Parameter-dependent Coefficients Research Articles

Non-Intrusive Reduced-Order Modeling Based on Parametrized Proper Orthogonal Decomposition

A new non-intrusive reduced-order modeling method based on space-time parameter decoupling for parametrized time-dependent problems is proposed. This method requires the preparation of a database comprising high-fidelity solutions. The spatial bases are extracted from the database through first-level proper orthogonal decomposition (POD). The algebraic relationship between the time trajectory/parameter positions and the projection coefficient is described by the linear superposition of the second-level POD bases (temporal bases) and the second-level projection coefficients (parameter-dependent coefficients). This decomposition strategy decouples the space-time parameter effects, providing a stable foundation for fast predictions of parametrized time-dependent problems. The mappings between the parameter locations and the parameter-dependent coefficients are approximated as Gaussian process regression (GPR) models. The accuracy and efficiency of the PPOD-ROM are demonstrated through two numerical examples: flows past a cylinder and turbine flows with a clocking effect.

A reduced basis super-localized orthogonal decomposition for reaction-convection-diffusion problems

This paper presents a method for the numerical treatment of reaction-convection-diffusion problems with parameter-dependent coefficients that are arbitrary rough and possibly varying at a very fine scale. The presented technique combines the reduced basis (RB) framework with the recently proposed super-localized orthogonal decomposition (SLOD). More specifically, the RB is used for accelerating the typically costly SLOD basis computation, while the SLOD is employed for an efficient compression of the problem's solution operator requiring coarse solves only. The combined advantages of both methods allow one to tackle the challenges arising from parametric heterogeneous coefficients. Given a value of the parameter vector, the method outputs a corresponding compressed solution operator which can be used to efficiently treat multiple, possibly non-affine, right-hand sides at the same time, requiring only one coarse solve per right-hand side.

The importance of parameter-dependent coefficient of restitution in discrete element method simulations

The coefficient of restitution (COR) is an important constant that represents the energy dissipation during contact between two objects. Simulation using the conventional discrete element method (DEM) involves a constant COR. This study presents a DEM simulation method that uses a parameter-dependent COR. The parameter-dependent COR was obtained from a collision incident between spherical particles and a plate surface using a drop-test apparatus. Glass and polypropylene beads of 3–6-mm diameter were used while acrylic and steel were used as the plate surfaces. The particle trajectories were captured by a high-speed camera and analyzed by an image analyzer. The COR was then correlated to a parameter-dependent COR function that depends on the material, impact velocity, and temperature. Free-fall DEM simulations using a constant COR and parameter-dependent COR were compared. The parameter-dependent COR approach obtained better agreement with experimental results than the constant-COR approach. The proposed concept could be applied for other material combinations with a wide range of operating conditions to obtain a database of parameter-dependent COR values for the simulation of solid handling applications.

Block Preconditioning of Stochastic Galerkin Problems: New Two-sided Guaranteed Spectral Bounds

The paper focuses on numerical solution of parametrized diffusion equations with scalar parameter-dependent coefficient function by the stochastic (spectral) Galerkin method. We study preconditioning of the related discretized problems using preconditioners obtained by modifying the stochastic part of the partial differential equation. We present a simple but general approach for obtaining two-sided bounds to the spectrum of the resulting matrices, based on a particular splitting of the discretized operator. Using this tool and considering the stochastic approximation space formed by classical orthogonal polynomials, we obtain new spectral bounds depending solely on the properties of the coefficient function and the type of the approximation polynomials for several classes of block-diagonal preconditioners. These bounds are guaranteed and applicable to various distributions of parameters. Moreover, the conditions on the parameter-dependent coefficient function are only local, and therefore less restrictive than those usually assumed in the literature.

