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Herschel Rabitz

Publications and source records attributed to Herschel Rabitz.

5 recordsLinked to original sources
PubMed2003

High-dimensional model representations generated from low order terms--lp-RS-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. RS-HDMR is a particular form of HDMR based on random sampling (RS) of the input variables. The component functions in an HDMR expansion are optimal choices tailored to the n-variate function f(x) being represented over the desired domain of the n-dimensional vector x. The high-order terms (usually larger than second order, or equivalently beyond cooperativity between pairs of variables) in the expansion are often negligible. When it is necessary to go beyond the first and the second order RS-HDMR, this article introduces a modified low-order term product (lp)-RS-HDMR method to approximately represent the high-order RS-HDMR component functions as products of low-order functions. Using this method the high-order truncated RS-HDMR expansions may be constructed without directly computing the original high-order terms. The mathematical foundations of lp-RS-HDMR are presented along with an illustration of its utility in an atmospheric chemical kinetics model.

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PubMed2003

Correlation method for variance reduction of Monte Carlo integration in RS-HDMR.

The High Dimensional Model Representation (HDMR) technique is a procedure for efficiently representing high-dimensional functions. A practical form of the technique, RS-HDMR, is based on randomly sampling the overall function and utilizing orthonormal polynomial expansions. The determination of expansion coefficients employs Monte Carlo integration, which controls the accuracy of RS-HDMR expansions. In this article, a correlation method is used to reduce the Monte Carlo integration error. The determination of the expansion coefficients becomes an iteration procedure, and the resultant RS-HDMR expansion has much better accuracy than that achieved by direct Monte Carlo integration. For an illustration in four dimensions a few hundred random samples are sufficient to construct an RS-HDMR expansion by the correlation method with an accuracy comparable to that obtained by direct Monte Carlo integration with thousands of samples.

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PubMed2002

Optimal identification of Hamiltonian information by closed-loop laser control of quantum systems.

A closed loop learning control concept is introduced for teaching lasers to manipulate quantum systems for the purpose of optimally identifying Hamiltonian information. The closed loop optimal identification algorithm operates by revealing the distribution of Hamiltonians consistent with the data. The control laser is guided to perform additional experiments, based on minimizing the dispersion of the distribution. Operation of such an apparatus is simulated for two model finite dimensional quantum systems.

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PubMed2002

Rational approximation with multidimensional scattered data.

Accurate and efficient rational approximation schemes are presented for interpolating multidimensional scattered data with a novel weighted least-squares procedure including domain decomposition. Two particular representations of the method are formulated and the corresponding algorithms are implemented. Numerical tests on three- and six-dimensional model systems are carried out, demonstrating high efficiency and accuracy. This work was motivated by the need for multidimensional function approximation using irregular grids when solving quantum fluid dynamics equations, and the method should have broader physical applications.

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