Numerical Functional Analysis and Optimization

Published by Taylor & Francis
Online ISSN: 1532-2467
Print ISSN: 0163-0563
The spectral analysis of discretized one-dimensional Schr\"{o}dinger operators is a very difficult problem which has been studied by numerous mathematicians. A natural problem at the interface of numerical analysis and operator theory is that of finding finite dimensional matrices whose eigenvalues approximate the spectrum of an infinite dimensional operator. In this note we observe that the seminal work of Pimsner-Voiculescu on AF embeddings of irrational rotation algebras provides a nice answer to the finite dimensional spectral approximation problem for a broad class of operators including the quasiperiodic case of the Schr\"{o}dinger operators mentioned above. Indeed, the theory of continued fractions not only provides good matrix models for spectral computations (i.e. the Pimsner-Voiculescu construction) but also yields {\em sharp} rates of convergence for spectral approximations of operators in irrational rotation algebras.
We propose a discontinuous finite element approximation for a model of quasi-static growth of brittle fractures in linearly elastic bodies formulated by Francfort and Marigo, and based on the classical Griffith's criterion. We restrict our analysis to the case of anti-planar shear and we consider discontinuous displacements which are piecewise affine with respect to a regular triangulation.
Let $X$ be a reflexive Banach space. In this paper we give a necessary and sufficient condition for an operator $T\in \mathcal{K}(X)$ to have the best approximation in numerical radius from the convex subset $\mathcal{U} \subset \mathcal{K}(X),$ where $\mathcal{K}(X)$ denotes the set of all linear, compact operators from $X$ into $X.$ We will also present an application to minimal extensions with respect to the numerical radius. In particular some results on best approximation in norm will be generalized to the case of the numerical radius. Comment: 13 pages
Kuhn-Tucker points play a fundamental role in the analysis and the numerical solution of monotone inclusion problems, providing in particular both primal and dual solutions. We propose a class of strongly convergent algorithms for constructing the best approximation to a reference point from the set of Kuhn-Tucker points of a general Hilbertian composite monotone inclusion problem. Applications to systems of coupled monotone inclusions are presented. Our framework does not impose additional assumptions on the operators present in the formulation, and it does not require knowledge of the norm of the linear operators involved in the compositions or the inversion of linear operators.
Example in dimension 2. Let a 1 , a 2 , a 3 be a dictionary on 2 . On the drawing, the sets P ⊥ i (B ·· (()), for i = 1, 2, 3, are shifted by an element of i . The dotted sets represent  
Example of an intersection in dimension 3. 1,2 is in between two planes, parallel to 1,2 . Same remark for 3,4 . The set 1,2 ∩ 3,4 is of the form W + P W ⊥ B ˜ g ((), where˜gwhere˜ where˜g is a norm for W = 1,2 ∩ 3,4 . We also have dim( 1,2 ∩ 3,4 ) < dim( 1,2 ) = dim( 3,4 ).  
Upper and lower bounds of (val( d ) ≤ K ) in the context of Euclidean norms.  
We consider the minimization of the number of non-zero coefficients (the $\ell_0$ "norm") of the representation of a data set in terms of a dictionary under a fidelity constraint. (Both the dictionary and the norm defining the constraint are arbitrary.) This (nonconvex) optimization problem naturally leads to the sparsest representations, compared with other functionals instead of the $\ell_0$ "norm". Our goal is to measure the sets of data yielding a $K$-sparse solution--i.e. involving $K$ non-zero components. Data are assumed uniformly distributed on a domain defined by any norm--to be chosen by the user. A precise description of these sets of data is given and relevant bounds on the Lebesgue measure of these sets are derived. They naturally lead to bound the probability of getting a $K$-sparse solution. We also express the expectation of the number of non-zero components. We further specify these results in the case of the Euclidean norm, the dictionary being arbitrary.
Well-posed models and computational algorithms are developed and analyzed for control of a class of partial differential equations that describe the motions of thermo-viscoelastic structures. An abstract (state space) framework and a general well-posedness result are presented that can be applied to a large class of thermo-elastic and thermo-viscoelastic models. This state space framework is used in the development of a computational scheme to be used in the solution of a linear quadratic regulator (LQR) control problem. A detailed convergence proof is provided for the viscoelastic model and several numerical results are presented to illustrate the theory and to analyze problems for which the theory is incomplete.
