Efficient use of parallelism in algorithmic parameter optimization applications

In the context of algorithmic parameter optimization, there is much room for efficient usage of computational resources. We consider the Opal framework in which a nonsmooth optimization problem models the parameter identification task, and is solved by a mesh adaptive direct search solver. Each evaluation of trial parameters requires the processing of a potentially large number of independent tasks. We describe and evaluate several strategies for using parallelism in this setting. Our test scenario consists in optimizing five parameters of a trust-region method for smooth unconstrained minimization.     

Gradient‐based optimization of a viscoelastic Ogden type model for cellular polymers

The parameter identification is the interface between a theoretical material model and its application in numerical computations. Only by an accurate identification of the theoretically introduced material parameters, an applicable simulation of the material is achieved. An increasing standard of the parameter identification is set by the requirements of complex material models used in computer‐aided engineering. A common identification strategy is a gradient‐based optimization of a least‐squares functional, e. g. the sequential quadratic programming (SQP) technique. In this paper, the SQP method is used to optimize material models of cellular polymers. In particular, the optimization is shown for a viscoelastic polyurethane (PU) foam. Due to the high‐grade nonlinear material behaviour, the foam is modelled by a finite viscoelastic Ogden type law in the framework of the Theory of Porous Media (TPM).     

Parameter–identification of macroscopic material models based on virtual testing of given material mesostructures

For many heterogeneous materials such as composites and polycrystals, the material modeling for the constituents on a representative mesoscale can be considered as known, including concrete values of their inherent material parameters. Typical examples are isotropic elastic–plastic models for the constituents of composites or anisotropic crystal–plasticity models for the grains of polycrystals. This knowledge can be exploited with regard to the modeling of the homogenized macroscopic response. In particular, parameters in macroscopic models may be identified by virtual experiments provided by a computational deformation–driving of representative mesostructures. This paper outlines the general concept for the parameter–identification of macroscopic materialmodels based on the virtual testing of given material mesostructures. The virtual test data are obtained in the form of multi–dimensional stress–strain paths by applying different deformation gradients to a given mesostructure. After specifying a corresponding macroscopic material model covering the observed effects on the macroscale, the material parameters are identified by a least–square–type optimization procedure that optimizes the macroscopic material parameters. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Adaptive inexact fast augmented Lagrangian methods for constrained convex optimization

In this paper we study two inexact fast augmented Lagrangian algorithms for solving linearly constrained convex optimization problems. Our methods rely on a combination of the excessive-gap-like smoothing technique introduced in Nesterov (SIAM J Optim 16(1):235–249, 2005) and the general inexact oracle framework studied in Devolder (Math Program 146:37–75, 2014). We develop and analyze two augmented based algorithmic instances with constant and adaptive smoothness parameters, and derive a total computational complexity estimate in terms of projections on a simple primal feasible set for each algorithm. For the constant parameter algorithm we obtain the overall computational complexity of order \(\mathcal {O}(\frac{1}{\epsilon ^{5/4}})\), while for the adaptive one we obtain \(\mathcal {O}(\frac{1}{\epsilon })\) total number of projections onto the primal feasible set in order to achieve an \(\epsilon \)-optimal solution for the original problem.     

Higher-order conditions for strict efficiency

Luu and Kien (On higher order conditions for strict efficiency, Soochow J. Math. 33 (2007), pp. 17–31), proposed higher-order conditions for strict efficiency of vector optimization problems based on the derivatives introduced in Ginchev (Higher order optimality conditions in nonsmooth optimization, Optimization 51 (2002), pp. 47–72). These derivatives are defined for scalar functions and in their terms necessary and sufficient conditions are obtained a point to be strictly efficient (isolated) minimizer of a given order for quite arbitrary scalar function. Passing to vector functions, Luu and Kien lose the peculiarity that the optimality conditions work with arbitrary functions. In this article, applying the mentioned derivatives for the scalarized problem and restoring the original idea, optimality conditions for strict efficiency of a given order are proposed, which work with quite arbitrary vector functions. It is shown that the results of Luu and Kien are corollaries of the given conditions and generalizations are discussed.     

