C. M. Rosenthal

High dimensional model representations generated from low dimensional data samples. I. mp-Cut-HDMR

High dimensional model representation (HDMR) is a general set of quantitative model assessment and analysis tools for improving the efficiency of deducing high dimensional input–output system behavior. For a high dimensional system, an output f(x) is commonly a function of many input variables x=|x1,x2,...,xn} with n∼102 or larger. HDMR describes f(x) by a finite hierarchical correlated function expansion in terms of the input variables. Various forms of HDMR can be constructed for different purposes. Cut- and RS-HDMR are two particular HDMR expansions. Since the correlated functions in an HDMR expansion are optimal choices tailored to f(x) over the entire domain of x, the high order terms (usually larger than second order, or beyond pair cooperativity) in the expansion are often negligible. When the approximations given by the first and the second order Cut-HDMR correlated functions are not adequate, this paper presents a monomial based preconditioned HDMR method to represent the higher order terms of a Cut-HDMR expansion by expressions similar to the lower order ones with monomial multipliers. The accuracy of the Cut-HDMR expansion can be significantly improved using preconditioning with a minimal number of additional input–output samples without directly invoking the determination of higher order terms. The mathematical foundations of monomial based preconditioned Cut-HDMR is presented along with an illustration of its applicability to an atmospheric chemical kinetics model.

Optimal control landscapes for quantum observables

The optimal control of quantum systems provides the means to achieve the best outcome from redirecting dynamical behavior. Quantum systems for optimal control are characterized by an evolving density matrix and a Hermitian operator associated with the observable of interest. The optimal control landscape is the observable as a functional of the control field. The features of interest over this control landscape consist of the extremum values and their topological character. For controllable finite dimensional quantum systems with no constraints placed on the controls, it is shown that there is only a finite number of distinct values for the extrema, dependent on the spectral degeneracy of the initial and target density matrices. The consequences of these findings for the practical discovery of effective quantum controls in the laboratory is discussed.

Constructing global functional maps between molecular potentials and quantum observables

The relationships that connect potential energy surfaces to quantum observables can be complex and nonlinear. In this paper, an approach toward globally representing and exploring potential-observable relationships using a functional mapping procedure is developed. Based on selected solutions of the Schrodinger equation, it is demonstrated that an observable’s behavior can be learned as a function of the potential and any other variables needed to specify the quantum system. Once such a map for the observable is in hand, it is available for use in a host of future applications without further need for solving the Schrodinger equation. As formulated here, maps provide explicit information about the global response of the observable to the potential. In this paper, we develop the mapping concept, estimate its scaling behavior (measured as the number of times the Schrodinger equation must be solved during the learning process), and numerically illustrate the technique’s globality and nonlinearity using well-...

Topological and statistical properties of quantum control transition landscapes

A puzzle arising in the control of quantum dynamics is to explain the relative ease with which high-quality control solutions can be found in the laboratory and in simulations. The emerging explanation appears to lie in the nature of the quantum control landscape, which is an observable as a function of the control variables. This work considers the common case of the observable being the transition probability between an initial and a target state. For any controllable quantum system, this landscape contains only global maxima and minima, and no local extrema traps. The probability distribution function for the landscape value is used to calculate the relative volume of the region of the landscape corresponding to good control solutions. The topology of the global optima of the landscape is analysed and the optima are shown to have inherent robustness to variations in the controls. Although the relative landscape volume of good control solutions is found to shrink rapidly as the system Hilbert space dimension increases, the highly favourable landscape topology at and away from the global optima provides a rationale for understanding the relative ease of finding high-quality, stable quantum optimal control solutions.

Theoretical prediction of geometries and vibrational infrared spectra of ruthenium oxide molecules

Abstract We present computations of the optimized geometries and the corresponding vibrational frequencies of the molecules RuO2, RuO3, and RuO4. The computations utilize the Gaussian 90 Program Package, and they are based on the use of effective core potentials. In the case of RuO2, we obtain a closed shell singlet configuration with a bond angle of 150.6° and also a close and, possibly, lower lying triplet state with a bond angle of 133.7°. The trioxide is trigonal planar and the tetroxide is tetrahedral. On the whole, the calculated vibrational frequencies and geometries agree well with experimental values.

Multiple Solutions in the Tracking Control of Quantum Systems

This paper demonstrates the existence of multiple solutions at each time point in tracking control of quantum systems. These solutions are shown to arise from the nonlinear dependence of the short-time propagators U(t + delta t,t) on the control field. The multiplicity of solutions depends on the parameters of the controlled system and the nature of the imposed track. Multiple solutions necessitate that a choice be made at each time point, resulting in an exponentially expanding space of distinct control fields that maintain the prescribed track. This behavior is illustrated by application to a small model system. The presence of multiple tracking control fields is consistent with behavior observed from quantum control landscape theory.

On the Inversion of Quantum Mechanical Systems: Determining the Amount and Type of Data for a Unique Solution

The inverse problem of extracting a quantum mechanical potential from laboratory data is studied from the perspective of determining the amount and type of data capable of giving a unique answer. Bound state spectral information and expectation values of time-independent operators are used as data. The Schrödinger equation is treated as finite dimensional and for these types of data there are algebraic equations relating the unknowns in the system to the experimental data (e.g., the spectrum of a matrix is related algebraically to the elements of the matrix). As these equations are polynomials in the unknown parameters of the system, it is possible to determine the multiplicity of the solution set. With a fixed number of unknowns the effect of increasing the number of equations on the multiplicity of solutions is assessed. In general, if the number of the equations matches the number of the unknowns, the solution set is denumerable. A result on the solvability of polynomial equations is extended to the case where the number of equations exceeds the number of unknowns. We show that if one has more equations than the number of unknowns, generically a unique solution exists. Several examples illustrating these results are provided.

Two center variationally determined scattering amplitudes

Varitionally determined elastic scattering amplitudes arising from the interaction of a structureless particle with two separated Yukawa force centers are reported in this study. The parameters involved include R, the vector separating the centers; k02/2, the energy of the particle; kj and kf, the wave vectors of the incoming the outgoing particles, repsectively; and g, the interaction strength. Numerical values for the T matrix elements are tabulated and plotted for a variety of these parameters. Comparison is also made with three other nonvariational approixmations to this problem.

Optimum two-body scattering T matrix elements on and off the energy shell for various potentials

Variationally optimized T matrices and phase shifts are reported for three potential scattering problems: (1) elastic scattering from a square well with a P state resonance, (2) elastic scattering from a Morse potential parameterized to fit S state nucleon–nucleon scattering data, and (3) off shell diagonal and nondiagonal scattering from the Reid potential. This last problem, involving a potential with long range attraction and short range repulsion, was selected for study because solving the three body problem within the context of the Faddeev formalism requires for input off shell two body T matrices arising from such a potential.

The use of Lanczos tridiagonalization in bounding eigenvalues

Lanczos tridiagonalization is applied to several problems: the one dimensional quartic oscillator, the two dimensional quartic oscillator, and a finite difference version of the latter. In each case high accuracy upper and lower bounds are established in excellent agreement with previous calculations.