Formulae for the value of a harmonic function at the center of a rectangle are found that involve boundary integrals. The central value of a harmonic function is shown to be well approximated by the mean value of the function on the boundary plus a very small number (often just 1 or 2) of additional boundary integrals. The formulae are consequences of Steklov (spectral) representations of the functions that converge exponentially at the center. Similar approximation are found for the central values of solutions of Robin and Neumann boundary value problems. The results are based on explicit expressions for the Steklov eigenvalues and eigenfunctions.
For some fractal measures it is a very difficult problem in general to prove the existence of spectrum (respectively, frame, Riesz and Bessel spectrum). In fact there are examples of extremely sparse sets that are not even Bessel spectra. In this paper we investigate this problem for general fractal measures induced by iterated function systems (IFS). We prove some existence results of spectra associated with Hadamard pairs. We also obtain some characterizations of Bessel spectrum in terms of finite matrices for affine IFS measures, and one sufficient condition of frame spectrum in the case that the affine IFS has no overlap.
Recently, Martin Hutzenthaler pointed out that the explicit Euler method fails to converge strongly to the exact solution of a stochastic differential equation (SDE) with superlinearly growing and globally one sided Lipschitz drift coefficient. Afterwards, he proposed an explicit and easily implementable Euler method, i.e tamed Euler method, for such an SDE and showed that this method converges strongly with order of one half. In this paper, we use the tamed Euler method to solve the stochastic differential equations with piecewise continuous arguments (SEPCAs) with superlinearly growing coefficients and prove that this method is convergent with strong order one half.
We consider the diffusion equation in the setting of operator theory. In particular, we study the characterization of the limit of the diffusion operator for diffusivities approaching zero on a subdomain $\Omega_1$ of the domain of integration of $\Omega$. We generalize Lions' results to covering the case of diffusivities which are piecewise $C^1$ up to the boundary of $\Omega_1$ and $\Omega_2$, where $\Omega_2 := \Omega \setminus \overline{\Omega}_1$ instead of piecewise constant coefficients. In addition, we extend both Lions' and our previous results by providing the strong convergence of $(A_{\bar{p}_\nu}^{-1})_{\nu \in \mathbb{N}^\ast},$ for a monotonically decreasing sequence of diffusivities $(\bar{p}_\nu )_{\nu \in \mathbb{N}^\ast}$.
In this paper we apply methods of proof mining to obtain a uniform effective rate of asymptotic regularity for the Mann iteration associated to $\kappa$-strict pseudo-contractions on convex subsets of Hilbert spaces.
Function spaces are central topic in analysis. Often those spaces and related analysis involves symmetries in form of an action of a Lie group. Coorbit theory as introduced by Feichtinger and Gr\"ochenig and then later extended in [3] gives a unified method to construct Banach spaces of functions based on representations of Lie groups. In this article we identify the homogeneous Besov spaces on stratified Lie groups introduced in [13] as coorbit spaces in the sense of [3] and use this to derive atomic decompositions for the Besov spaces.
Kuhn-Tucker conditions for mathematical programming problems in Banach spaces partially ordered by cone with empty interior are obtained under strong simultaneity condition. If partial ordered cone has interior point, it is proved that Slater and strong simultaneity conditions are equivalent.
In this paper, we extend the Moreau (Riesz) decomposition theorem, from Hilbert spaces to Banach spaces. Criteria for a closed subspace to be (strongly) orthogonally complemented in a Banach space are given. We prove that every closed subspace of a Banach space X with dimX⩾3(dimX⩽2) is strongly orthogonally complemented if and only if the Banach space X is isometric to a Hilbert space (resp. strictly convex Banach space), which is complementary to the well-known result saying that every closed subspace of a Banach space X is topologically complemented if and only if the Banach space X is isomorphic to a Hilbert space.
