Function Space Setting Research Articles

A duality based semismooth Newton framework for solving variational inequalities of the second kind

In an appropriate function space setting, semismooth Newton methods are proposed for iteratively computing the solution of a rather general class of variational inequalities (VIs) of the second kind. The Newton scheme is based on the Fenchel dual of the original VI problem which is regularized if necessary. In the latter case, consistency of the regularization with respect to the original problem is studied. The application of the general framework to specific model problems including Bingham flows, simplified friction, or total variation regularization in mathematical imaging is described in detail. Finally, numerical experiments are presented in order to verify the theoretical results.

Banach operator pairs and common fixed points in modular function spaces

Abstract In this article, we introduce the concept of a Banach operator pair in the setting of modular function spaces. We prove some common fixed point results for such type of operators satisfying a more general condition of nonexpansiveness. We also establish a version of the well-known De Marr's theorem for an arbitrary family of symmetric Banach operator pairs in modular function spaces without Δ2-condition. MSC(2000) primary 06F30; 47H09; secondary 46B20; 47E10; 47H10.

On a PDE involving the [formula omitted]-Laplace operator

This paper establishes the existence of solutions for a partial differential equation in which a differential operator involving variable exponent growth conditions is present. This operator represents a generalization of the p ( ⋅ ) -Laplace operator, i.e. Δ p ( ⋅ ) u = div ( ∣ ∇ u ∣ p ( ⋅ ) − 2 ∇ u ) , where p ( ⋅ ) is a continuous function. The proof of the main result is based on Schauder’s fixed point theorem combined with adequate variational arguments. The function space setting used here makes appeal to the variable exponent Lebesgue and Sobolev spaces.

Fragmentability of groups and metric-valued function spaces

Let ( X , τ ) be a topological space and let ρ be a metric defined on X. We shall say that ( X , τ ) is fragmented by ρ if whenever ε > 0 and A is a nonempty subset of X there is a τ-open set U such that U ∩ A ≠ ∅ and ρ − diam ( U ∩ A ) < ε . In this paper we consider the notion of fragmentability, and its generalisation σ-fragmentability, in the setting of topological groups and metric-valued function spaces. We show that in the presence of Baireness fragmentability of a topological group is very close to metrizability of that group. We also show that for a compact Hausdorff space X, σ-fragmentability of ( C ( X ) , ‖ ⋅ ‖ ∞ ) implies that the space C p ( X ; M ) of all continuous functions from X into a metric space M, endowed with the topology of pointwise convergence on X, is fragmented by a metric whose topology is at least as strong as the uniform topology on C ( X ; M ) . The primary tool used is that of topological games.

Equations involving a variable exponent Grushin-type operator

In this paper we define a Grushin-type operator with a variable exponent growth and establish existence results for an equation involving such an operator. A suitable function space setting is introduced. Regarding the tools used in proving the existence of solutions for the equation analysed here, they rely on the critical point theory combined with adequate variational techniques.

QUASI-MONTE CARLO METHODS FOR HIGH-DIMENSIONAL INTEGRATION: THE STANDARD (WEIGHTED HILBERT SPACE) SETTING AND BEYOND

AbstractThis paper is a contemporary review of quasi-Monte Carlo (QMC) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0,1]s. It first introduces the by-now standard setting of weighted Hilbert spaces of functions with square-integrable mixed first derivatives, and then indicates alternative settings, such as non-Hilbert spaces, that can sometimes be more suitable. Original contributions include the extension of the fast component-by-component (CBC) construction of lattice rules that achieve the optimal convergence order (a rate of almost 1/N, whereNis the number of points, independently of dimension) to so-called “product and order dependent” (POD) weights, as seen in some recent applications. Although the paper has a strong focus on lattice rules, the function space settings are applicable to all QMC methods. Furthermore, the error analysis and construction of lattice rules can be adapted to polynomial lattice rules from the family of digital nets.

Function spaces and optimal currents in impedance tomography

Abstract The main objective of this paper is to compare – analytically as well as numerically – different approaches for obtaining optimal input currents in impedance tomography. Following the approaches described in, e.g. [Cheney and Isaacson, IEEE Trans. Biomed. Eng. 39: 852–860, 1992, Isaacson, IEEE Trans. Med. Imag. 5: 91–95, 1986, Ito and Kunisch, SIAM J. Contr. Optim. 33: 643–666., 1995, Knowles, An optimal current functional for electrical impedance tomography, 2004], we aim at constructing input currents j, which contain the most information about the difference between the unknown physical conductivity σ* and a given approximation σ 0. The differences can be measured by different discrepancy functionals and the optimal input currents which maximize these functionals depend on the function spaces chosen for defining j and on the norm for measuring the discrepancy. Moreover, the definition of the appropriately weighted Sobolev spaces depends on σ and this subsequently influences the iteration for maximizing the functionals. Numerical experiments illustrate features of the optimal input currents obtained for several combinations of function spaces. The reconstructions with these optimal currents are compared with those with standard input currents (sinusoid and dipole). The differences between the optimal currents obtained by different function space settings are significant. Two newly developed optimal currents can yield qualitatively better reconstructions.

