A canonical decomposition for quadratic forms with applications to monotone convergence theorems

We prove monotone convergence theorems for quadratic forms on a Hilbert space which improve existing results. The main tool is a canonical decomposition for any positive quadratic form h = h_r + h_s where h_r is characterized as the largest closable form smaller than h. There is also a systematic discussion of nondensely defined forms.     

Convergence of operator semigroups associated with generalised elliptic forms

In a recent article, Arendt and ter Elst have shown that every sectorial form is in a natural way associated with the generator of an analytic strongly continuous semigroup, even if the form fails to be closable. As an intermediate step, they have introduced so-called j?elliptic forms, which generalise the concept of elliptic forms in the sense of Lions. We push their analysis forward in that we discuss some perturbation and convergence results for semigroups associated with j?elliptic forms. In particular, we study convergence with respect to the trace norm or other Schatten norms. We apply our results to Laplace operators and Dirichlet-to-Neumann-type operators.     

Quantum Fields as Pointlike Localized Objects

We show that Wightman fields are always well defined objects at each space-time point in the meaning of sesquilinear forms. These sesquilinear forms “sandwiched” with e^–cH (H is the energy operator) are closable operators. We use these results to extend some assertions of Fredenhagen and Hertel about the recovering of Wightman fields from a Haag -Kastler theory of local observables.     

Freeness of linear and quadratic forms in von Neumann algebras

We characterize the semicircular distribution by freeness of linear and quadratic forms in noncommutative random variables from tracial W^?-probability spaces with relaxed moment conditions.     

On the Kolmogorov inequalities for quadratic forms of dependent uniformly bounded random variables

Some Kolmogorov probability inequalities for quadratic forms and weighted quadratic forms of negative superadditive dependent (NSD) uniformly bounded random variables are provided. Using these inequalities, some complete convergence of randomized quadratic forms under some suitable conditions are evaluated. Moreover, various examples are presented in which the given conditions of our results are satisfied.     

Tensor products and discriminants of unital quadratic forms over commutative rings

A tensor product for unital quadratic forms is introduced which extends the product of separable quadratic algebras and is naturally associative and commutative. It admits a multiplicative functor vdis, the vector discriminant, with values in symmetric bilinear forms. We also compute the usual (signed) discriminant of the tensor product in terms of the discriminants of the factors. The orthogonal group scheme of a nonsingular unital quadratic formQ of even rank is isomorphic toZ ₂×SO(Q ⁰) whereQ ⁰ is the restriction of –Q to the space of trace zero elements. We use cohomology to interpret the action of separable quadratic algebras on unital quadratic forms, and to determine which forms of odd rank can be realized asQ ⁰.     

On the quadratic convergence of the Aitken Δ<Superscript>2</Superscript> process

The Aitken Δ² method for finding fixed points of scalar mappings is interpreted as a modification of the Wegstein method. Based on this approach, conditions for the quadratic convergence of this method are obtained for various situations of convergence/divergence of simple iteration. An algorithm for calculating fixed points that keeps track of these situations is presented.     

Quadratic systems with a rational first integral of degree three: a complete classification in the coefficient space ℝ12

A quadratic polynomial differential systemcan be identified with a single point of ?¹² through its coefficients. The phase portrait of the quadratic systems having a rational first integral of degree 3 have been studied using normal forms. Here using the algebraic invariant theory, we characterize all the non-degenerate quadratic polynomial differential systems in ?¹² having a rational first integral of degree 3. We show that there are only 31 different topological phase portraits in the Poincaré disc associated to this family of quadratic systems up to a reversal of the sense of their orbits, and we provide representatives of every class modulo an affine change of variables and a rescaling of the time variable. Moreover, each one of these 31 representatives is determined by a set of algebraic invariant conditions and we provide for it a first integral.     

