Generalized Tractability for Multivariate Problems Part II: Linear Tensor Product Problems, Linear Information, and Unrestricted Tractability

([formula omitted])-weak tractability of linear problems

EC-(t1,t2)-tractability of approximation in weighted Korobov spaces in the worst case setting

Quasi-polynomial tractability

This paper is concerned with the mixed linear complementarity problem and the role it and its variants play in the stability analysis of the nonlinear complementarity problem and the Karush-Kuhn-Tucker system of a variational inequality problem. Under a nonsingular assumption, the mixed linear complementarity problem can be converted to the standard problem; in this case, the rich theory of the latter can be directly applied to the former. In this work, we employ degree theory to derive some sufficient conditions for the existence of a solution to the mixed linear complementarity problem in the absence of the nonsingularity property. Next, we extend this existence theory to the mixed nonlinear complementarity problem and establish a main stability result under a certain degree-theoretic assumption concerning the linearized problem. We then specialize this stability result and its consequences to the parametric variational inequality problem under the assumption of a unique set of multipliers. Finally, we consider the latter problem with the uniqueness assumption of the multipliers replaced by a convexity assumption and obtain stability results under some weak second-order conditions. In addition to the new existence results for the mixed linear complementarity problem, the main contributions of this paper in the stability category are the following: a resolution to a conjecture concerning the local solvability of a parametric variational inequality, the use of the generalized linear complementarity problem as a tool to broaden the second-order conditions, the characterization of the solution stability of the linear complementarity problem and the affine variational inequality problem in terms of the solution isolatedness under some weak hypotheses, and various stability theorems under some weak second-order conditions.

We study a relatively new notion of tractability called “uniform weak tractability” that was recently introduced in (Siedlecki, J. Complex. 29:438–453, 2013 [5]). This notion holds for a multivariable problem iff the information complexity \(n(\varepsilon , d)\) of its d-variate component to be solved to within \(\varepsilon \) is not an exponential function of any positive power of \(\varepsilon ^{-1}\) and/or d. We are interested in necessary and sufficient conditions on uniform weak tractability for weighted integration. Weights are used to control the “role” or “importance” of successive variables and groups of variables. We consider here product weights. We present necessary and sufficient conditions on product weights for uniform weak tractability for two Sobolev spaces of functions defined over the whole Euclidean space with arbitrary smoothness, and of functions defined over the unit cube with smoothness 1. We also briefly consider (s, t)-weak tractability introduced in (Siedlecki and Weimar, J. Approx. Theory 200:227–258, 2015 [6]), and show that as long as \(t>1\) then this notion holds for weighted integration defined over quite general tensor product Hilbert spaces with arbitrary bounded product weights.

Notes on [formula omitted]-weak tractability: A refined classification of problems with (sub)exponential information complexity

Singular value decomposition is the key tool in the analysis and understanding of linear regularization methods in Hilbert spaces. Besides simplifying computations it allows to provide a good understanding of properties of the forward problem compared to the prior information introduced by the regularization methods. In the last decade nonlinear variational approaches such as ` or total variation regularizations became quite prominent regularization techniques with certain properties being superior to standard methods. In the analysis of those, singular values and vectors did not play any role so far, for the obvious reason that these problems are nonlinear, together with the issue of defining singular values and singular vectors in the first place. In this paper however we want to start a study of singular values and vectors for nonlinear variational regularization of linear inverse problems, with particular focus on singular onehomogeneous regularization functionals. A major role is played by the smallest singular value, which we define as the ground state of an appropriate functional combining the (semi)norm introduced by the forward operator and the regularization functional. The optimality condition for the ground state further yields a natural generalization to higher singular values and vectors involving the subdifferential of the regularization functional, although we shall see that the Rayleigh principle may fail for higher singular values. Using those definitions of singular values and vectors, we shall carry over two main properties from the world of linear regularization. The first one is gaining information about scale, respectively the behavior of regularization techniques at different scales. This also leads to novel estimates at different scales, generalizing the estimates for the coefficients in the linear singular value expansion. The second one is to provide classes of exact solutions for variational regularization methods. We will show that all singular vectors can be reconstructed up to a scalar factor by the standard Tikhonov-type regularization approach even in the presence of (small) noise. Moreover, we will show that they can even be reconstructed without any bias by the recently popularized inverse scale space method.

