Dimensional Function Space Research Articles

The Generalized Moment Problem with Complexity Constraint

In this paper, we present a synthesis of our differentiable approach to the generalized moment problem, an approach which begins with a reformulation in terms of differential forms and which ultimately ends up with a canonically derived, strictly convex optimization problem. Engineering applications typically demand a solution that is the ratio of functions in certain finite dimensional vector space of functions, usually the same vector space that is prescribed in the generalized moment problem. Solutions of this type are hinted at in the classical text by Krein and Nudelman and stated in the vast generalization of interpolation problems by Sarason. In this paper, formulated as generalized moment problems with complexity constraint, we give a complete parameterization of such solutions, in harmony with the above mentioned results and the engineering applications. While our previously announced results required some differentiability hypotheses, this paper uses a weak form involving integrability and measurability hypotheses that are more in the spirit of the classical treatment of the generalized moment problem. Because of this generality, we can extend the existence and well-posedness of solutions to this problem to nonnegative, rather than positive, initial data in the complexity constraint. This has nontrivial implications in the engineering applications of this theory. We also extend this more general result to the case where the numerator can be an arbitrary positive absolutely integrable function that determines a unique denominator in this finite-dimensional vector space. Finally, we conclude with four examples illustrating our results.

VAR Modeling for Dynamic Semiparametric Factors of Volatility Strings

The implied volatility of a European option as a function of strike price and time to maturity forms a volatility surface. Traders price according to the dynamics of this high dimensional surface. Recent developments that employ semiparametric models approximate the implied volatility surface (IVS) in a finite dimensional function space, allowing for a low dimensional factor representation of these dynamics. This paper presents an investigation into the stochastic properties of the factor loading times series using the vector autoregressive (VAR) framework and analyzes associated movements of these factors with movements in some macroeconomic variables of the Euro-economy.

VAR Modeling for Dynamic Semiparametric Factors of Volatility Strings

The implied volatility of a European option as a function of strike price and time to maturity forms a volatility surface. Traders price according to the dynamics of this high dimensional surface. Recent developments that employ semiparametric models approximate the implied volatility surface (IVS) in a finite dimensional function space, allowing for a low dimensional factor representation of these dynamics. This paper presents an investigation into the stochastic properties of the factor loading times series using the vector autoregressive (VAR) framework and analyzes associated movements of these factors with movements in some macroeconomic variables of the Euro - economy.

A Dynamic Semiparametric Factor Model for Implied Volatility String Dynamics

A primary goal in modelling the implied volatility surface (IVS) for pricing and hedging aims at reducing complexity. For this purpose one fits the IVS each day and applies a principal component analysis using a functional norm. This approach, however, neglects the degenerated string structure of the implied volatility data and may result in a modelling bias. We propose a dynamic semiparametric factor model (DSFM), which approximates the IVS in a finite dimensional function space. The key feature is that we only fit in the local neighborhood of the design points. Our approach is a combination of methods from functional principal component analysis and backfitting techniques for additive models. The model is found to have an approximate 10% better performance than a sticky moneyness model. Finally, based on the DSFM, we devise a generalized vega-hedging strategy for exotic options that are priced in the local volatility framework. The generalized vega-hedging extends the usual approaches employed in the local volatility framework.

B-series and Order Conditions for Exponential Integrators

We introduce a general format of numerical ODE-solvers which include many of the recently proposed exponential integrators. We derive a general order theory for these schemes in terms of $B$-series and bicolored rooted trees. To ease the construction of specific schemes we generalize an idea of Zennaro [{\em {Math. Comp.,}} 46 (1986), pp. 119--133] and define natural continuous extensions in the context of exponential integrators. This leads to a relatively easy derivation of some of the most popular recently proposed schemes. The general format of schemes considered here makes use of coefficient functions which will usually be selected from some finite dimensional function spaces. We will derive lower bounds for the dimension of these spaces in terms of the order of the resulting schemes. Finally, we illustrate the presented ideas by giving examples of new exponential integrators of orders 4 and 5.

