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Results for “Nonlinear approximation”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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DOE OSTI2026

Direct Nonlinear Approximation for Security Region Boundary of Integrated Energy Systems: A Polynomial Chaos Expansion Solution

The strong interdependence of electricity, gas, and heating systems can facilitate fault propagation within integrated energy systems (IESs), posing significant challenges to secure operation. This paper proposes a polynomial chaos expansion (PCE)-based approximation method to accurately characterize the IES security region boundary (IES–SRB). By integrating the Karush-Kuhn-Tucker conditions with PCE theory, the IES-SRB approximation problem is reformulated as a set of nonlinear equations concerning the approximation coefficients. Using the Galerkin projection method, these equations are further transformed into a system of projection equations that govern the polynomial approximation coefficients in the IES-SRB approximation. To reduce computational complexity while maintaining high approximation accuracy, a piecewise polynomial approximation method is proposed. Numerical studies on the E39-G20-H6 and E118-G96-H52 IES test systems demonstrate that the proposed method can accurately and effectively construct IES security regions.

Wu, Chenghao [Northeast Electric Power University]↗
NASA NTRS2019

Efficient method for approximating nonlinear dynamics: applications to uncertainty propagation and estimation

High-order Taylor series expansions can be used to model nonlinear dynamics at the cost of integrating a large set of variational equations to obtain high-order state-transition tensors (STTs). This paper presents an innovative technique for approximating the high-order STTs that reduces significantly the computational cost by retaining only the dominant secular terms. We propagate the low-order partial derivatives of Kepler’s equation, which only requires the integration of six additional equations to extend an n-th order approximation to order (n + 1). The approximation stems from the Lindstedt-Poincare procedure and exploits the stability properties of orbital motion. Since the method makes no dynamical assumptions, it can accommodate any source of orbital perturbations. We show how the approximation of the second-order STT significantly increases the accuracy of the linear method for uncertainty propagation with only a small computational overhead. Finally, we derive a high-order approximate extended Kalman filter that implements the proposed approximation of the STT and improves the performance of linear filters. Examples of application with different perturbation sources include the heliocentric orbit of an asteroid, an orbiter around Europa, and an Earth-orbiting satellite.

Park, Ryan S↗
NASA NTRS1990

Conservative buffering of approximate nonlinear constraints

In engineering design practice behavior is usually predicted based on some known nominal design. However, when the design is fabricated it will differ from the nominal design because of manufacturing tolerances. In order to generate nominal designs that will still satisfy behavior constraints in the presence of manufacturing tolerances, engineers resort to the use of safety factors, over and above those introduced to account for other uncertainties (e.g., in load conditions, material properties, analysis modeling). The accurate selection of the values of these manufacturing tolerances safety factors is dependent on the capability of the engineer to determine the sensitivity of the critical constraints to changes in the design variables. This process usually leads to overly conservative designs. The task of choosing safety factors is much more difficult in structural synthesis because: (1) it is not known which constraints will be active at the final design, (2) as the design changes during the synthesis process the sensitivities of the constraints with respect to the design variables also change, and (3) the imposition of the safety factors themselves may change the set of critical constraints. These difficulties can be overcome with the approximation concepts approach to structural synthesis by buffering the approximate constraints with quantities that are related to the design variable tolerances and the accurate sensitivities of the constraints with respect to the design variable. Designs generated by this approach tend to be feasible but not overly conservative.

Thomas, H. L.↗
NASA NTRS1987

Nonlinear system approximations

The approximation of a nonlinear system by a feedback linearizable system instead of a linear system is considered. Two approaches presently exist in the literature. One involves the concept of involutivity to a certain order, and the other considers a canonical expansion and pure feedback approximation. The relationship between these two methods is shown. This provides insight into lower-order invariants in the equivalence problem for two nonlinear systems. The output time responses for a nonlinear system and its feedback linearizable approximation are outlined.

Goldthwait, Richard G.↗
NASA NTRS1982

Difference equation state approximations for nonlinear hereditary control problems

Discrete approximation schemes for the solution of nonlinear hereditary control problems are constructed. The methods involve approximation by a sequence of optimal control problems in which the original infinite dimensional state equation has been approximated by a finite dimensional discrete difference equation. Convergence of the state approximations is argued using linear semigroup theory and is then used to demonstrate that solutions to the approximating optimal control problems in some sense approximate solutions to the original control problem. Two schemes, one based upon piecewise constant approximation, and the other involving spline functions are discussed. Numerical results are presented, analyzed and used to compare the schemes to other available approximation methods for the solution of hereditary control problems.