Balanced Model Order Reduction Method for Systems Depending on a Parameter

Abstract We provide an analytical framework for balanced realization model order reduction of linear control systems which depend on an unknown parameter. Besides recovering known results for the first order corrections, we obtain explicit novel expressions for the form of second order corrections for singular values and singular vectors. The final result of our procedure is an order reduced model which incorporates the uncertain parameter. We apply our algorithm to a system of masses and springs with parameter dependent coefficients.

Homoclinic and Heteroclinic Orbits for a Liénard System

A 2D dynamical system exhibiting a double-zero bifurcation with symmetry of order two is considered. This bifurcation involves the presence in the parameter space of a curve corresponding either to double homoclinic or to heteroclinic bifurcations. In this paper we derive second order approximations for the homoclinic orbits and for the curve of homoclinic bifurcation values considering the system truncated up to five order terms and parameter-dependent coefficients. These approximations were obtained using the regular perturbation method. These formulae are applied to a Lienard system, which develops a double-zero bifurcation with symmetry of order two for some parameters values. Second order approximations for the heteroclinic orbits of this system are also given. The analytical results are very accurate and they are in good accordance with the numerical ones.

Direct tensor-product solution of one-dimensional elliptic equations with parameter-dependent coefficients

We consider a one-dimensional second-order elliptic equation with a high-dimensional parameter in a hypercube as a parametric domain. Such a problem arises, for example, from the Karhunen–Loève expansion of a stochastic PDE posed in a one-dimensional physical domain. For the discretization in the parametric domain we use the collocation on a tensor-product grid. The paper is focused on the tensor-structured solution of the resulting multiparametric problem, which allows to avoid the curse of dimensionality owing to the use of the separation of parametric variables in the tensor train and quantized tensor train formats.We suggest an efficient tensor-structured preconditioning of the entire multiparametric family of one-dimensional elliptic problems and arrive at a direct solution formula. We compare this method to a tensor-structured preconditioned GMRES solver in a series of numerical experiments.

Team Optimal Control of Stochastically Switched Systems With Local Parameter Knowledge

This paper considers the optimal decentralized control of linear systems with stochastically switched cost and system matrices depending on local parameters. Two types of dynamic switched problems are considered, with partially nested and one-step delayed sharing information structures. For the former case, parameters and measurements follow a partially nested structure with the parameters possibly being correlated across all stages. For the latter case, parameters are assumed to be Markov processes, with their values along with measurements available instantaneously to local controllers, but with a one time step delay to others. The solution to both these problems rely on the optimal solution to a static (one-stage) stochastic-parameter problem with local type dependent Gaussian measurements, and for this purpose the static quadratic team problem is examined in some generality. Using an operator theoretic framework, it is shown that the sequential update scheme always converges to the team optimal strategy for general static quadratic team setups. This, when applied to the static stochastic-parameter problem, yields the team optimal strategy. The corresponding strategy is affine in the measurements with the parameter dependent coefficients obtained by solving a set of linear equations. These equations are immediately solvable when the total number of parameter values is finite. However, for the case of infinite parameter values, the update scheme also provides a mechanism to determine an approximation to the team optimal strategy.

Parametric Reduced Order Models Using Adaptive Sampling and Interpolation

Over the past decade, a number of approaches have been put forth to improve the accuracy of projection-based reduced order models over parameter ranges. These can be classified as either i.) building a global basis that is suitable for a large parameter set by applying sampling strategies, ii.) identifying parameter dependent coefficient functions in the reduced order model, or iii.) changing the basis as parameters change. We propose a strategy that combines sampling with basis interpolation. We apply sampling strategies that identify suitable parameter values from which associated basis functions are interpolated at any parameter value in a region. While our approach has practical limits to roughly a handful of parameters, it has the advantage of achieving a desired level of accuracy in parametric reduced-order models of relatively small size. We present this method using a proper orthogonal decomposition model of a nonlinear partial differential equation with variable coefficients and initial conditions.

