Nonintrusive reduced‐order modeling of parametrized time‐dependent partial differential equations

We propose a nonintrusive reduced‐order modeling method based on the notion of space‐time‐parameter proper orthogonal decomposition (POD) for approximating the solution of nonlinear parametrized time‐dependent partial differential equations. A two‐level POD method is introduced for constructing spatial and temporal basis functions with special properties such that the reduced‐order model satisfies the boundary and initial conditions by construction. A radial basis function approximation method is used to estimate the undetermined coefficients in the reduced‐order model without resorting to Galerkin projection. This nonintrusive approach enables the application of our approach to general problems with complicated nonlinearity terms. Numerical studies are presented for the parametrized Burgers' equation and a parametrized convection‐reaction‐diffusion problem. We demonstrate that our approach leads to reduced‐order models that accurately capture the behavior of the field variables as a function of the spatial coordinates, the parameter vector and time. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

A new local reduced basis discontinuous Galerkin approach for heterogeneous multiscale problems

Inspired by the reduced basis approach and modern numerical multiscale methods, we present a new framework for an efficient treatment of heterogeneous multiscale problems. The new approach is based on the idea of considering heterogeneous multiscale problems as parametrized partial differential equations where the parameters are smooth functions. We then construct, in an offline phase, a suitable localized reduced basis that is used in an online phase to efficiently compute approximations of the multiscale problem by means of a discontinuous Galerkin method on a coarse grid. We present our approach for elliptic multiscale problems and discuss an a posteriori error estimate that can be used in the construction process of the localized reduced basis. Numerical experiments are given to demonstrate the efficiency of the new approach.     

Reduced basis a posteriori error bounds for parametrized linear-quadratic elliptic optimal control problems

We employ the reduced basis method as a surrogate model for the solution of optimal control problems governed by parametrized partial differential equations (PDEs) and develop rigorous a posteriori error bounds for the error in the optimal control and the associated error in the cost functional. The proposed bounds can be efficiently evaluated in an offline–online computational procedure. We present numerical results that confirm the validity of our approach.     

A fully Sinc-Galerkin method for Euler–Bernoulli beam models

A fully Sinc-Galerkin method in both space and time is presented for fourth-order time-dependent partial differential equations with fixed and cantilever boundary conditions. The sine discretizations for the second-order temporal problem and the fourth-order spatial problems are presented. Alternate formulations for variable parameter fourth-order problems are given, which prove to be especially useful when applying the forward techniques of this article to parameter recovery problems. The discrete system that corresponds to the time-dependent partial differential equations of interest are then formulated. Computational issues are discussed and an accurate and efficient algorithm for solving the resulting matrix system is outlined. Numerical results that highlight the method are given for problems with both analytic and singular solutions as well as fixed and cantilever boundary conditions.     

On Efficient Computation of the Optimization Problem Arising in the Inverse Modeling of Non-Stationary Multiphase Multicomponent Flow Through Porous Media

In this paper we discuss a method of solving inverse problems in non-isothermal multiphase multicomponent flow through porous media. The conceptual model is described by a system of non-linear partial differential equations which involve unknown parameters. These parameters are to be determined using a set of observations at discrete points in space and time by an optimization method. It is based on a reduced Gauss-Newton iteration in combination with an efficient gradient computation which takes advantage of a recently developed efficient numerical simulation technique. A sensitivity analysis is carried out for the optimum parameter set. Numerical experiments are performed for a one dimensional column experiment carried out at the VEGAS, University of Stuttgart, Germany.     

Fractional-order attractors synthesis via parameter switchings

In this paper we provide numerical evidence, via graphics generated with the help of computer simulations, that switching the control parameter of a dynamical system belonging to a class of fractional-order systems in a deterministic way, one obtains an attractor which belongs to the class of all admissible attractors of the considered system. For this purpose, while a multistep numerical method for fractional-order differential equations approximates the solution to the mathematical model, the control parameter is switched periodically every few integration steps. The switch is made inside of a considered set of admissible parameter values. Moreover, the synthesized attractor matches the attractor obtained with the control parameter replaced with the averaged switched parameter values. The results are verified in this paper on a representative system, the fractional-order Lü system. In this way we were able to extend the applicability of the algorithm presented in earlier papers using a numerical method for fractional differential equations.     

3D Multiphase Piecewise Constant Level Set Method Based on Graph Cut Minimization

Segmentation of three-dimensional (3D) complicated structures is of great importance for many real applications. In this work we combine graph cut minimization method with a variant of the level set idea for 3D segmentation based on the Mumford-Shah model. Compared with the traditional approach for solving the Euler-Lagrange equation we do not need to solve any partial differential equations. Instead, the minimum cut on a special designed graph need to be computed. The method is tested on data with complicated structures. It is rather stable with respect to initial value and the algorithm is nearly parameter free. Experiments show that it can solve large problems much faster than traditional approaches.     