Adaptive Optimization of Convex Functionals in Banach Spaces

This paper is concerned with optimization or minimization problems that are governed by operator equations, such as partial differential or integral equations, and thus are naturally formulated in an infinite dimensional function space V. We first construct a prototype algorithm of steepest descent type in V and prove its convergence. By using a Riesz basis in V we can transform the minimization problem into an equivalent one posed in a sequence space of type $\ell_p$. We convert the prototype algorithm into an adaptive method in $\ell_p$. This algorithm is shown to be convergent under mild conditions on the parameters that appear in the algorithm. Under more restrictive assumptions we are also able to establish the rate of convergence of our algorithm and prove that the work/accuracy balance is asymptotically optimal. Finally, we give two particular examples.

Semismooth Newton and Augmented Lagrangian Methods for a Simplified Friction Problem

In this paper a simplified friction problem and iterative second-order algorithms for its solution are analyzed in infinite dimensional function spaces. Motivated from the dual formulation, a primal-dual active set strategy and a semismooth Newton method for a regularized problem as well as an augmented Lagrangian method for the original problem are presented and their close relation is analyzed. Local as well as global convergence results are given. By means of numerical tests, we discuss among others convergence properties, the dependence on the mesh, and the role of the regularization and illustrate the efficiency of the proposed methodologies.

Implied Volatility String Dynamics

A primary goal in modelling the dynamics of implied volatility surfaces (IVS) aims at reducing complexity. For this purpose one fits the IVS each day and applies a principal component analysis using a functional norm. This approach, however, neglects the degenerated string structure of the implied volatility data and may result in a severe modelling bias. We propose a dynamic semiparametric factor model, which approximates the IVS in a finite dimensional function space. The key feature is that we only fit in the local neighborhood of the design points. Our approach is a combination of methods from functional principal component analysis and backfitting techniques for additive models. The model is found to have an approximate 10% better performance than the typical naive trader models. The model can be a backbone in risk management serving for value at risk computations and scenario analysis.

Multiple algebraisations of an elliptic Calogero–Sutherland model

Recently, Gómez-Ullate et al. [Phys. Lett. B 511, 112 (2001)] have studied an N-particle quantum problem with elliptic-function potentials. They have shown that the Hamiltonian operator preserves a finite dimensional space of functions and as such is quasi-exactly solvable (QES). In this article we show that other types of invariant function spaces exist, which are in close relation to the algebraic properties of the elliptic functions. Accordingly, series of new algebraic eigenfunctions can be constructed.

Maximum entropy moment systems and Galilean invariance

Maximum entropy moment closure systems of gas dynamics are investigated. It is shown that polynomial weight functions growing super-quadratically at infinity lead to hyperbolic systems with an unpleasant state space: equilibrium states are boundary points with possibly singular fluxes. This in its generality previously unknown result applies to any moment system including, for example, the 26 or 35 moment case. One might try to avoid singular fluxes by choosing non-polynomial weight functions which grow sub-quadratically at infinity. This attempt, however, is shown to be incompatible with the Galilean invariance of the moment systems because rotational and translational invariant, finite dimensional function spaces necessarily consist of polynomials.

Functional Continuum of Regulatory Peptides (RPs): Vector Model of RP-Effects Representation

During the past decades, bioactive (regulatory) peptides have been identified as the major players in the regulation of many important biological processes. Dozens of peptides have found their application as pharmaceutical agents, which further stimulated research in this field making it one of the most rapidly developing areas on the edge of biological science and medicine. However, the fast accumulation of enormous amounts of experimental data has revealed a great difficulty in their analysis and demanded the development of a systematic approach for generalization of the obtained information. We propose a new computer-based algorithm for studying biological activities of regulatory peptides and their groups based on their representation as vectors in n -dimensional functional space. Our method allows the rapid analysis of databases containing thousands of polyfunctional regulatory peptides with overlapping spectra of physiological activity. The described method permits to perform several types of correlations which, when applied to the large databases, could reveal new important information about the system of regulatory peptides. It can select the groups of peptides with similar physiological role (peptide constellations) and search for the optimal peptide combinations with predetermined spectrum of effects and minimal side effects for their further pharmacological application. It can also reveal the role of regulatory peptides in induction of chain physiological reactions.

