Burcu Tunga

A probabilistic evolution approach trilogy, part 3: temporal variation of state variable expectation values from Liouville equation perspective

This is the third and therefore the final part of a trilogy on probabilistic evolution approach. The work presented here focuses on the probabilistic evolution determination for the state variables of a many particle system from classical mechanical point of view. Probabilistic evolution involves the expected value evolutions for all natural number Kronecker powers of the state variables, positions and momenta. We use the phase space distribution of the Liouville equation perspective to construct the expected values of the state variables’ Kronecker powers to define unknown temporal functions. The infinite number homogeneous linear ODEs with an infinite constant coefficient matrix are constructed by following the same steps as in the previous two works on quantum mechanics. The only difference is in the definitions of the expected values here. We also focus on a system of many harmonic oscillators to illustrate the block triangularity.

Constancy maximization based weight optimization in high dimensional model representation

This work focuses on the weight function optimization in high dimensional model representation (HDMR) via constancy maximization. There are a lot of circumstances where HDMR’s weight function becomes completely flexible in its factors. The univariate coordinate changes which can be constructed to produce nonnegative factors in the integrands of HDMR component, are perhaps the most important ones of these cases. Here, the weight function is considered as the square of a linear combination of certain basis functions spanning an appropriately chosen Hilbert space. Then, the coefficients of these linear combinations are determined to maximize the HDMR’s constant term contribution to the function. Although the resulting equations are nonlinear we could have been able to approximate the solutions by using recently proven fluctuationlessness theorem on matrix representations appearing in the equations.

Fluctuationlessness Approximation Based Multivariate Integration in Hybrid High Dimensional Model Representation

This paper focuses on multivariate integration via fluctuationlessness approximation in hybrid High Dimensional Model Representation (HHDMR) for multivariate functions. The basic idea here is to bypass the N—tuple integration with the help of the fluctuationlessness approximation as a quite powerful method for integral evaluations. This method decreases the computational complexity of the HHDMR method in computer based applications. Furthermore, by using this method, we are able to get rid of evaluating complicated integral structures.

A novel piecewise multivariate function approximation method via universal matrix representation

Multivariance in science and engineering causes problematic situations even for continous and discrete cases. One way to overcome this situation is to decrease the multivariance level of the problem by using a divide—and—conquer based method. In this sense, Enhanced Multivariance Product Representation (EMPR) plays a part in the considered scenario and acts successfully. This method brings up a finite expansion to represent a multivariate function in terms of less-variate functions with the assistance of univariate support functions. This work aims to propose a new EMPR based algorithm which has two new features that improves the determination process of each expansion component through Fluctuation Free Integration method, which is an efficient method in evaluating multiple integrals through a universal matrix representation, and increases the approximation quality through inserting a piecewise structure into the standard EMPR algorithm. This new method is called Fluctuation Free Integration based piecewise EMPR. Some numerical implementations are also given to examine the performance of this proposed method.

A novel hybrid high dimensional model representation (HHDMR) based on the combination of plain and logarithmic high dimensional model representations

This paper focuses on a new version of Hybrid High Dimensional Model Representation for multivariate functions. High Dimensional Model Representation (HDMR) was proposed to approximate the multivariate functions by the functions having less number of independent variables. Towards this end, HDMR disintegrates a multivariate function to components which are respectively constant, univariate, bivariate and so on in an ascending ordering of multivariance. HDMR method is a scheme truncating the representation at a prescribed multivariance. If the given multivariate function is purely additive then HDMR method spontaneously truncates at univariance, otherwise the highly multivariant terms are required. On the other hand, if the given function is dominantly multiplicative then the Logarithmic HDMR method which truncates the scheme at a prescribed multivariance of the HDMR of the logarithm of the given function is taken into consideration. In most cases the given multivariate function has both additive and multiplicative natures. If so then a new method is needed. Hybrid High Dimensional Model Representation method is used for these types of problems. This new representation method joins both plain High Dimensional Model Representation and Logarithmic High Dimensional Model Representation components via an hybridity parameter. This work focuses on the construction and certain details of this novel method.

Support function influences on the univariance of the enhanced multivariance product representation

This paper focuses on recently proposed Enhanced Multivariance Product Representation (EMPR) for multivariate functions. This method has been proposed to approximate the multivariate functions by certain additive less variate functions via support functions to get better quality than HDMRs. For this purpose, EMPR disintegrates a multivariate function to components which are respectively constant, univariate, bivariate and so on in ascending multivariance. This work aims at the investigation of the EMPR truncation qualities with respect to the selection of the support functions.

Probabilistic evolutions in classical dynamics: Conicalization and block triangularization of Lennard-Jones systems

This work aims at the solution of two-body problem with Lennard-Jones potential via recently developed Probabilistic Evolution Approach (PEA) proposed by M. Demiralp for explicit ODEs, as final target. PEA has been quite well developed and it is well-known that the conicality in descriptive functions facilitates the construction of the truncation approximants. Conicality, if does not exist, can be obtained by using the space extension method at the expense of an increase in the number of the unknowns and may bring the block triangularity which is another important facility to investigate and control the properties of PEA. In space extension we construct the Hamilton equations of the motion for the system first and then define new appropriate unknown functions depending on the existing unknowns such that the resulting ODEs to be used in PEA become conical. This attempt provides us with the block triangularity at the same time.

Basic Issues in Vector High Dimensional Model Representation

A multivariate modelling may involve a set of multivariate functions. A vector valued function structure can be used to mathematically express the given problem and each multivariate function can be considered as an element of this vector. This work aims to construct a new approach representing the elements of this vector structure in terms of less‐variate functions to reduce the computational complexity. For this purpose, a new method based on the plain High Dimensional Model Representation (HDMR) philosophy is developed. The basic concepts of this method and several illustrative numerical implementations are given here.

Hybrid HDMR method with an optimized hybridity parameter in multivariate function representation

High Dimensional Model Representation (HDMR) based methods are used to generate an approximation for a given multivariate function in terms of less variate functions. This paper focuses on Hybrid HDMR which is composed of Plain HDMR and Logarithmic HDMR. The Plain HDMR method works well for representing multivariate functions having additive nature. If the function under consideration has a multiplicative nature, then the Logarithmic HDMR method produces better approximation. Hybrid HDMR method aims to successfully represent a multivariate function having neither purely additive nor purely multiplicative nature under a hybridity parameter. The performance of the Hybrid HDMR method strongly depends on the value of this hybridity parameter because this parameter manages the contribution level of Plain and Logarithmic HDMR expansions. The main purpose of this work is to optimize the hybridity parameter to get the best approximations. Fluctuationlessness Approximation Theorem is used in this optimization process and in evaluating the multiple integrals appearing in HDMR based methods. A number of numerical implementations are given at the end of the paper to show the performance of our proposed method.

A hybrid algorithm with cluster analysis in modelling high dimensional data

Abstract Multivariate data modelling aims to predict unknown function values through an established mathematical model. It is essential to construct an analytical structure using the given set of high dimensional data points with corresponding function values. The level of multivariance directly affects the modelling process. Increase in the number of independent variables makes the standard numerical methods incapable of obtaining the sought analytical structure. This work aims to overcome the difficulties of high multivariance and to improve the modelling quality by carrying out two main steps: data clustering and data partitioning. Data clustering step deals with dividing the whole problem domain into several clusters by performing k-means clustering algorithm. Data partitioning step performs the Enhanced Multivariance Product Representation method to partition the high dimensional data set of each cluster. The analytical structure is obtained through the partitioned data for each cluster and can be used to predict the unknown function values.