Reliable gain‐scheduled control design for networked control systems

In this article, the problem of reliable gain‐scheduled H_∞ performance optimization and controller design for a class of discrete‐time networked control system (NCS) is discussed. The main aim of this work is to design a gain‐scheduled controller, which consists of not only the constant parameters but also the time‐varying parameter such that NCS is asymptotically stable. In particular, the proposed gain‐scheduled controller is not only based on fixed gains but also the measured time‐varying parameter. Further, the result is extended to obtain a robust reliable gain‐scheduled H_∞ control by considering both unknown disturbances and linear fractional transformation parametric uncertainties in the system model. By constructing a parameter‐dependent Lyapunov–Krasovskii functional, a new set of sufficient conditions are obtained in terms of linear matrix inequalities (LMIs). The existence conditions for controllers are formulated in the form of LMIs, and the controller design is cast into a convex optimization problem subject to LMI constraints. Finally, a numerical example based on a station‐keeping satellite system is given to demonstrate the effectiveness and applicability of the proposed reliable control law. © 2014 Wiley Periodicals, Inc. Complexity 21: 214–228, 2015     

On Topological Derivatives for Elastic Solids with Uncertain Input Data

In this paper, a new approach to the derivation of the worst scenario and the maximum range scenario methods is proposed. The derivation is based on the topological derivative concept for the boundary-value problems of elasticity in two and three spatial dimensions. It is shown that the topological derivatives can be applied to the shape and topology optimization problems within a certain range of input data including the Lamé coefficients and the boundary tractions. In other words, the topological derivatives are stable functions and the concept of topological sensitivity is robust with respect to the imperfections caused by uncertain data. Two classes of integral shape functionals are considered, the first for the displacement field and the second for the stresses. For such classes, the form of the topological derivatives is given and, for the second class, some restrictions on the shape functionals are introduced in order to assure the existence of topological derivatives. The results on topological derivatives are used for the mathematical analysis of the worst scenario and the maximum range scenario methods. The presented results can be extended to more realistic methods for some uncertain material parameters and with the optimality criteria including the shape and topological derivatives for a broad class of shape functionals. This research is partially supported by the Brazilian Agency CNPq under Grant 472182/2007-2, FAPERJ under Grant E-26/171.099/2006 (Rio de Janeiro) and Brazilian-French Research Program CAPES/COFECUB under Grant 604/08 between LNCC in Petrópolis and IECN in Nancy, and by the Research Grant CNRS-CSAV between Institut Elie Cartan in Nancy and the Institut of Mathematics in Prague. The support is gratefully acknowledged.     

Algorithmic Consistency in Computational Inelasticity – a Conceptual Completion

This paper communicates a new algorithmic concept, how higher-order Runge-Kutta (RK) methods for time integration of viscoelastic constitutive laws can be introduced into nonlinear finite element methods in order (i) to obtain the full nominal order p in time integration, (ii) to ensure that global equilibrium is only required at the end of time intervals Δt but not in the interior at RK-stages, and (iii) to obtain –based on (i) and (ii)– a considerable speed-up compared with Backward-Euler. The condition to realize (i)–(iii) is, that the approximation of total strain in time must be of the same order as the time-integration method, which is a completion of the concept of algorithmic consistency in computational inelasticity. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Goal-oriented r-adaptivity based on the evaluation of the physical and material residual

This contribution is concerned with goal–oriented r-adaptivity based on energy minimization principles for the primal and the dual problem. We obtain a material residual of the primal and of the dual problem, which are indicators for non–optimal finite element meshes. For goal–oriented r-adaptivity we have to optimize the mesh with respect to the dual solution, because the error of a local quantity of interest depends on the error in the corresponding dual solution. We use the material residual of the primal and dual problem in order to obtain a procedure for mesh optimization with respect to a local quantity of interest. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A space decomposition scheme for maximum eigenvalue functions and its applications