A recurrent theme in functional analysis is the interplay between the theory of positive definite functions, and their reproducing kernels, on the one hand, and Gaussian stochastic processes, on the other. This central theme is motivated by a host of applications, e.g., in mathematical physics, and in stochastic differential equations, and their use in financial models. In this paper, we show that, for three classes of cases in the correspondence, it is possible to obtain explicit formulas which are amenable to computations of the respective Gaussian stochastic processes. For achieving this, we first develop two functional analytic tools. They are: $(i)$ an identification of a universal sample space $\Omega$ where we may realize the particular Gaussian processes in the correspondence; and (ii) a procedure for discretizing computations in $\Omega$. The three classes of processes we study are as follows: Processes associated with: (a) arbitrarily given sigma finite regular measures on a fixed Borel measure space; (b) with Hilbert spaces of sigma-functions; and (c) with systems of self-similar measures arising in the theory of iterated function systems. Even our results in (a) go beyond what has been obtained previously, in that earlier studies have focused on more narrow classes of measures, typically Borel measures on $\mathbb R^n$. In our last theorem (section 10), starting with a non-degenerate positive definite function $K$ on some fixed set $T$, we show that there is a choice of a universal sample space $\Omega$, which can be realized as a "boundary" of $(T, K)$. Its boundary-theoretic properties are analyzed, and we point out their relevance to the study of electrical networks on countable infinite graphs.
We study the approximation of fixed points of nonexpansive mappings in CAT(k) spaces. We show that the iterative sequence generated by the Moudafi's viscosity type algorithm converges to one of the fixed points of the nonexpansive mapping depending on the contraction applied in the algorithm.
In this paper, error estimates are presented for a certain class of optimal control problems with elliptic PDE-constraints. It is assumed that in the cost functional the state is measured in terms of the energy norm generated by the state equation. The functional a posteriori error estimates developed by Repin in late 90's are applied to estimate the cost function value from both sides without requiring the exact solution of the state equation. Moreover, a lower bound for the minimal cost functional value is derived. A meaningful error quantity coinciding with the gap between the cost functional values of an arbitrary admissible control and the optimal control is introduced. This error quantity can be estimated from both sides using the estimates for the cost functional value. The theoretical results are confirmed by numerical tests.
An equilateral dimension of a normed space is a maximal number of pairwise equidistant points of this space. The aim of this paper is to study the equilateral dimension of certain classes of finite dimensional normed spaces. The well-known conjecture states that the equilateral dimension of any $n$-dimensional normed space is not less than $n+1$. By using an elementary continuity argument, we establish it in the following classes of spaces: permutation-invariant spaces, Orlicz-Musielak spaces and in one codimensional subspaces of $\ell^n_{\infty}$. For smooth and symmetric spaces, Orlicz-Musielak spaces satisfying an additional condition and every $(n-1)$-dimensional subspace of $\ell^{n}_{\infty}$ we also provide some weaker bounds on the equilateral dimension for every space which is sufficiently close to one of these. This generalizes the result of Swanepoel and Villa concerning the $\ell_p^n$ spaces.
Error in estimating B H p1q with H " 0.7 as a function of the number of coefficients in the sum (6.4). The dashed curve is the exact error obtained by (6.7). 
In this work we combine Wiener chaos expansion approach to study the dynamics of a stochastic system with the classical problem of the prediction of a Gaussian process based on part of its sample path. This is done by considering special bases for the Gaussian space $\mathcal G$ generated by the process, which allows us to obtain an orthogonal basis for the Fock space of $\mathcal G$ such that each basis element is either measurable or independent with respect to the given samples. This allows us to easily derive the chaos expansion of a random variable conditioned on part of the sample path. We provide a general method for the construction of such basis when the underlying process is Gaussian with stationary increment. We evaluate the bases elements in the case of the fractional Brownian motion, which leads to a prediction formula for this process.
We introduce new types of systems of generalized quasi-variational inequalities and we prove the existence of the solutions by using results of pair equilibrium existence for free abstract economies. We consider the fuzzy models and we also introduce the random free abstract economy and the random equilibrium pair. The existence of the solutions for the systems of quasi-variational inequalities comes as consequences of the existence of equilibrium pairs for the considered free abstract economies.