Improved Sobolev inequalities and Muckenhoupt weights on stratified Lie groups

We study in this article the improved Sobolev inequalities with Muckenhoupt weights within the framework of stratified Lie groups. This family of inequalities estimate the Lq norm of a function by the geometric mean of two norms corresponding to Sobolev spaces W˙s,p and Besov spaces B˙∞−β,∞. When the value p which characterizes Sobolev space is strictly larger than 1, the required result is well known in Rn and is classically obtained by a Littlewood–Paley dyadic blocks manipulation. For these inequalities we will develop here another totally different technique. When p=1, these two techniques are not available anymore and following M. Ledoux (2003) [12], in Rn, we will treat here the critical case p=1 for general stratified Lie groups in a weighted functional space setting. Finally, we will go a step further with a new generalization of improved Sobolev inequalities using weak-type Sobolev spaces.

Optimal control of transport-reaction system with time varying spatial domain

This paper considers the multiscale optimal control of crystal growth. The optimal control is realized for the crystal pulling arm modeled by the standard rigid body dynamics, while the underlying dynamics of the diffusion of heat in the crystal growth region is given by parabolic partial differential equations (PDEs) with time varying spatial domain. The underlying transport-reaction system is developed from the first principles and the associated dynamics is analyzed in appropriate functional state space setting. The complete description of the evolutionary parabolic domain time varying PDE is provided and explored within the coupled master-slave control setting. Numerical simulations demonstrate an optimal pulling evolution rate and its effects on the temperature profile in the crystal with the time varying domain.

Existence and concentration of solitary waves for a class of quasilinear Schrödinger equations

In this paper we prove the existence and qualitative properties of positive bound state solutions for a class of quasilinear Schrödinger equations in dimension $N\ge 3$: we investigate the case of unboundedpotentials which can not be covered in a standard function space setting. This leads us to usea nonstandard penalization technique, in a borderline Orlicz space framework, which enable us to build up a one parameter family of classical solutions which have finite energy and exhibit, as the parameter goes to zero, a concentratingbehavior around some point which is localized.

Randomly shifted lattice rules with the optimal rate of convergence for unbounded integrands

We study the multivariate integration problem ∫ R d f ( x ) ρ ( x ) d x , with ρ being a product of univariate probability density functions. We assume that f belongs to a weighted tensor-product reproducing kernel Hilbert space of functions whose mixed first derivatives, when multiplied by a weight function ψ , have bounded L 2 -norms. After mapping into the unit cube [ 0 , 1 ] d , the transformed integrands are typically unbounded or have huge derivatives near the boundary, and thus fail to lie in the usual function space setting where many good results have been established. In our previous work, we have shown that randomly shifted lattice rules can be constructed component-by-component to achieve a worst case error bound of order O ( n − 1 / 2 ) in this new function space setting. Using a more clever proof technique together with more restrictive assumptions, in this article we improve the results by proving that a rate of convergence close to the optimal order O ( n − 1 ) can be achieved with an appropriate choice of parameters for the function space. The implied constants in the big- O bounds can be independent of d under appropriate conditions on the weights of the function space.

Boundary model predictive control of Kuramoto–Sivashinsky equation with input and state constraints

In this work an asymptotic stabilization of highly dissipative Kuramoto–Sivashinsky equation (KSE) by means of boundary model modal predictive control (MMPC) in the presence of input and state constraints is demonstrated. The KS equation is initially defined in an appropriate functional space setting and an exact transformation is used to reformulate the original boundary control problem as an abstract boundary control problem of the KSE partial differential equation (PDE). An appropriate discrete infinite-dimensional representation of the abstract boundary control problem is used for synthesis of low dimensional model modal predictive controller (MMPC) incorporating both the pointwise enforced KSE state constraints and input constraints. The proposed control problem formulation and the performance of the closed-loop system in the full state feedback controller realization have been evaluated through simulations.

Integral Geometry on Discrete Grassmannians in Z n

We study the Radon transform R on the discrete Grassmannian of rank-d affine sublattices of Zn for 0 < d < n, extending and building on previous work of the first- and third-named authors in codimension 1. By analogy with the integral geometry on Grassmannians in Rn, various natural questions are treated, such as definition and properties of R and its dual transform R*, function space setting, support theorems and inversion formulas.

Fixed Points of Multivalued Maps in Modular Function Spaces

AbstractThe purpose of this paper is to study the existence of fixed points for contractive-type and nonexpansive-type multivalued maps in the setting of modular function spaces. We also discuss the concept of "Equation missing"-modular function and prove fixed point results for weakly-modular contractive maps in modular function spaces. These results extend several similar results proved in metric and Banach spaces settings.