Rosen, I. G.↗
NASA NTRS1984

Difference equation state approximations for nonlinear hereditary control problems

Discrete approximation schemes for the solution of nonlinear hereditary control problems are constructed. The methods involve approximation by a sequence of optimal control problems in which the original infinite dimensional state equation has been approximated by a finite dimensional discrete difference equation. Convergence of the state approximations is argued using linear semigroup theory and is then used to demonstrate that solutions to the approximating optimal control problems in some sense approximate solutions to the original control problem. Two schemes, one based upon piecewise constant approximation, and the other involving spline functions are discussed. Numerical results are presented, analyzed and used to compare the schemes to other available approximation methods for the solution of hereditary control problems. Previously announced in STAR as N83-33589

Rosen, I. G.↗
NASA NTRS1988

Backward differentiation approximations of nonlinear differential/algebraic systems

Finite-difference approximations of dynamical systems modeled by nonlinear, semiexplicit, differential/algebraic equations are analyzed. Convergence for the backward differentiation method is proved for index two and index three problems when the numerical initial values obey certain constraints. The appropriate asymptotic convergence rates and the leading error terms are determined.

Brenan, Kathryn E.↗
NASA NTRS1982

Spline approximations for nonlinear hereditary control systems

A sline-based approximation scheme is discussed for optimal control problems governed by nonlinear nonautonomous delay differential equations. The approximating framework reduces the original control problem to a sequence of optimization problems governed by ordinary differential equations. Convergence proofs, which appeal directly to dissipative-type estimates for the underlying nonlinear operator, are given and numerical findings are summarized.

Daniel, P. L.↗
NASA NTRS1992

Nonlinear functional approximation with networks using adaptive neurons

A novel mathematical framework for the rapid learning of nonlinear mappings and topological transformations is presented. It is based on allowing the neuron's parameters to adapt as a function of learning. This fully recurrent adaptive neuron model (ANM) has been successfully applied to complex nonlinear function approximation problems such as the highly degenerate inverse kinematics problem in robotics.

Tawel, Raoul↗
NASA NTRS1988

Semi-discrete approximations to nonlinear systems of conservation laws; consistency and L(infinity)-stability imply convergence

A convergence theory for semi-discrete approximations to nonlinear systems of conservation laws is developed. It is shown, by a series of scalar counter-examples, that consistency with the conservation law alone does not guarantee convergence. Instead, a notion of consistency which takes into account both the conservation law and its augmenting entropy condition is introduced. In this context it is concluded that consistency and L(infinity)-stability guarantee for a relevant class of admissible entropy functions, that their entropy production rate belongs to a compact subset of H(loc)sup -1 (x,t). One can now use compensated compactness arguments in order to turn this conclusion into a convergence proof. The current state of the art for these arguments includes the scalar and a wide class of 2 x 2 systems of conservation laws. The general framework of the vanishing viscosity method is studied as an effective way to meet the consistency and L(infinity)-stability requirements. How this method is utilized to enforce consistency and stability for scalar conservation laws is shown. In this context we prove, under the appropriate assumptions, the convergence of finite difference approximations (e.g., the high resolution TVD and UNO methods), finite element approximations (e.g., the Streamline-Diffusion methods) and spectral and pseudospectral approximations (e.g., the Spectral Viscosity methods).

Tadmor, Eitan↗
NASA NTRS1983

Numerical approximations to nonlinear conservation laws with locally varying time and space grids

Numerical approximations to the initial value problem for nonlinear systems of conservation laws are considered. The considered system is said to be hyperbolic when all eigenvalues of every real linear combination of the Jacobian matrices are real. Solutions may develop discontinuities in finite time, even when the initial data are smooth. In the investigation, explicit finite difference methods which use locally varying time grids are considered. The global CFL restriction is replaced by a local restriction. The numerical flux function is studied from a finite volume viewpoint, and a differencing technique is developed at interface points between regions of distinct time increments.

Osher, S.↗
DOE OSTI2025

Massively parallel and universal approximation of nonlinear functions using diffractive processors

Nonlinear computation is essential for a wide range of information processing tasks, yet implementing nonlinear functions using optical systems remains a challenge due to the weak and power-intensive nature of optical nonlinearities. Overcoming this limitation without relying on nonlinear optical materials could unlock unprecedented opportunities for ultrafast and parallel optical computing systems. Here, we demonstrate that large-scale nonlinear computation can be performed using linear optics through optimized diffractive processors composed of passive phase-only surfaces. In this framework, the input variables of nonlinear functions are encoded into the phase of an optical wavefront—e.g., via a spatial light modulator (SLM)—and transformed by an optimized diffractive structure with spatially varying point-spread functions to yield output intensities that approximate a large set of unique nonlinear functions–all in parallel. We provide proof establishing that this architecture serves as a universal function approximator for an arbitrary set of bandlimited nonlinear functions, also covering wavelength-multiplexed nonlinear functions as well as multi-variate and complex-valued functions that are all-optically cascadable. Our analysis also indicates the successful approximation of typical nonlinear activation functions commonly used in neural networks, including the sigmoid, tanh, ReLU (rectified linear unit), and softplus. We numerically demonstrate the parallel computation of one million distinct nonlinear functions, accurately executed at wavelength-scale spatial density at the output of a diffractive optical processor. Furthermore, we experimentally validated this framework using in situ optical learning and approximated 35 unique nonlinear functions in a single shot using a compact setup consisting of an SLM and an image sensor. These results establish diffractive optical processors as a scalable platform for massively parallel universal nonlinear function approximation, paving the way for new capabilities in analog optical computing based on linear materials.

Rahman, Md Sadman Sakib [University of California,↗