Parameter estimation with model order reduction and global measurements

We present a new method for parameter estimation for elliptic partial differential equations. Parameter estimation requires the evaluation of the partial differential equation for many different parameter sets. Therefore, model order reduction is reasonable. Model order reduction is composed of an offline phase and an online phase. In the offline phase the reduced model is constructed using snapshots. In this paper we use the given measurement as only snapshot. Hence, the computational costs of the offline phase are reduced. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An asymptotic approach to analysis of 3D beam stress-strain elastic states

A regular asymptotic approach to analysis of 3D beam stress–strain states is proposed based on the linear theory of elasticity and the method of integrodifferential relations. Using the integral formulation of Hooke's law and polynomial expansions of unknown stress and displacement functions with respect to transversal Cartesian coordinates the initial system of partial differential equations is reduced to a countable system of ordinary differential equations with constant coefficients. For rectilinear beams with rectangular cross–sections the consistent boundary value problems describing independently the compression and stretch, bends, and torsion states are derived. To find equilibrium stress and admissible displacement fields satisfying boundary conditions an effective numerical algorithm is worked out. Integral and local criteria for explicit bilateral estimates of resulted solution quality are proposed. The numerical results are presented and discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

General Parameter Mapping Technique--A Procedure for Solution of Non-linear Boundary Value Problems Depending on an Actual Parameter

A new general method is developed to obtain the dependence ofthe solution of a non-linear boundary value problem on an arbitraryactual parameter of the problem under consideration. This techniqueis essentially an application of the implicit function theorem(Davidenko's approach) to the solution of non-linear boundaryvalue problems. The procedure suggested requires the integrationof sets of ordinary differential equations (initial value problems)with the right-hand sides obtained from a solution of a certainset of auxiliary differential equations.     

Domain Decomposition,Optimal Control of Systems Governed by Partial Differential Equations,and Synthesis of Feedback Laws

We present an iterative domain decomposition method for the optimal control of systems governed by linear partial differential equations. The equations can be of elliptic, parabolic, or hyperbolic type. The space region supporting the partial differential equations is decomposed and the original global optimal control problem is reduced to a sequence of similar local optimal control problems set on the subdomains. The local problems communicate through transmission conditions, which take the form of carefully chosen boundary conditions on the interfaces between the subdomains. This domain decomposition method can be combined with any suitable numerical procedure to solve the local optimal control problems. We remark that it offers a good potential for using feedback laws (synthesis) in the case of time-dependent partial differential equations. A test problem for the wave equation is solved using this combination of synthesis and domain decomposition methods. Numerical results are presented and discussed. Details on discretization and implementation can be found in Ref. 1.     

Determination of a control parameter in a one-dimensional parabolic equation using the method of radial basis functions

In this work, the method of radial basis functions is used for finding the solution of an inverse problem with source control parameter. Because a much wider range of physical phenomena are modelled by nonclassical parabolic initial-boundary value problems, theoretical behavior and numerical approximation of these problems have been active areas of research. The radial basis functions (RBF) method is an efficient mesh free technique for the numerical solution of partial differential equations. The main advantage of numerical methods which use radial basis functions over traditional techniques is the meshless property of these methods. In a meshless method, a set of scattered nodes are used instead of meshing the domain of the problem. The results of numerical experiments are presented and some comparisons are made with several well-known finite difference schemes.     

A Dirichlet–Neumann reduced basis method for homogeneous domain decomposition problems

Reduced basis methods allow efficient model reduction of parametrized partial differential equations. In the current paper, we consider a reduced basis method based on an iterative Dirichlet–Neumann coupling for homogeneous domain decomposition of elliptic PDEʼs. We gain very small basis sizes by an efficient treatment of problems with a-priori known geometry. Moreover iterative schemes may offer advantages over other approaches in the context of parallelization. We prove convergence of the iterative reduced scheme, derive rigorous a-posteriori error bounds and provide a full offline/online decomposition. Different methods for basis generation are investigated, in particular a variant of the POD-Greedy procedure. Experiments confirm the rigor of the error estimators and identify beneficial basis construction procedures.     