This thesis is concerned with the analysis and synthesis of positive systems under ℓ1 and L1 performance. Two classes of systems are considered: positive linear systems and positive Takagi-Sugeno (T-S) fuzzy systems. For positive linear systems, the controller, state-bounding observer and filter design problems are considered. Due to the special structures and unique features of positive systems, some previous approach used for general systems, such as similarity transformation, are no longer applicable to positive systems. First, the stabilization problem for positive linear systems is studied. In detail, analytical formulae to compute the exact values of ℓ1-induced and L1-induced norms are presented. Moreover, it is shown how the necessary and sufficient conditions can be constructed such that the closed-loop system is stable and satisfies a prescribed L1-induced performance. For single-input multiple-output (SIMO) positive systems, analytical solutions are established to show how the optimal ℓ1-induced and L1-induced controllers are designed. In addition, the L1-induced sparse state-feedback controller is investigated for continuous-time interval positive systems. Then, to estimate the state of positive systems at all times, the problem of positive state-bounding observers for interval positive systems is studied under the L1-induced performance. Necessary and sufficient conditions are presented to design a pair of state-bounding positive observers. Finally, the positive filtering problem is addressed for positive systems under the L1-induced performance. A pair of positive filters with error-bounding feature is designed to estimate the output of positive systems and the obtained results are expressed in terms of linear programming problems.
\nFor positive T-S fuzzy systems, the controller and filter design problems are investigated under the ℓ1-induced performance. First, novel performance characterization of positive fuzzy systems is established. Sufficient conditions are further established for the existence of state-feedback controller. An iterative convex optimization algorithm is developed to solve the design conditions. Furthermore, a pair of error-bounding positive filters are constructed to estimate the output of positive T-S fuzzy systems. A new performance characterization is first established to guarantee the asymptotic stability of the filtering error system with the ℓ1-induced performance. Then, sufficient conditions expressed by linear programming problems are derived to design the required filters.

Termination Criteria for Linear Problems

We have obtained the conditions of existence and a scheme for constructing solutions of a weakly nonlinear boundary value problem for a degenerate differential-algebraic system in the noncritical case. The boundary condition is determined by a weakly nonlinear vector functional. The linear part of the problem is a linear boundary value problem for a degenerate differential-algebraic system. Linear differential-algebraic boundary value problems have been studied in monographs by S. Campbell, J.R. Magnus, A.M. Samoilenko and V.P. Yakovets. In the works of A.M. Samoilenko and O.A. Boichuk, using the central canonical form, the necessary and sufficient conditions for the existence of solutions of nonlinear differential-algebraic boundary value problems were obtained. We have obtained necessary and sufficient conditions for the existence of solutions of nonlinear differential-algebraic systems without using the central canonical form, which allows us to study the solvability of differential-algebraic boundary value problems that depend on arbitrary continuous functions. This approach significantly varies the classification of nonlinear differential-algebraic boundary value problems in critical and noncritical cases. Our formulation of the weakly nonlinear differential-algebraic boundary value problem generalises the boundary value problems studied in the works of Yu.O. Mitropolsky, A.M. Samoilenko, and O.A. Boichuk. The case when a differential-algebraic system is not solvable with respect to the derivative is considered, and substitutions of the unknown are proposed. It leads the original system to a nonlinear differential-algebraic system solvable with respect to the derivative. Finally, we present an example of a nonlinear differential-algebraic antiperiodic boundary value problem for a Riccati-type equation, which demonstrates the constructiveness of the obtained necessary and sufficient conditions for the existence of solutions of nonlinear differential-algebraic systems. The obtained results can be transferred to the problems of finding conditions for the existence and schemes for constructing solutions of nonlinear degenerate differential-algebraic boundary value problems in critical cases, as well as to the problems of finding conditions for the stability of such solutions.

