volume discretization of the all-speed Navier-Stokes equations.
42 ENGINEERING↗
Optimization of Desalination Systems with Detailed Water Chemistry through Integration of Reaktoro in WaterTAP
Chemistry predictions are critical for an accurate estimation of performance and costs in desalination process models, which allows for the estimation of the value of new technologies and the viability of treating new water sources. Herein, we present how an implicit function formulation can be used to integrate the chemical modeling package, Reaktoro, into the techno-economic assessment and modeling platform, WaterTAP. This approach resolves the critical issues of integrating large-scale thermodynamic models and databases into equation-oriented process models while allowing more flexibility relative to previously presented surrogate-based methods. We describe how this integration into Pyomo and WaterTAP models is implemented and used through the open-source package Reaktoro-PSE . We first validate this integration approach by performing optimization on a previously presented desalination treatment train with softening and acid addition as the pretreatment steps. Then, to demonstrate the value of this approach, we extend the cost-optimization problem to include the simultaneous addition of lime and soda ash for softening, and HCl and H 2 SO 4 in the acidification steps. Finally, we were able to confirm the previously established results that were obtained by using surrogate models and demonstrate that the implicit function approach enables exploration of different feedwater compositions and a larger number of chemicals and their combinations.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Bayesian calibration of thermodynamic models for the uptake of CO{sub 2} in supported amine sorbents using ab initio priors
A statistical methodology was applied to the simultaneous calibration and validation of thermodynamic models for the uptake of CO{sub 2} in mesoporous silica-supported amines. The methodology is Bayesian, and follows the procedure introduced by Kennedy and O'Hagan. One key aspect of the application presented is the use of quantum chemical calculations to define prior probability distributions for physical model parameters. Inclusion of this prior information proved to be crucial to the identifiability of model parameters against experimental thermogravimetric data. Through the statistical analysis, a quantitative assessment of the accuracy of various quantum chemical methods is produced. Another important aspect of the current approach is the conditioning of the model form discrepancy – a critical component of the Kennedy and O'Hagan methodology – to the experimental data in such a manner that it becomes an implicit function of the model parameters and thereby connected with the posterior distribution. It is shown that the inclusion of prior information in the analysis leads to a shifting of uncertainty from the posterior distribution for model parameters to this conditioned model form discrepancy. Prospects for more accurate model predictions and propagation of uncertainty in upscaling and extrapolation through a “model-plus-discrepancy” approach are discussed. The synthesis methods and thermogravimetric characterization of hybrid grafted/impregnated mesoporous silica-supported amine sorbents are presented, along with the details of the quantum chemical study, which shows that a carbamic acid–base acceptor complex is the most stable form of adsorbed CO{sub 2} in both alkanol- and ethyleneamines.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Multireference Density Functional Theory with Generalized Auxiliary Systems for Ground and Excited States
To describe static correlation, we create a new approach to density functional theory (DFT), which uses a generalized auxiliary system that is of a different symmetry, such as particle number or spin, from that of the physical system. The total energy of the physical system consists of two parts: the energy of the auxiliary system, which is determined with a chosen density functional approximation (DFA), and the excitation energy from an approximate linear response theory that restores the symmetry to that of the physical system, thus rigorously leading to a multideterminant description of the physical system. The electron density of the physical system is different from that of the auxiliary system and is uniquely determined from the functional derivative of the total energy with respect to the external potential. Our energy functional is thus an implicit functional of the physical system density, but an explicit functional of the auxiliary system density. We demonstrate that the total energy minimum and stationary states, describing the ground and excited states of the physical system, can be obtained by a self-consistent optimization with respect to the explicit variable, the generalized Kohn–Sham noninteracting density matrix. We have developed the generalized optimized effective potential method for the self-consistent optimization. Among options of the auxiliary system and the associated linear response theory, reformulated versions of the particle–particle random phase approximation (pp-RPA) and the spin-flip time-dependent density functional theory (SF-TDDFT) are selected for illustration of principle. Numerical findings show that our multireference DFT successfully describes static correlation in bond dissociation and double bond rotation.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Advanced probabilistic structural analysis method for implicit performance functions
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Wu, Y.-T.↗
Task-based detectability in CT image reconstruction by filtered backprojection and penalized likelihood estimation
Purpose: Nonstationarity is an important aspect of imaging performance in CT and cone-beam CT (CBCT), especially for systems employing iterative reconstruction. This work presents a theoretical framework for both filtered-backprojection (FBP) and penalized-likelihood (PL) reconstruction that includes explicit descriptions of nonstationary noise, spatial resolution, and task-based detectability index. Potential utility of the model was demonstrated in the optimal selection of regularization parameters in PL reconstruction. Methods: Analytical models for local modulation transfer function (MTF) and noise-power spectrum (NPS) were investigated for both FBP and PL reconstruction, including explicit dependence on the object and spatial location. For FBP, a cascaded systems analysis framework was adapted to account for nonstationarity by separately calculating fluence and system gains for each ray passing through any given voxel. For PL, the point-spread function and covariance were derived using the implicit function theorem and first-order Taylor expansion according toFessler [“Mean and variance of implicitly defined biased estimators (such as penalized maximum likelihood): Applications to tomography,” IEEE Trans. Image Process. 5(3), 493–506 (1996)]. Detectability index was calculated for a variety of simple tasks. The model for PL was used in selecting the regularization strength parameter to optimize task-based performance, with both a constant and a spatially varying regularization map. Results: Theoretical models of FBP and PL were validated in 2D simulated fan-beam data and found to yield accurate predictions of local MTF and NPS as a function of the object and the spatial location. The NPS for both FBP and PL exhibit similar anisotropic nature depending on the pathlength (and therefore, the object and spatial location within the object) traversed by each ray, with the PL NPS experiencing greater smoothing along directions with higher noise. The MTF of FBP is isotropic and independent of location to a first order approximation, whereas the MTF of PL is anisotropic in a manner complementary to the NPS. Task-based detectability demonstrates dependence on the task, object, spatial location, and smoothing parameters. A spatially varying regularization “map” designed from locally optimal regularization can improve overall detectability beyond that achievable with the commonly used constant regularization parameter. Conclusions: Analytical models for task-based FBP and PL reconstruction are predictive of nonstationary noise and resolution characteristics, providing a valuable framework for understanding and optimizing system performance in CT and CBCT.