The Sinc-Galerkin method for parameter-dependent self-adjoint problems

The Sinc-Galerkin method is being applied to a growing number of diverse problems in ordinary and partial differential equations including both forward and inverse (parameter recovery) problems. As a result of these continuing extensions, the treatment of parameter-dependent problems needs to be thoroughly investigated. Two specific questions considered here are the incorporation of various nonhomogeneous boundary conditions and the treatment of a variable parameter. The latter topic is particularly important for inverse problems that arise when numerically estimating physical parameters. The point of view taken emphasizes the maintenance of the classical exponential convergence rate. The techniques described are suitable both for the direct problem and for the parameter estimation problem. Numerical results are presented to substantiate the accuracy of the method.     

Adaptive POD basis computation for parametrized nonlinear systems using optimal snapshot location

The construction of reduced-order models for parametrized partial differential systems using proper orthogonal decomposition (POD) is based on the information of the so-called snapshots. These provide the spatial distribution of the nonlinear system at discrete parameter and/or time instances. In this work a strategy is used, where the POD reduced-order model is improved by choosing additional snapshot locations in an optimal way; see Kunisch and Volkwein (ESAIM: M2AN, 44:509–529, 2010). These optimal snapshot locations influences the POD basis functions and therefore the POD reduced-order model. This strategy is used to build up a POD basis on a parameter set in an adaptive way. The approach is illustrated by the construction of the POD reduced-order model for the complex-valued Helmholtz equation.     

Transverse free vibration and stability of axially moving nanoplates based on nonlocal elasticity theory

Transverse dynamical behaviors of axially moving nanoplates which could be used to model the graphene nanosheets or other plate-like nanostructures with axial motion are examined based on the nonlocal elasticity theory. The Hamilton's principle is employed to derive the multivariable coupling partial differential equations governing the transverse motion of the axially moving nanoplates. Subsequently, the equations are transformed into a set of ordinary differential equations by the method of separation of variables. The effects of dimensionless small-scale parameter, axial speed and boundary conditions on the natural frequencies in sub-critical region are discussed by the method of complex mode. Then the Galerkin method is employed to analyze the effects of small-scale parameter on divergent instability and coupled-mode flutter in super-critical region. It is shown that the existence of small-scale parameter contributes to strengthen the stability in the super-critical region, but the stability of the sub-critical region is weakened. The regions of divergent instability and coupled-mode flutter decrease even disappear with an increase in the small-scale parameter. The natural frequencies in sub-critical region show different tendencies with different boundary effects, while the natural frequencies in super-critical region keep constants with the increase of axial speed.     

Learning patient-specific parameters for a diffuse interface glioblastoma model from neuroimaging data

Parameters in mathematical models for glioblastoma multiforme (GBM) tumour growth are highly patient specific. Here, we aim to estimate parameters in a Cahn–Hilliard type diffuse interface model in an optimised way using model order reduction (MOR) based on proper orthogonal decomposition (POD). Based on snapshots derived from finite element simulations for the full-order model (FOM), we use POD for dimension reduction and solve the parameter estimation for the reduced-order model (ROM). Neuroimaging data are used to define the highly inhomogeneous diffusion tensors as well as to define a target functional in a patient-specific manner. The ROM heavily relies on the discrete empirical interpolation method, which has to be appropriately adapted in order to deal with the highly nonlinear and degenerate parabolic partial differential equations. A feature of the approach is that we iterate between full order solvers with new parameters to compute a POD basis function and sensitivity-based parameter estimation for the ROM problems. The algorithm is applied using neuroimaging data for two clinical test cases, and we can demonstrate that the reduced-order approach drastically decreases the computational effort.     

Adaptive Inverse Control Using a Gradient-Projection Optimization Technique

As an application of an optimization technique, a gradient-projection method is employed to derive an adaptive algorithm for updating the parameters of an inverse which is designed to cancel the effects of actuator uncertainties in a control system. The actuator uncertainty is parametrized by a set of unknown parameters which belong to a parameter region. A desirable inverse is implemented with adaptive estimates of the actuator parameters. Minimizing an estimation error, a gradient algorithm is used to update such parameter estimates. To ensure that the parameter estimates also belong to the parameter region, the adaptive update law is designed with parameter projection. With such an adaptive inverse, desired control system performance can be achieved despite the presence of the actuator uncertainties.     

A numerical scheme based on Bernoulli wavelets and collocation method for solving fractional partial differential equations with Dirichlet boundary conditions

In this paper, an efficient and accurate numerical method is presented for solving two types of fractional partial differential equations. The fractional derivative is described in the Caputo sense. Our approach is based on Bernoulli wavelets collocation techniques together with the fractional integral operator, described in the Riemann‐Liouville sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations, which greatly simplifies the problem. By using Newton's iterative method, this system is solved and the solution of fractional partial differential equations is achieved. Some results concerning the error analysis are obtained. The validity and applicability of the method are demonstrated by solving four numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions much easier.