We study the initial-boundary value problem for a nonlinear wave equation given by u_{tt}-u_{xx}+\int_{0}^{t}k(t-s)u_{xx}(s)ds+ u_{t}^{q-2}u_{t}=f(x,t,u) , 0 < x < 1, 0 < t < T, u_{x}(0,t)=u(0,t), u_{x}(1,t)+\eta u(1,t)=g(t), u(x,0)=\^u_{0}(x), u_{t}(x,0)={\^u}_{1}(x), where \eta \geq 0, q\geq 2 are given constants {\^u}_{0}, {\^u}_{1}, g, k, f are given functions. In part I under a certain local Lipschitzian condition on f, a global existence and uniqueness theorem is proved. The proof is based on the paper [10] associated to a contraction mapping theorem and standard arguments of density. In Part} 2, under more restrictive conditions it is proved that the solution u(t) and its derivative u_{x}(t) decay exponentially to 0 as t tends to infinity. Comment: 26 pages
The Rayleigh conjecture on the representation of the scattered field in the exterior of an obstacle $D$ is widely used in applications. However this conjecture is false for some obstacles. AGR introduced the Modified Rayleigh Conjecture (MRC), and in this paper we present successful numerical algorithms based on the MRC for various 2D and 3D obstacle scattering problems. The 3D obstacles include a cube and an ellipsoid. The MRC method is easy to implement for both simple and complex geometries. It is shown to be a viable alternative for other obstacle scattering methods.
In this paper, a kind of non regular constraints and a principle for seeking critical point under the constraint are presented, where no Lagrange multiplier is involved. Let $E, F$ be two Banach spaces, $g: E\rightarrow F$ a $c^1$ map defined on an open set $U$ in $E,$ and the constraint $S=$ the preimage $g^{-1}(y_0), y_0\in F.$ A main deference between the non regular constraint and regular constraint is that $g'(x)$ at any $x\in S$ is not surjective. Recently, the critical point theory under the non regular constraint is a concerned focus in optimization theory. The principle also suits the case of regular constraint. Coordinately, the generalized regular constraint is introduced, and the critical point principle on generalized regular constraint is established. Let $f: U \rightarrow \mathbb{R}$ be a nonlinear functional. While the Lagrange multiplier $L$ in classical critical point principle is considered, and its expression is given by using generalized inverse ${g'}^+(x)$ of $g'(x)$ as follows : if $x\in S$ is a critical point of $f|_S,$ then $L=f'(x)\circ {g'}^+(x) \in F^*.$ Moreover, it is proved that if $S$ is a regular constraint, then the Lagrange multiplier $L$ is unique; otherwise, $L$ is ill-posed. Hence, in case of the non regular constraint, it is very difficult to solve Euler equations, however, it is often the case in optimization theory. So the principle here seems to be new and applicable. By the way, the following theorem is proved; if $A\in B(E,F)$ is double split, then the set of all generalized inverses of $A,$ $GI(A)$ is smooth diffeomorphic to certain Banach space. This is a new and interesting result in generalized inverse analysis.
In recent years, a series of convergence rates conditions for regularization methods has been developed. Mainly, the motivations for developing novel conditions came from the desire to carry over convergence rates results from the Hilbert space setting to generalized Tikhonov regularization in Banach spaces. For instance, variational source conditions have been developed, and they were expected to be equivalent to standard source conditions for linear inverse problems in a Hilbert space setting (see Schuster et al. [1313. T. Schuster , B. Kaltenbacher , B. Hofmann , and K. S. Kazimierski ( 2012 ). Regularization Methods in Banach Spaces . Radon Series on Computational and Applied Mathematics, Vol. 10. Walter de Gruyter GmbH & Co. KG, Berlin. View all references]). We show that this expectation does not hold. However, in the standard Hilbert space setting these novel conditions are optimal, which we prove by using some deep results from Neubauer [1111. A. Neubauer ( 1997 ). On converse and saturation results for Tikhonov regularization of linear ill-posed problems . SIAM J. Numer. Anal. 34 : 517 – 527 . View all references], and generalize existing convergence rates results. The key tool in our analysis is a homogeneous source condition, which we put into relation to the other existing source conditions from the literature. As a positive by-product, convergence rates results can be proven without spectral theory, which is the standard technique for proving convergence rates for linear inverse problems in Hilbert spaces (see Groetsch [77. Groetsch , C. W. ( 1984 ). The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind . Pitman , Boston . View all references]).