The dependence of the radius of a ball centered at zero inscribed in the values of the integral of a set-valued mapping on the upper integration limit is studied. For some types of integrals, exact asymptotics of the radius with respect to the upper limit are found when the upper limit tends to zero. Examples of finding this radius are considered. The results obtained are used to derive new sufficient conditions for the uniformly continuous dependence of the minimum time and solution-point in the linear minimum time control problem on the initial data. We also consider applications in some algorithms with a reachability set of a linear control system

We first determine the kernels of those unitary representations of a locally compact group which are obtained by integrating, inducing and tensoring. We then describe a broad class of group extensions for which the previous results can be used to find (1) the kernel of any representation and (2) necessary and sufficient conditions for these extensions to be maximally almost periodic. Introduction. The primary purpose of this paper is to determine the kernel of any (unitary) representation of a reasonably general locally compact group extension. (See ?4 for a specific description of the exten- sion.) Roughly speaking, the representations of such an extension are obtained by successively tensoring, inducing and integrating (at least in the separable case) representations of certain subgroups. In ?1 we deter- mine the kernel of a direct integral. (This is the only place where we find it necessary to introduce separability assumptions.) This determination has the effect of reducing our problem to the consideration of irreducible representations only. In ?2 we describe the kernel of an induced repre- sentation, thus extending Lemma 2.1 of (11) to the nonseparable case. Regarding the tensor product of representations, it is well known (see (9) and (6, ?17)) that the most useful kernel to determine is that of the tensor product of projective representations or, more accurately, of representa- tions of corresponding central group extensions of the circle. We do this in ?3. Finally, in ?4 we describe the type of extension for which the preceding results solve the problem stated at the beginning. This section is essentially a combination and augmentation of R. J. Blattner's work in (3) and the relevant results of J. M. G. Fell in ?17 of (6). We conclude by indicating how the previous results can be used to find necessary and sufficient conditions for the extension to be maximally almost periodic. Throughout this paper, G will be a locally compact group and all representations will be unitary. If h is a group homomorphism then

SUMMARY The LQ+ problem, i.e. the finite-horizon linear quadratic optimal control problem with nonnegative state constraints, is studied for positive linear systems in continuous time and in discrete time. Necessary and sufficient optimality conditions are obtained by using the maximum principle. These conditions lead to a computational method for the solution of the LQ+ problem by means of a corresponding Hamiltonian system. In addition, the necessary and sufficient conditions are proved for the LQ+-optimal control to be given by the standard LQ-optimal state feedback law. Then sufficient conditions are established for the positivity of the LQ-optimal closed-loop system. In particular, such conditions are obtained for the problem of minimal energy control with penalization of the final state. Moreover, a positivity criterion for the LQ-optimal closed-loop system is derived for positive discrete-time systems with a positively invertible (dynamics) generator. The main results are illustrated by numerical examples. Copyright q 2010 John Wiley & Sons, Ltd.

A very natural question raised about products of graphs is the following. Do there exist necessary and sufficient conditions on a pair (G,H) of graphs for their product to have some specified property? In particular, for which graphs is the product one-factorizable? Sufficient conditions have been investigated for the cartesian, lexicographic, and tensor products in [3], [4], and [5]. However, the conditions for the tensor product to be one-factorizable are significantly more scanty than those for the other products. The purpose of this paper is to correct this situation somewhat.

This chapter extends the concept of polynomially paranormal operators on Hilbert spaces and their tensor products to the broader class of polynomially *-paranormal and uncertainty-based operators.We introduce new definitions and establish sufficient conditions under which the tensor product of such operators preserves paranormality and spectral stability.Moreover, the single-valued extension property (SVEP) and spectral radius inequalities are investigated for these new operator classes.The results unify operator theoretic structures with uncertainty theory and provide a framework for future extensions to Banach spaces and C *algebras.