Search

There are plenty of methods to approximate a multi-variate function arising from different concepts. Some of them are based on the truncation of infinite series like Taylor series or orthogonal expansions. These do not explicitly reflect the singularities of the function to be approximated. One of the infinite series explicitly showing singularities is the Laurent series, which include finite or infinite simple fractions to represent polar singularities. Since any rational function can be uniquely represented by a Laurent series, it is common practice to use rational approximants for approximating the functions to explicitly reflect all polar singularities. Padé approximants are perhaps the most frequently used ones in this category. Functions having strong singularities like branch points or essential singularities need special attention and can be approximated by using more specific techniques. Rational approximation techniques are perhaps the leading and easiest ones used to handle nonlinearities or singularities in the approximation of univariate or multivariate functions in science and engineering. They can be used in model-reduction of multidimensional Linear Shift Invariant (LSI) recursive systems, computing packet loss probabilities in multiplexer models, and shape reconstruction. In this work, we construct a new multivariate rational approximation method and discuss its ad-vantages and disadvantages by comparing with the Padé technique, which is at the moment the leading rational approximation technique. The rational approximation scheme proposed here is formulated by using HDMR (High Dimensional Model Representation) truncations. HDMR has been developed in the last 15 years to approximate multivariate functions by a sum of less variate functions. Here, we insert certain flexibilities into the approximation, in order to improve its quality. The most general class of this type of HDMR is called "Transformational High Dimensional Model Representation" (THDMR). In THDMR, the image of the original multivariate function is approximated by a sum of less variate functions. The coefficients of the transformation are considered as optimization parameters, which make the error of the method as small as possible. In this work, a first degree polynomial structure is used as THDMR's operator and the coefficients of this first degree transformation are assumed to be varying with independent variables. This is for giving flexibilities to the transformation and fixing them to approximate the HDMR expansion optimally. Although THDMR has a more general structure than we used here, we chose to work with the first degree polynomial structure for unique invertibility. Once an approximation for this first order polynomial structure constructed by HDMR method, the approximation for the original function can be obtained by inverting the polynomial structure. The rational structure of the method comes from this special choice of THDMR operator. The univariate and bivariate results are quite impressive and imply that we can approximate a multi-variate function over a quite larger interval when compared with the Padé method. This is caused by the fact that these two methods involve different strategies of extracting information from the original function. In THDMR method, the integral of the original function is computed over a specified interval to extract information from it, while Padé approximation method uses the derivatives of the function in a specific point. So THDMR method approximates the function more uniformly compared with Padé approximation technique which exhibits a lo-cal character. Moreover, the quality of the approximation can be improved by changing the integration intervals in THDMR. An important defect of the rational approximation techniques is that they can produce some poles over the focused interval, although the original function has no singularity over the same interval at all. The singularity problem exists also in THDMR method. However, in THDMR, in order to obtain the rational approximant, certain integrals are needed to be computed and the location of the possible poles of the approximant can be altered by changing the interval of the integration. This facility of changing the locations of the poles with the integration interval makes the method attractive and efficient. In order to see the behavior of the method on physical problems, we have applied it on LSI systems to reduce the system. The transfer function of the LSI system has been approximated by THDMR and Padé methods. The results show that, the newly developed THDMR method can be used on LSI systems with high level approximation quality. [ABSTRACT FROM AUTHOR]

This work focuses on the evaluation of critical control times in the optimal control of linear quantum systems. The linearity is used to imply the existence of only linear intra forces in the system. This means that the system under consideration is composed of harmonic oscillators. The linearity is not only in the forces. The external field's dipole polarizability function is also taken linear in spatial variables. Beyond that the objective operator whose expectation value's deviation from its target value should be minimized in square and the penalty operator whose expectation value is to be suppressed are also linear in position and momentum operators. We assume that the system has 3N degree of freedom and the cost functional for the optimal control is composed of four terms: (1) objective term which is the square of the objective operator whose expectation value's deviation from its prescribed target; (2) penalty term for getting finiteness in external field; (3) penalty term for an operator whose norm is to be suppressed ; (4) dynamical constraint term which enters the quantum motion of the system to the optimisation. This cost functional depends on five unkowns: (1-2) wave function and its complex conjugate; (3-4) costate function which is in fact the Lagrange multiplier of the dynamical constraint term. It varies in time and space and somehow describes the backward motion of the system from the final instant of the control. It is a complex valued function hence its complex conjugate is the companion unknown to itself; (5) The external field amplitude within the dipole the polarizability approximation. It is temporal and real valued. There may be a shorthand notation for the deviation of the expectation value of the objective operator from its target value. In that case, this entity can be considered as another unknown and can be handled by adding its definition as an extra condition. The equations of motion can be optained by setting the first variation of the cost functional to zero. Four partial differential equations are obtained for them-selves and complex conjugates of the wave and costate functions. The equations for the wave function are accompanied by initial conditions while the costate function related partial differential equations are right hand sided and accompanied by final conditions. This for equations two by two, describes forward and backward evolutions for the system and these evolutions are connected by the algebraic however functional relation obtained for the external field amplitude. All these equations are non line-ar for the general case. However for the present linearity assumptions they become linear. The equations of motion mentioned above are not attempted to be solved. Instead, ordinary differential equations for the expectation values of position and momentum operators on the wave function and the transition terms between the states decscribed by the wave and costate functions are constructed. These temporal ODEs are linear but may be timevariant in the coefficients if the weight functions of the penalty terms are temporally varying. The simplest case involves the cases where these weights are constansts and the ODEs can be analytically solved in terms of an exponentials matrix. This results in analytic expressions for the unknowns. These expressions de-pend not only on time but also control time (T). As can be shown by rigorous analytical analysis and verified by the numerical implementations, there are certain critical values of the control time where the extressions grow up to infinity or vanishes. Vanishing values mean exact achievement of the target value by the expectation value of the objective operator. These are the best control instants where the goal is exactly achieved. On the other hand, control times causing infinite jumps, are just the time instants the control becomes infeasible. The critical control times creating infinite jumps are called "Critical Control Times for Uncontrollability" and it is important to know these values before attempting the experimentation to avoid uncontrollability. Otherwise, experiment is wasted out. The critical control times which make the deviation of the expectation values of the objective operator from its target value are called "Critical Control Times for Exact Achievement" and they are useful to get the best control. Paper contains all important issues about these points. [ABSTRACT FROM AUTHOR]