Rates of convergence of solutions of various two-dimensional $\alpha-$regularization models, subject to periodic boundary conditions, toward solutions of the exact Navier-Stokes equations are given in the $L^\infty$-$L^2$ time-space norm, in terms of the regularization parameter $ \alpha$, when $\alpha$ approaches zero. Furthermore, as a paradigm, error estimates for the Galerkin approximation of the exact two-dimensional Leray-$\alpha$ model are also presented in the $L^\infty$-$L^2$ time-space norm. Simply by the triangle inequality, one can reach the error estimates of the solutions of Galerkin approximation of the $\alpha$-regularization models toward the exact solutions of the Navier-Stokes equations in the two-dimensional periodic boundary conditions case. Comment: 29 pages
This work focus on convergence analysis of the projected gradient method for solving constrained convex minimization problem in Hilbert spaces. We show that the sequence of points generated by the method employing the Armijo linesearch converges weakly to a solution of the considered convex optimization problem. Weak convergence is established by assuming convexity and G\^ateaux differentiability of the objective function, whose G\^ateaux derivative is supposed to be uniformly continuous on bounded sets. Furthermore, we propose some modifications in the classical projected gradient method in order to obtain strong convergence. The new variant has the following desirable properties: the sequence of generated points is entirely contained in a ball with diameter equal to the distance between the initial point and the solution set, and the whole sequence converges strongly to the solution of the problem that lies closest to the initial iterate. Convergence analysis of both methods is presented without Lipschitz continuity assumption.
Analytic vs. numerical approximation for the FDE (6) with one initial condition. 
We obtain series expansion formulas for the Hadamard fractional integral and fractional derivative of a smooth function. When considering finite sums only, an upper bound for the error is given. Numerical simulations show the efficiency of the approximation method.
Numerical result of functions for the ODE (36)
Components of our algorithm
Based on functional analysis, we propose an algorithm for finite-norm solutions of higher-order linear Fuchsian-type ordinary differential equations (ODEs) P(x,d/dx)f(x)=0 with P(x,d/dx):=[\sum_m p_m (x) (d/dx)^m] by using only the four arithmetical operations on integers. This algorithm is based on a band-diagonal matrix representation of the differential operator P(x,d/dx), though it is quite different from the usual Galerkin methods. This representation is made for the respective CONSs of the input Hilbert space H and the output Hilbert space H' of P(x,d/dx). This band-diagonal matrix enables the construction of a recursive algorithm for solving the ODE. However, a solution of the simultaneous linear equations represented by this matrix does not necessarily correspond to the true solution of ODE. We show that when this solution is an l^2 sequence, it corresponds to the true solution of ODE. We invent a method based on an integer-type algorithm for extracting only l^2 components. Further, the concrete choice of Hilbert spaces H and H' is also given for our algorithm when p_m is a polynomial or a rational function with rational coefficients. We check how our algorithm works based on several numerical demonstrations related to special functions, where the results show that the accuracy of our method is extremely high.
This paper is devoted to the study of a nonlinear heat equation associated with Dirichlet-Robin conditions. At first, we use the Faedo -- Galerkin and the compactness method to prove existence and uniqueness results. Next, we consider the properties of solutions. We obtain that if the initial condition is bounded then so is the solution and we also get asymptotic behavior of solutions as. Finally, we give numerical results Comment: 20 pages
1. Errors and estimator for Example 1
In this article, an abstract framework for the error analysis of discontinuous Galerkin methods for control constrained optimal control problems is developed. The analysis establishes the best approximation result from a priori analysis point of view and delivers reliable and efficient a posteriori error estimators. The results are applicable to a variety of problems just under the minimal regularity possessed by the well-posed ness of the problem. Subsequently, applications of $C^0$ interior penalty methods for a boundary control problem as well as a distributed control problem governed by the biharmonic equation subject to simply supported boundary conditions are discussed through the abstract analysis. Numerical experiments illustrate the theoretical findings. Finally, we also discuss the variational discontinuous discretization method (without discretizing the control) and its corresponding error estimates.