62 RADIOLOGY AND NUCLEAR MEDICINE↗
Explicit dispersion relations for warm fluid waves in a uniform plasma (invited)
Classical dispersion relations for waves in a fluid plasma are expressed as implicit functions of wave frequency and wave- number. The implicit dispersion relation must in general be solved numerically. This work introduces an astute method for obtaining an explicit dispersion relation for waves in a fluid plasma. The explicit expression has an advantage of providing eigenmodes without numerical computations and giving a dispersion relation more detailed than the implicit expression. As in the case of cold waves, the wavenumber for an arbitrary frequency can be directly obtained from the explicit formula. The explicit formula also enables an accurate evaluation of finite-temperature effects on a dispersion relation even near the cyclotron resonance, where a numerical analysis of the implicit relation is impractical. As a result, the analytic formula can be used to investigate temperature effects on electromagnetic waves.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Unstable density distribution associated with equatorial plasma bubble
In this work, we present a simulation study of equatorial plasma bubble (EPB) in the evening time ionosphere. The fluid simulation is performed with a high grid resolution, enabling us to probe the steepened updrafting density structures inside EPB. Inside the density depletion that eventually evolves as EPB, both density and updraft are functions of space from which the density as implicit function of updraft velocity or the density distribution function is constructed. In the present study, this distribution function and the corresponding probability distribution function are found to evolve from Maxwellian to non-Maxwellian as the initial small depletion grows to EPB. This non-Maxwellian distribution is of a gentle-bump type, in confirmation with the recently reported distribution within EPB from space-borne measurements that offer favorable condition for small scale kinetic instabilities.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Semi-Analytic Functions to Calculate the Deposition Coefficients for Ice Crystal Vapor Growth in Bin and Bulk Microphysical Models
Numerical cloud models require estimates of the vapor growth rate for ice crystals. Current bulk and bin microphysical parameterizations generally assume that vapor growth is diffusion limited, though some parameterizations include the influence of surface attachment kinetics through a constant deposition coefficient. A parameterization for variable deposition coefficients is provided herein. The parameterization is an explicit function of the ambient ice supersaturation and temperature, and an implicit function of crystal dimensions and pressure. The parameterization is valid for variable surface types including growth by dislocations and growth by step nucleation. Deposition coefficients are predicted for the two primary growth directions of crystals, allowing for the evolution of the primary habits. Comparisons with benchmark calculations of instantaneous mass growth indicate that the parameterization is accurate to within a relative error of 1%. Parcel model simulations using Lagrangian microphysics as a benchmark indicate that the bulk parameterization captures the evolution of mass mixing ratio and fall speed with typical relative errors of less than 10%, whereas the average axis lengths can have errors of up to 20%. The bin model produces greater accuracy with relative errors often less that 10%. The deposition coefficient parameterization can be used in any bulk and bin scheme, with low error, if an equivalent volume spherical radius is provided.
54 ENVIRONMENTAL SCIENCES↗
A search for periodic variations of solar radio emission at 3.7- and 11.1-cm wavelengths
Periodic time variations in the emission of quiet solar regions were sought with the NRAO three-element interferometer at 3.7 and 11.1 cm. The visibility function of a source as obtained with a tracking interferometer is an implicit function of time, because of projected baseline changes, and therefore only observations taken when the baseline changes vary slowly with time were used. The visibility function shows significant time variations, but a power-spectrum analysis of the data revealed no unique periodicity.
Kundu, M. R.↗