Approximation by quadrilateral finite elements

We consider the approximation properties of finite element spaces on quadrilateral meshes. The finite element spaces are constructed starting with a given finite dimensional space of functions on a square reference element, which is then transformed to a space of functions on each convex quadrilateral element via a bilinear isomorphism of the square onto the element. It is known that for affine isomorphisms, a necessary and sufficient condition for approximation of order r+1 in L2 and order r in H1 is that the given space of functions on the reference element contain all polynomial functions of total degree at most r. In the case of bilinear isomorphisms, it is known that the same estimates hold if the function space contains all polynomial functions of separate degree r. We show, by means of a counterexample, that this latter condition is also necessary. As applications we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for serendipity finite elements and for various mixed and nonconforming finite elements.

Classification of type A -fold supersymmetry

Type A N-fold supersymmetry of one-dimensional quantum mechanics can be constructed by using sl(2) generators represented on a finite dimensional functional space. Using this sl(2) formalism we show a general method of constructing Type A N-fold supersymmetric models. We also present systematic generation of known models and several new models using this method.

Mapped Finite Elements on Hexahedra. Necessary and Sufficient Conditions for Optimal Interpolation Errors

This paper considers finite elements which are defined on hexahedral cells via a reference transformation which is in general trilinear. For affine reference mappings, the necessary and sufficient condition for an interpolation order O(h k+1) in the L 2-norm and O(h k ) in the H 1-norm is that the finite dimensional function space on the reference cell contains all polynomials of degree less than or equal to k. The situation changes in the case of a general trilinear reference transformation. We will show that on general meshes the necessary and sufficient condition for an optimal order for the interpolation error is that the space of polynomials of degree less than or equal to k in each variable separately is contained in the function space on the reference cell. Furthermore, we will show that this condition can be weakened on special families of meshes. These families which are obtained by applying usual refinement techniques can be characterized by the asymptotic behaviour of the semi-norms of the reference mapping.

Quantum mechanical origin of QSAR: theory and applications

In this paper, search is carried out on how to develop the formalism where quantum similarity measures (QSM) become a natural product of the theoretical framework. This fact is later used to establish a fundamental connection between quantum theory and QSAR, which is analysed in turn within the realm of discrete quantum chemistry. In order to perform such a task, several theoretical tools are revised in a previous step. The first section is devoted to construct the concept of tagged set. Next, the definition of quantum object (QO) is clarified by means of ideas from a quantum theory background and the previous tagged set formalism. In the definition of QO, density functions (DF) play a fundamental role and a possible simplified mathematical picture is presented. On the road to prepare the problem solving tools, convex sets result to be prominent, and the notion of vector semispace appears as a consequence. The transformation rule, a device to connect wavefunctions with DF, is defined in a new step. Various products of this preliminary discussion are described, among others the concept of kinetic energy and other observable distributions. QSM as a source of discrete representation of molecular structures is made evident in this context. Further theoretical development intends to study discretisation, the transformation of infinite dimensional functional spaces into n-dimensional ones. This result adds new perspectives to the discrete representation of QO, because it (a) provides a source of new QO descriptors, like a generalisation of scalar product and new similarity indices, (b) describes the QSPR theoretical background enabling the construction of the adequate mathematical tools in order to discuss connected problems (limitations of linear models, tuned-QSAR, p-valued classification of QO, among others), (c) allows the construction of sound and general alternatives of Hammet's σ or log P parameters. All the steps above summarised are completed and illustrated, when possible, with practical application examples and visualisation pictures.