Strategies for time-dependent PDE control with inequality constraints using an integrated modeling and simulation environment

In Neitzel et al. (Strategies for time-dependent PDE control using an integrated modeling and simulation environment. Part one: problems without inequality constraints. Technical Report 408, Matheon, Berlin, 2007) we have shown how time-dependent optimal control for partial differential equations can be realized in a modern high-level modeling and simulation package. In this article we extend our approach to (state) constrained problems. “Pure” state constraints in a function space setting lead to non-regular Lagrange multipliers (if they exist), i.e. the Lagrange multipliers are in general Borel measures. This will be overcome by different regularization techniques. To implement inequality constraints, active set methods and barrier methods are widely in use. We show how these techniques can be realized in a modeling and simulation package. We implement a projection method based on active sets as well as a barrier method and a Moreau Yosida regularization, and compare these methods by a program that optimizes the discrete version of the given problem.

Radon–Nikodým derivatives for vector measures belonging to Köthe function spaces

Let m, n be a couple of vector measures with values on a Banach space. We develop a separation argument which provides a characterization of when the Radon–Nikodým derivative of n with respect to m—in the sense of the Bartle–Dunford–Schwartz integral—exists and belongs to a particular sublattice Z ( μ ) of the space of integrable functions L 1 ( m ) . We show that this theorem is in fact a particular feature of our separation argument, which can be applied to prove other results in both the vector measure and the function space settings.

Boundary Integral Equations on Uubounded Rough Surfaces: Fredholmness and the Finite Section Method

We consider a class of boundary integral equations that arise in the study of strongly elliptic BVPs in unbounded domains of the form $D = \{(x, z)\in \mathbb{R}^{n+1} : x\in \mathbb{R}^n, z > f(x)\}$ where $f : \mathbb{R}^n \to\mathbb{R}$ is a sufficiently smooth bounded and continuous function. A number of specific problems of this type, for example acoustic scattering problems, problems involving elastic waves, and problems in potential theory, have been reformulated as second kind integral equations $u+Ku = v$ in the space $BC$ of bounded, continuous functions. Having recourse to the so-called limit operator method, we address two questions for the operator $A = I + K$ under consideration, with an emphasis on the function space setting $BC$. Firstly, under which conditions is $A$ a Fredholm operator, and, secondly, when is the finite section method applicable to $A$?

A Framework for the Adaptive Finite Element Solution of Large-Scale Inverse Problems

Since problems involving the estimation of distributed coefficients in partial differential equations are numerically very challenging, efficient methods are indispensable. In this paper, we will introduce a framework for the efficient solution of such problems. This comprises the use of adaptive finite element schemes, solvers for the large linear systems arising from discretization, and methods to treat additional information in the form of inequality constraints on the parameter to be recovered. The methods to be developed will be based on an all-at-once approach, in which the inverse problem is solved through a Lagrangian formulation. The main feature of the paper is the use of a continuous (function space) setting to formulate algorithms, in order to allow for discretizations that are adaptively refined as nonlinear iterations proceed. This entails that steps such as the description of a Newton step or a line search are first formulated on continuous functions and only then evaluated for discrete functions. On the other hand, this approach avoids the dependence of finite dimensional norms on the mesh size, making individual steps of the algorithm comparable even if they used differently refined meshes. Numerical examples will demonstrate the applicability and efficiency of the method for problems with several million unknowns and more than 10,000 parameters.

An interior point method for a parabolic optimal control problem with regularized pointwise state constraints

AbstractA primal‐dual interior point method for state‐constrained parabolic optimal control problems is considered. By a Lavrentiev type regularization, the state constraints are transformed to mixed control‐state constraints which, after a simple transformation, can be handled as control constraints. Existence and convergence of the central path are shown. Moreover, the convergence of a short step interior point algorithm is proven in a function space setting. The theoretical properties of the algorithm are confirmed by numerical examples.

Preconditioning and convergence in the right norm

The convergence of numerical approximations to the solutions of differential equations is a key aspect of numerical analysis and scientific computing. Iterative solution methods for the systems of linear(ized) equations which often result are also underpinned by analyses of convergence. In the function space setting, it is widely appreciated that there are appropriate ways in which to assess convergence and it is well-known that different norms are not equivalent. In the finite dimensional linear algebra setting, however, all norms are equivalent and little attention is often paid to the norms used. In this paper, we highlight this consideration in the context of preconditioning for minimum residual methods (MINRES and GMRES/GCR/ORTHOMIN) and argue that even in the linear algebra setting there is a ‘right’ norm in which to consider convergence: stopping an iteration which is rapidly converging in an irrelevant or highly scaled norm at some tolerance level may still give a poor answer.