In this paper, we study nonlinear optimization problems involving eigenvalues of symmetric matrices. One of the difficulties in solving these problems is that the eigenvalue functions are not differentiable when the multiplicity of the function is not one. We apply the \({\mathcal {U}}\)-Lagrangian theory to analyze the largest eigenvalue function of a convex matrix-valued mapping which extends the corresponding results for linear mapping in the literature. We also provides the formula of first-and second-order derivatives of the \({\mathcal {U}}\)-Lagrangian under mild assumptions. These theoretical results provide us new second-order information about the largest eigenvalue function along a suitable smooth manifold, and leads to a new algorithmic framework for analyzing the underlying optimization problem.     

A layer adaptive B‐spline collocation method for singularly perturbed one‐dimensional parabolic problem with a boundary turning point

In this article, we develop a parameter uniform numerical method for a class of singularly perturbed parabolic equations with a multiple boundary turning point on a rectangular domain. The coefficient of the first derivative with respect to x is given by the formula a₀(x, t)x^p, where a₀(x, t) ≥ α > 0 and the parameter p ∈ [1,∞) takes the arbitrary value. For small values of the parameter ε, the solution of this particular class of problem exhibits the parabolic boundary layer in a neighborhood of the boundary x = 0 of the domain. We use the implicit Euler method to discretize the temporal variable on uniform mesh and a B‐spline collocation method defined on piecewise uniform Shishkin mesh to discretize the spatial variable. Asymptotic bounds for the derivatives of the solution are established by decomposing the solution into smooth and singular component. These bounds are applied in the convergence analysis of the proposed scheme on Shishkin mesh. The resulting method is boundary layer resolving and has been shown almost second‐order accurate in space and first‐order accurate in time. It is also shown that the proposed method is uniformly convergent with respect to the singular perturbation parameter ε. Some numerical results are given to confirm the predicted theory and comparison of numerical results made with a scheme consisting of a standard upwind finite difference operator on a piecewise uniform Shishkin mesh. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1143–1164, 2011     

Gibbs measures for fertile hard-core models on the Cayley tree

We study fertile hard-core models with the activity parameter λ > 0 and four states on the Cayley tree. It is known that there are three types of such models. For each of these models, we prove the uniqueness of the translation-invariant Gibbs measure for any value of the parameter λ on the Cayley tree of order three. Moreover, for one of the models, we obtain critical values of λ at which the translation-invariant Gibbs measure is nonunique on the Cayley tree of order five. In this case, we verify a sufficient condition (the Kesten–Stigum condition) for a measure not to be extreme.     

A level set approach for the solution of a state-constrained optimal control problem

Summary. State constrained optimal control problems for linear elliptic partial differential equations are considered. The corresponding first order optimality conditions in primal-dual form are analyzed and linked to a free boundary problem resulting in a novel algorithmic approach with the boundary (interface) between the active and inactive sets as optimization variable. The new algorithm is based on the level set methodology. The speed function involved in the level set equation for propagating the interface is computed by utilizing techniques from shape optimization. Encouraging numerical results attained by the new algorithm are reported on.Mathematics Subject Classification (1991): 35R35, 49K20, 49Q10, 65K10Revised version received March 19, 2003     

Pitfalls of the Fourier Transform Method in Affine Models,and Remedies

We study sources of potentially serious errors of popular numerical realizations of the Fourier method in affine models and explain that, in many cases, a calibration procedure based on such a realization will be able to find a “correct parameter set” only in a rather small region of the parameter space, with a blind spot: an interval of strikes depending on the model and time to maturity, where accurate calculations are extremely time-consuming. We explain how to construct more accurate and faster pricing and calibration procedures. An important ingredient of our method is the study of the analytic continuation of the solution of the associated system of generalized Riccati equations, and contour deformation techniques. As a byproduct, we show that the straightforward application of the Runge–Kutta method may lead to sizable errors, and suggest certain remedies. In the paper, the method is applied to a wide class of stochastic volatility models with stochastic interest rate and interest rate models of A_n(n) class. The methodology of the paper can be applied to other models (e.g., quadratic term structure models, Wishart dynamics, 3/2-model).     