We study the finite-dimensional spaces V, that are invariant under the action of the finite differences operator . Concretely, we prove that if V is such an space, there exists a finite-dimensional translation invariant space W such that V ⊆ W. In particular, all elements of V are exponential polynomials. Furthermore, V admits a decomposition V = P ⊕ E with P a space of polynomials and E a translation invariant space. As a consequence of this study, we prove a generalization of a famous result by Montel [77. M. P. Montel ( 1937 ). Sur quelques extensions d'un théorème de Jacobi . Prace Matematyczno-Fizyczne 44 ( 1 ): 315 – 329 . View all references], which states that, if f: ℝ → ℂ is a continuous function satisfying for all t ∈ ℝ and certain h 1, h 2 ∈ ℝ∖{0} such that h 1/h 2 ∉ ℚ, then f(t) = a 0 + a 1t + … +a m−1t m−1 for all t ∈ ℝ and certain complex numbers a 0, a 1,…, a m−1. We demonstrate, with quite different arguments, the same result not only for ordinary functions f(t) but also for complex valued distributions. Finally, we also consider the subspaces V that are Δh 1h 2…h m -invariant for all h 1,…, h m ∈ ℝ.
Weak sharp minimality is a notion emerged in optimization, whose utility is largeley recognized in the convergence analysis of algorithms for solving extremum problems as well as in the study of the perturbation behaviour of such problems. In the present paper some dual constructions of nonsmooth analysis, mainly related to quasidifferential calculus and its recent developments, are employed in formulating sufficient conditions for global weak sharp minimality. They extend to nonconvex functions a condition, which is known to be valid in the convex case. A feature distinguishing the results here proposed is that they avoid to assume the Asplund property on the underlying space.
In this paper, we give sufficient conditions for the existence of solutions of a general model which includes as special cases many generalized vector quasi-equilibrium problems with set-valued maps. The obtained results generalize and improve several earlier results.
We incorporate inertial terms in the hybrid proximal-extragradient algorithm and investigate the convergence properties of the resulting iterative scheme designed for finding the zeros of a maximally monotone operator in real Hilbert spaces. The convergence analysis relies on extended Fej\'er monotonicity techniques combined with the celebrated Opial Lemma. We also show that the classical hybrid proximal-extragradient algorithm and the inertial versions of the proximal point, the forward-backward and the forward-backward-forward algorithms can be embedded in the framework of the proposed iterative scheme.
Certain Bernoulli convolution measures (\mu) are known to be spectral. Recently, much work has concentrated on determining conditions under which orthonormal Fourier bases (i.e. spectral bases) exist. For a fixed measure known to be spectral, the orthonormal basis need not be unique; indeed, there are often families of such spectral bases. Let \lambda = 1/(2n) for a natural number n and consider the Bernoulli measure (\mu) with scale factor \lambda. It is known that L^2(\mu) has a Fourier basis. We first show that there are Cuntz operators acting on this Hilbert space which create an orthogonal decomposition, thereby offering powerful algorithms for computations for Fourier expansions. When L^2(\mu) has more than one Fourier basis, there are natural unitary operators U, indexed by a subset of odd scaling factors p; each U is defined by mapping one ONB to another. We show that the unitary operator U can also be orthogonally decomposed according to the Cuntz relations. Moreover, this operator-fractal U exhibits its own self-similarity.
The classical frame potential in a finite dimensional Hilbert space has been introduced by Benedetto and Fickus, who showed that all finite unit-norm tight frames can be characterized as the minimizers of this energy functional. This was the start point of a series of new results in frame theory, related to finding tight frames with determined length. The frame potential has been studied in the traditional setting as well as in the finite-dimensional fusion frame context. In this work we introduce the concept of {\sl mixed frame potential}, which generalizes the notion of the Benedetto-Fickus frame potential. We study properties of this new potential, and give the structure of its critical pairs of sequences on a suitable restricted domain. For a given sequence ${\alpha_m}_{m=1,...,N}$ in $K,$ where $K$ is $\mathbb{R}$ or $\mathbb{C},$ we obtain necessary and sufficient conditions in order to have a dual pair of frames ${f_m}_{m=1,...,N}$, ${g_m}_{m=1,...,N}$ such that $<f_m, g_m>=\alpha_m$ for all $m=1,..., N.$
We will show that tight frames satisfying the restricted isometry property give rise to nearly tight fusion frames which are nearly orthogonal and hence are nearly equi-isoclinic. We will also show how to replace parts of the RIP frame with orthonormal sets while maintaining the RIP property.