Renormalised Steepest Descent in Hilbert Space Converges to a Two-Point Attractor

The result that for quadratic functions the classical steepest descent algorithm in R ^d converges locally to a two-point attractor was proved by Akaike. In this paper this result is proved for bounded quadratic operators in Hilbert space. The asymptotic rate of convergence is shown to depend on the starting point while, as expected, confirming the Kantorovich bounds. The introduction of a relaxation coefficient in the steepest-descent algorithm completely changes its behaviour, which may become chaotic. Different attractors are presented. We show that relaxation allows a significantly improved rate of convergence.     

Densely defined non‐closable curl on carpet‐like metric measure spaces

The paper deals with the possibly degenerate behaviour of the exterior derivative operator defined on 1‐forms on metric measure spaces. The main examples we consider are the non self‐similar Sierpinski carpets recently introduced by Mackay, Tyson and Wildrick. Although topologically one‐dimensional, they may have positive two‐dimensional Lebesgue measure and carry nontrivial 2‐forms. We prove that in this case the curl operator (and therefore also the exterior derivative on 1‐forms) is not closable, and that its adjoint operator has a trivial domain. We also formulate a similar more abstract result. It states that for spaces that are, in a certain way, structurally similar to Sierpinski carpets, the exterior derivative operator taking 1‐forms into 2‐forms cannot be closable if the martingale dimension is larger than one.     

A NEW TRUST REGION DOGLEG METHOD FOR UNCONSTRAINED OPTIMIZATION

Abstract. This paper presents a new trust region dogleg method for unconstrained optimization.The method can deal with the case when the Hessian B of quadratic models is indefinite. It isproved that the method is globally convergent and has a quadratic convergence rate if Under certain conditions, the solution obtained by the method is even a second order     

The rate of convergence in the functional central limit theorem for random quadratic forms with some applications to the law of the iterated logarithm

We prove a convergence rate in the functional central limit theorem for quadratic forms in independent random variables satisfying a fourth moment condition. Using this result we get a law of the iterated logarithm as well as an analogue of Chung's law of the iterated logarithm for random quadratic forms.     

Continuous piecewise linear finite elements for the Kirchhoff–Love plate equation

A family of continuous piecewise linear finite elements for thin plate problems is presented. We use standard linear interpolation of the deflection field to reconstruct a discontinuous piecewise quadratic deflection field. This allows us to use discontinuous Galerkin methods for the Kirchhoff–Love plate equation. Three example reconstructions of quadratic functions from linear interpolation triangles are presented: a reconstruction using Morley basis functions, a fully quadratic reconstruction, and a more general least squares approach to a fully quadratic reconstruction. The Morley reconstruction is shown to be equivalent to the basic plate triangle (BPT). Given a condition on the reconstruction operator, a priori error estimates are proved in energy norm and L ² norm. Numerical results indicate that the Morley reconstruction/BPT does not converge on unstructured meshes while the fully quadratic reconstruction show optimal convergence.     

A quadratically convergent method for minimizing a sum of euclidean norms

We consider the problem of minimizing a sum of Euclidean norms. \(F(x) = \sum\nolimits_{i = 1}^m {||r_i } (x)||\) here the residuals {r _i(x)} are affine functions fromR ⁿ toR ¹ (n≥1≥2,m>-2). This arises in a number of applications, including single-and multi-facility location problems. The functionF is, in general, not differentiable atx if at least oner _i (x) is zero. Computational methods described in the literature converge quite slowly if the solution is at such a point. We present a new method which, at each iteration, computes a direction of search by solving the Newton system of equations, projected, if necessary, into a linear manifold along whichF is locally differentiable. A special line search is used to obtain the next iterate. The algorithm is closely related to a method recently described by Calamai and Conn. The new method has quadratic convergence to a solutionx under given conditions. The reason for this property depends on the nature of the solution. If none of the residuals is zero at^* x, thenF is differentiable at^* x and the quadratic convergence follows from standard properties of Newton's method. If one of the residuals, sayr _i ^* x), is zero, then, as the iteration proceeds, the Hessian ofF becomes extremely ill-conditioned. It is proved that this illconditioning, instead of creating difficulties, actually causes quadratic convergence to the manifold (x?r _i (x)=0}. If this is a single point, the solution is thus identified. Otherwise it is necessary to continue the iteration restricted to this manifold, where the usual quadratic convergence for Newton's method applies. If several residuals are zero at^* x, several stages of quadratic convergence take place as the correct index set is constructed. Thus the ill-conditioning property accelerates the identification of the residuals which are zero at the solution. Numerical experiments are presented, illustrating these results.     