Singular Hopf Bifurcations in DAE Models of Power Systems

We investigate an important relationship that exists between the Hopf bifurcation in the singularly perturbed nonlinear power systems and the singularity induced bifurcations (SIBs) in the corresponding different- tial-algebraic equations (DAEs). In a generic case, the SIB phenomenon in a system of DAEs signals Hopf bifurcation in the singularly perturbed systems of ODEs. The analysis is based on the linear matrix pencil theory and polynomials with parameter dependent coefficients. A few numerical examples are included.

A Multilevel Jacobi–Davidson Method for Polynomial PDE Eigenvalue Problems Arising in Plasma Physics

The simulation of drift instabilities in the plasma edge leads to cubic polynomial PDE eigenvalue problems with parameter dependent coefficients. The aim is to determine the wave number which leads to the maximum growth rate of the amplitude of the wave. This requires the solution of a large number of PDE eigenvalue problems. Since we are only interested in a smooth eigenfunction corresponding to the eigenvalue with largest imaginary part, the Jacobi–Davidson method can be applied. Unfortunately, a naive implementation of this method is much too expensive for the large number of problems that have to be solved. In this paper we will present a multilevel approach for the construction of an appropriate initial search space. We will also discuss the efficient solution of the correction equation, and we will show how optimal scaling helps to accelerate the convergence.

Birth-and-Death Type Systems with Parameter and Chaotic Dynamics of some Linear Kinetic Models

J. Banasiak and M. Lachowicz proved in [Math. Models Methods Appl. Sci. 12 (2002), 755 – 775] that, under certain conditions on the coefficients, the dynamics generated by birth-and-death type systems with proliferation was chaotic. In this paper we extend this result to systems with parameter-dependent coefficients and present an application to a linear Boltzmann equation describing inelastic collisions.

2-D stable polynomials with parameter-dependent coefficients: generalizations and new results

Stability of multidimensional systems is a field of intensive research. In this context, different classes of Hurwitz polynomials (in the continuous case) and Schur polynomials (in the discrete case) are in the focus of interest. Although there exist various methods for testing whether a given polynomial belongs to a certain class of the afore mentioned. The type of converse problem, namely the design of stable polynomials is much more tedious. In this paper, a parametric model for the characterization of real or complex two-variable scattering Schur polynomials is given. In other words, the coefficients of the two-dimensional (2-D) polynomial model are functions of real parameters. The following features make it best suited for the design of 2-D systems: no dependencies between the real valued parameters, coverage of the whole class of 2-D scattering Schur polynomials, and the coefficients of the polynomial are rational functions of the parameters. The synthesis of 2-D lossless networks and unitary matrices play a key role in our considerations.

Remarks on recurrences on streams

The notion of sequences on streams intuitively present in constructions of [2] and [3] is here defined and then a new kind of recurrences for these sequences labelled by elements (“streams”) of linearly ordered set are studied. A peculiar set of solutions of such recurrences consists of Tchebyshev-like special functions, which are polynomials in co-ordinates of a certain curve on hypersurface in ▪ determined by quasi-numbers of determinant one. These Tchebyshev-like special functions are shown to satisfy an ordinary m-th order recurrence with parameter dependent coefficients. For m = 2 one gets classical Tchebyshev polynomials of both kinds. The main content of this note is accompanied with an overview comparison with other “non-trigonometric” approaches.

Stability determination of two-dimensional discrete-time systems

In determining root distribution of univariate polynomials with real or complex-valued coefficients, the Bistritz tabular form offers a significant computational advantage. Stability studies of two-dimensional (2-D) discrete-time systems involve univariate polynomials possessing parameter-dependent coefficients, where the parameter takes values on the unit circle in the complex plane. This paper investigates the application of Bistritz tabular form in determining stability of 2-D discrete-time systems, and for this purpose we present two algorithms. Both algorithms utilize a recent result that has established the relationship between Schur-Cohn minors and the entries of the Bistritz tabular form corresponding to a given polynomial. A comparison between the use of the modified Jury table and the Bistritz table in stability checking of 2-D discrete-time systems is also presented.