Given a total sequence in a Hilbert space, we speak of an upper (resp. lower) semi-frame if only the upper (resp. lower) frame bound is valid. Equivalently, for an upper semi-frame, the frame operator is bounded, but has an unbounded inverse, whereas a lower semi-frame has an unbounded frame operator, with bounded inverse. For upper semi-frames, in the discrete and the continuous case, we build two natural Hilbert scales which may yield a novel characterization of certain function spaces of interest in signal processing. We present some examples and, in addition, some results concerning the duality between lower and upper semi-frames, as well as some generalizations, including fusion semi-frames and Banach semi-frames.
Given a parametrized family of finite frames, we consider the optimization problem of finding the member of this family whose coefficient space most closely contains a given data vector. This nonlinear least squares problem arises naturally in the context of a certain type of radar system. We derive analytic expressions for the first and second partial derivatives of the objective function in question, permitting this optimization problem to be efficiently solved using Newton's method. We also consider how sensitive the location of this minimizer is to noise in the data vector. We further provide conditions under which one should expect the minimizer of this objective function to be unique. We conclude by discussing a related variational-calculus-based approach for solving this frame optimization problem over an interval of time.
A generalization with singular weights of Moore-Penrose generalized inverses of closed range operators in Hilbert spaces is studied using the notion of compatibility of subspaces and positive operators.
Given $A$ and $B$ two nonempty subsets in a metric space, a mapping $T : A \cup B \rightarrow A \cup B$ is relatively nonexpansive if $d(Tx,Ty) \leq d(x,y) \text{for every} x\in A, y\in B.$ A best proximity point for such a mapping is a point $x \in A \cup B$ such that $d(x,Tx)=\text{dist}(A,B)$. In this work, we extend the results given in [A.A. Eldred, W.A. Kirk, P. Veeramani, Proximal normal structure and relatively nonexpansive mappings, Studia Math., 171 (2005), 283-293] for relatively nonexpansive mappings in Banach spaces to more general metric spaces. Namely, we give existence results of best proximity points for cyclic and noncyclic relatively nonexpansive mappings in the context of Busemann convex reflexive metric spaces. Moreover, particular results are proved in the setting of CAT(0) and uniformly convex geodesic spaces. Finally, we show that proximal normal structure is a sufficient but not necessary condition for the existence in $A \times B$ of a pair of best proximity points.
approximated solution For the second problem ∆v = 0, x ∈ R, 0 < y < 1 v(x, 1) = ϕ(x), v y (x, 1) = 0 (58) whose exact solution is v(x, y) = 1 2 √ 2π 3 − y x 2 + (3 − y) 2 + 1 + y x 2 + (1 + y) 2 , we calculate the regularized solution v ε (x, y) of (58) for ε = 10 −2 by the formula (51). Finally for the third problem ∆w = f (x, y, v + w), x ∈ R, 0 < y < 1 w(x, 1) = 0, w y (x, 1) = 0 (59) the regularized solution w ε (x, y) of the problem (59) is calculated from its definition (52). So in Fig.2 we have drawed the regularized solution of the problem 2 i.e. u ε (x, y) = v ε (x, y) + w ε (x, y) on the interval [−4, 4] × [0, 1].
We consider the problem of reconstructing, from the interior data $u(x,1)$, a function $u$ satisfying a nonlinear elliptic equation $$ \Delta u = f(x,y,u(x,y)), x \in \RR, y > 0. $$ Comment: 24 pages
In this paper, we consider localized integral operators whose kernels have mild singularity near the diagonal and certain Holder regularity and decay off the diagonal. Our model example is the Bessel potential operator ${\mathcal J}_\gamma, \gamma>0$. We show that if such a localized integral operator has stability on a weighted function space $L^p_w$ for some $p\in [1, \infty)$ and Muckenhoupt $A_p$-weight $w$, then it has stability on weighted function spaces $L^{p'}_{w'}$ for all $1\le p'<\infty$ and Muckenhoupt $A_{p'}$-weights $w'$.
We set up a multiresolution analysis on fractal sets derived from limit sets of Markov Interval Maps. For this we consider the $\mathbb{Z}$-convolution of a non-atomic measure supported on the limit set of such systems and give a thorough investigation of the space of square integrable functions with respect to this measure. We define an abstract multiresolution analysis, prove the existence of mother wavelets, and then apply these abstract results to Markov Interval Maps. Even though, in our setting the corresponding scaling operators are in general not unitary we are able to give a complete description of the multiresolution analysis in terms of multiwavelets.