A smoothing Newton method for a type of inverse semi-definite quadratic programming problem

We consider an inverse problem arising from the semi-definite quadratic programming (SDQP) problem. We represent this problem as a cone-constrained minimization problem and its dual (denoted ISDQD) is a semismoothly differentiable (SC¹${\text{SC}}^1$) convex programming problem with fewer variables than the original one. The Karush–Kuhn–Tucker conditions of the dual problem (ISDQD) can be formulated as a system of semismooth equations which involves the projection onto the cone of positive semi-definite matrices. A smoothing Newton method is given for getting a Karush–Kuhn–Tucker point of ISDQD. The proposed method needs to compute the directional derivative of the smoothing projector at the corresponding point and to solve one linear system per iteration. The quadratic convergence of the smoothing Newton method is proved under a suitable condition. Numerical experiments are reported to show that the smoothing Newton method is very effective for solving this type of inverse quadratic programming problems.     

Regularized Newton Methods for Convex Minimization Problems with Singular Solutions

This paper studies convergence properties of regularized Newton methods for minimizing a convex function whose Hessian matrix may be singular everywhere. We show that if the objective function is LC², then the methods possess local quadratic convergence under a local error bound condition without the requirement of isolated nonsingular solutions. By using a backtracking line search, we globalize an inexact regularized Newton method. We show that the unit stepsize is accepted eventually. Limited numerical experiments are presented, which show the practical advantage of the method.     

A Joint Tikhonov Regularization and Augmented Lagrange Approach for Ill-Posed State Constrained Control Problems with Sparse Controls

Abstract

We provide a modified augmented Lagrange method coupled with a Tikhonov regularization for solving ill-posed state constrained elliptic optimal control problems with sparse controls. We consider a linear quadratic optimal control problem without any additional L² regularization terms. The sparsity is guaranteed by an additional L¹ term. Here, the modification of the classical augmented Lagrange method guarantees us uniform boundedness of the multiplier that corresponds to the state constraints. We present a coupling between the regularization parameter introduced by the Tikhonov regularization and the penalty parameter from the augmented Lagrange method, which allows us to prove strong convergence of the controls and their corresponding states. Moreover, convergence results proving the weak convergence of the adjoint state and weak*-convergence of the multiplier are provided. Finally, we demonstrate our method in several numerical examples.     

A class of infinite dimensional stochastic processes with unbounded diffusion

The paper studies Dirichlet forms on the classical Wiener space and the Wiener space over non-compact complete Riemannian manifolds. The diffusion operator is almost everywhere an unbounded operator on the Cameron–Martin space. In particular, it is shown that under a class of changes of the reference measure, quasi-regularity of the form is preserved. We also show that under these changes of the reference measure, derivative and divergence are closable with certain closable inverses. We first treat the case of the classical Wiener space and then we transfer the results to the Wiener space over a Riemannian manifold.     

Efficient Sequential Quadratic Programming Implementations for Equality-Constrained Discrete-Time Optimal Control

Efficient sequential quadratic programming (SQP) implementations are presented for equality-constrained, discrete-time, optimal control problems. The algorithm developed calculates the search direction for the equality-based variant of SQP and is applicable to problems with either fixed or free final time. Problem solutions are obtained by solving iteratively a series of constrained quadratic programs. The number of mathematical operations required for each iteration is proportional to the number of discrete times N. This is contrasted by conventional methods in which this number is proportional to N ³. The algorithm results in quadratic convergence of the iterates under the same conditions as those for SQP and simplifies to an existing dynamic programming approach when there are no constraints and the final time is fixed. A simple test problem and two application problems are presented. The application examples include a satellite dynamics problem and a set of brachistochrone problems involving viscous friction.