A class of extremum problems related to agency models with imperfect monitoring

The cost minimization problem in an agency model with imperfect monitoring is considered. Under the first order approach, this can be stated as a convex minimization problem with linear inequality and equality constraints in a generally infinite dimensional function space. We apply the Fenchel Duality Theorem, and obtain as a dual problem a concave maximization problem of finite dimension. In particular, a Lagrange multiplier description of the optimal solution to the cost minimization problem is derived, justifying and extending thus the approach of Kim (1995). By the duality, the dependence of the minimum cost value on the information system used becomes particularly visible. The minimum cost value behaves monotonically w.r.t. the convex ordering of certain distributions induced by the competing information systems. Under the standard inequality constraint, one is led to the distributions of the score functions of the information systems and their convex order relation. It is shown that also for multivariate actions, Blackwell sufficiency implies the convex order relation of the score function distributions. A further result refers to a multi-agents model recently considered by Budde (1997), when the maximum of n independent and identically distributed (i.i.d.) univariate output variables is focussed. If two univariate information systems have monotone likelihood ratios, then the convex ordering between the two score function distributions implies the weaker convex increasing ordering between the distributions of the same score functions under the maximum distributions.

次世代超音速輸送機主翼の超音速と遷音速巡航の多目的空力最適化

This paper describes the design optimization of a wing for supersonic transport (SST) using Multi-Objective Genetic Algorithm (MOGA). The objective functions are to minimize the drag for transonic cruise, the drag for supersonic cruise and the bending moment at the wing root for supersonic cruise. The wing shape is defined by planform and warp shapes in total of 66 design variables. An Euler code is used to evaluate supersonic performance, and a potential code is used to evaluate transonic performance. To reduce the enormous total calculation time, the CFD calculations are parallelized on NEC SX-4 (32 PE) at Computer Center of Tohoku University. The Pareto optimal solutions are obtained in the three dimensional objective function space by the present approach. The resulting Pareto solutions identify that arrow wing configuration is one of the important factors for the wing performance.

Notes on the Elliptic Ruijsenaars Operators

We construct the spaces that the elliptic Ruijsenaars operators act on. It is shown that they are extensible to nonnegative selfadjoint operators on a space of square integrable functions, or preserve a finite dimensional vector space of entire functions. These facts are shown in terms of the R-operators which satisfy the Yang–lBaxter equation. The elliptic Ruijsenaars operators are considered as the elliptic analogues of the Macdonald operators or the difference analogues of the Lame operators.

Trajectory Optimization for Nonholonomic Systems. Space Robot Case.

In this paper, we deal with trajectory optimization problems for free-floating space robots, where the bases are not controlled by external moments and forces. Robot arms are mounted on free-floating bases-for example, spacecrafts, space stations, and so on. First, we formulate the trajectory optimization problems as nonlinear optimal control problems. Next, applying optimal control algorithms, we numerically obtain optimal solutions in infinite dimensional function spaces. Finally, we demonstrate how to find global solutions through local solutions by some numerical simulations.

A function estimation approach for determining temperature-dependent thermophysical properties

The estimation of the temperature dependence of thermophysical properties has been generally treated as a parameter estimation problem in the literature. In this paper, we apply a function estimation approach to the inverse problem of determining the temperature dependence of either the volumetric heat capacity or the thermal conductivity. No information regarding the functional form of the unknown property is required in the present approach, and the minimization is performed in an infinite dimensional space of functions. The Conjugate Gradient Method with Adjoint Equation is used in the inverse analysis. Results obtained by using simulated temperature measurements of a single sensor, slow that the present approach is capable of recovering functions containing discontinuities, which are the most difficult to be recovered by an inverse analysis. The effects of sensor location on the inverse problem solution are also addressed on the paper.