Retrieving the time-dependent thermal conductivity of an orthotropic rectangular conductor

The aim of this paper is to determine the thermal properties of an orthotropic planar structure characterized by the thermal conductivity tensor in the coordinate system of the main directions (Oxy) being diagonal. In particular, we consider retrieving the time-dependent thermal conductivity components of an orthotropic rectangular conductor from nonlocal overspecified heat flux conditions. Since only boundary measurements are considered, this inverse formulation belongs to the desirable approach of non-destructive testing of materials. The unique solvability of this inverse coefficient problem is proved based on the Schauder fixed point theorem and the theory of Volterra integral equations of the second kind. Furthermore, the numerical reconstruction based on a nonlinear least-squares minimization is performed using the MATLAB optimization toolbox routine lsqnonlin. Numerical results are presented and discussed in order to illustrate the performance of the inversion for orthotropic parameter identification.     

Complexity Penalized M-Estimation

We present very fast algorithms for the exact computation of estimators for time series, based on complexity penalized log-likelihood or M-functions. The algorithms apply to a wide range of functionals with morphological constraints, in particular to Potts or Blake–Zisserman functionals. The latter are the discrete versions of the celebrated Mumford–Shah functionals. All such functionals contain model parameters. Our algorithms allow for optimization not only for each separate parameter, but even for all parameters simultaneously. This allows for the examination of the models in the sense of a family approach. The algorithms are accompanied by a series of illustrative examples from molecular biology.     

High-Order Approximation to Caputo Derivatives and Caputo-type Advection–Diffusion Equations: Revisited

In this paper, a series of new high-order numerical approximations to α-th Caputo derivatives (0<α<2) is derived based on a compound of shift operators and high-order approximations to Riemann–Liouville derivatives. The convergence order is independent of the derivative order α, rather than the previous error estimates. Several numerical examples including the Caputo-type advection–diffusion equation are displayed, which support the derived numerical schemes.     

On a Regularized Family of Models for Homogeneous Incompressible Two-Phase Flows

We consider a general family of regularized models for incompressible two-phase flows based on the Allen–Cahn formulation in \(n\) -dimensional compact Riemannian manifolds for \(n=2,3\) . The system we consider consists of a regularized family of Navier–Stokes equations (including the Navier–Stokes- \(\alpha \) -like model, the Leray- \(\alpha \) model, the modified Leray- \(\alpha \) model, the simplified Bardina model, the Navier–Stokes–Voight model, and the Navier–Stokes model) for the fluid velocity \(u\) suitably coupled with a convective Allen–Cahn equation for the order (phase) parameter \(\phi \) . We give a unified analysis of the entire three-parameter family of two-phase models using only abstract mapping properties of the principal dissipation and smoothing operators and then use assumptions about the specific form of the parameterizations, leading to specific models, only when necessary to obtain the sharpest results. We establish existence, stability, and regularity results and some results for singular perturbations, which as special cases include the inviscid limit of viscous models and the \(\alpha \rightarrow 0\) limit in \(\alpha \) models. Then we show the existence of a global attractor and exponential attractor for our general model and establish precise conditions under which each trajectory \(\left( u,\phi \right) \) converges to a single equilibrium by means of a Lojasiewicz–Simon inequality. We also derive new results on the existence of global and exponential attractors for the regularized family of Navier–Stokes equations and magnetohydrodynamics models that improve and complement the results of Holst et al. (J Nonlinear Sci 20(5):523–567, 2010). Finally, our analysis is applied to certain regularized Ericksen–Leslie models for the hydrodynamics of liquid crystals in \(n\) -dimensional compact Riemannian manifolds.