The problem of the minimization of least squares functionals with $\ell^1$ penalties is considered in an infinite dimensional Hilbert space setting. While there are several algorithms available in the finite dimensional setting there are only a few of them which come with a proper convergence analysis in the infinite dimensional setting. In this work we provide an algorithm from a class which have not been considered for $\ell^1$ minimization before, namely a proximal-point method in combination with a projection step. We show that this idea gives a simple and easy to implement algorithm. We present experiments which indicate that the algorithm may perform better than other algorithms if we employ them without any special tricks. Hence, we may conclude that the projection proximal-point idea is a promising idea in the context of $\ell^1$-minimization.
We study numerical methods for the solution of general linear moment problems, where the solution belongs to a family of nested subspaces of a Hilbert space. Multi-level algorithms, based on the conjugate gradient method and the Landweber--Richardson method are proposed that determine the "optimal" reconstruction level a posteriori from quantities that arise during the numerical calculations. As an important example we discuss the reconstruction of band-limited signals from irregularly spaced noisy samples, when the actual bandwidth of the signal is not available. Numerical examples show the usefulness of the proposed algorithms.
An adaptive regularization strategy for stabilizing Newton-like iterations on a coarse mesh is developed in the context of adaptive finite element methods for nonlinear PDE. Existence, uniqueness and approximation properties are known for finite element solutions of quasilinear problems assuming the initial mesh is fine enough. Here, an adaptive method is started on a coarse mesh where the finite element discretization and quadrature error produce a sequence of approximate problems with indefinite and ill-conditioned Jacobians. The methods of Tikhonov regularization and pseudo-transient continuation are related and used to define a regularized iteration using a positive semidefinite penalty term. The regularization matrix is adapted with the mesh refinements and its scaling is adapted with the iterations to find an approximate sequence of coarse mesh solutions leading to an efficient approximation of the PDE solution. Local q-linear convergence is shown for the error and the residual in the asymptotic regime and numerical examples of a model problem illustrate distinct phases of the solution process and support the convergence theory.
In this paper we study a general family of multivariable Gaussian stochastic processes. Each process is prescribed by a fixed Borel measure $\sigma$ on $\mathbb R^n$. The case when $\sigma$ is assumed absolutely continuous with respect to Lebesgue measure was studied earlier in the literature, when $n=1$. Our focus here is on showing how different equivalence classes (defined from relative absolute continuity for pairs of measures) translate into concrete spectral decompositions of the corresponding stochastic processes under study. The measures $\sigma$ we consider are typically purely singular. Our proofs rely on the theory of (singular) unbounded operators in Hilbert space, and their spectral theory.
This paper addresses Tikhonov like regularization methods with convex penalty functionals for solving nonlinear ill-posed operator equations formulated in Banach or, more general, topological spaces. We present an approach for proving convergence rates which combines advantages of approximate source conditions and variational inequalities. Precisely, our technique provides both a wide range of convergence rates and the capability to handle general and not necessarily convex residual terms as well as nonsmooth operators. Initially formulated for topological spaces, the approach is extensively discussed for Banach and Hilbert space situations, showing that it generalizes some well-known convergence rates results. Comment: 22 pages, submitted to "Numerical Functional Analysis and Optimization"
The theorem of Gopengauz guarantees the existence of a polynomial which well approximates a function f 2 C q [Gamma1; 1], while at the same time its kth derivative (k q) well approximates the kth derivative of the function, and moreover the polynomial and its derivatives respectively interpolate the function and its derivatives at Sigma1. With more generality, we shall prescribe that the polynomial interpolate the function at up to q + 1 points near 1 and up to q + 1 points near Gamma1. The points may coalesce, in which case one also interpolates at the coalescent point a number of derivatives one less than the multiplicity of coalescence. Aside from intrinsic theoretical interest, our results are clearly applicable in describing more precisely the error incurred in certain linear processes of simultaneous approximation, such as interpolation with added nodes near Sigma1. The original theorem of Gopengauz will be shown to follow as a special case. AMS Subject Classification: Pr...
Top-cited authors
B. D. Craven
  • RMIT University
Salih Aytar
  • T.C. Süleyman Demirel Üniversitesi
Changpin Li
  • Shanghai University
Fanhai Zeng
  • Shandong University
Thorsten Hohage
  • Georg-August-Universität Göttingen