Optimization by Direct Search: New Perspectives on Some Classical and Modern Methods

Kolda, Tamara G.; Lewis, Robert; Torczon, Virginia

doi:10.1137/s003614450242889

Cited by 1,387 publications

(997 citation statements)

References 183 publications

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“…To try to further optimize the LS_NUFFT method, we used the Nelder-Mead simplex method [11] to minimize E max (s) in (17) over s numerically. We then recomputed the LS_NUFFT interpolator using the resulting "optimal" scaling factors s. Fig.…”

Section: Resultsmentioning

confidence: 99%

“…Define the integer offset as follows: (10) where ⌊·⌋ denotes the integer floor function, and [·] denotes rounding to the nearest integer. This offset satisfies the following (integer) shift invariance property: (11) Then we replace (5) by the equivalent expression (12) where u(ω) = (u 1 (ω), …, u J (ω)) denotes the vector of length J of interpolation coefficients associated with frequency ω.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On NUFFT-based gridding for non-Cartesian MRI

Fessler

2007

Journal of Magnetic Resonance

195

167

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For MRI with non-Cartesian sampling, the conventional approach to reconstructing images is to use the gridding method with a Kaiser-Bessel (KB) interpolation kernel. Recently, Sha et al. [1] proposed an alternative method based on a nonuniform FFT (NUFFT) with least-squares (LS) design of the interpolation coefficients. They described this LS_NUFFT method as shift variant and reported that it yielded smaller reconstruction approximation errors than the conventional shift-invariant KB approach. This paper analyzes the LS_NUFFT approach in detail. We show that when one accounts for a certain linear phase factor, the core of the LS_NUFFT interpolator is in fact real and shift invariant. Furthermore, we find that the KB approach yields smaller errors than the original LS_NUFFT approach. We show that optimizing certain scaling factors can lead to a somewhat improved LS_NUFFT approach, but the high computation cost seems to outweigh the modest reduction in reconstruction error. We conclude that the standard KB approach, with appropriate parameters as described in the literature, remains the practical method of choice for gridding reconstruction in MRI.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On NUFFT-based gridding for non-Cartesian MRI

Fessler

2007

Journal of Magnetic Resonance

195

167

View full text Add to dashboard Cite

show abstract

“…These methods have a long and rich history in the scientific and engineering communities where they have been applied to numerous problems. An excellent introduction and survey of these methods can be found in [21], which also contains numerous references. The main attraction of direct search methods is their ability to find optimal solutions without the need for computing derivatives in contrast to the more familiar gradient-based methods.…”

Section: Pattern Search Methodsmentioning

confidence: 99%

Using pattern search methods for surface structure determination of nanomaterials

Zhao

Meza

Hove

2006

J. Phys.: Condens. Matter

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Atomic scale surface structure plays an important role in describing many properties of materials, especially in the case of nanomaterials. One of the most effective techniques for surface structure determination is low-energy electron diffraction (LEED), which can be used in conjunction with optimization to fit simulated LEED intensities to experimental data. This optimization problem has a number of characteristics that make it challenging: it has many local minima, the optimization variables can be either continuous or categorical, the objective function can be discontinuous, there are no exact analytic derivatives (and no derivatives at all for categorical variables), and function evaluations are expensive. In this study, we show how to apply a particular class of optimization methods known as pattern search methods to address these challenges. These methods do not explicitly use derivatives, and are particularly appropriate when categorical variables are present, an important feature that has not been addressed in previous LEED studies. We have found that pattern search methods can produce excellent results, compared to previously used methods, both in terms of performance and locating optimal results.

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“…If the exact scalar field is known, we can use standard optimization techniques to find the best parameter values for that specific scalar field in the presence of the specific known flow field. Although gradient descent methods might be applicable here, an easier approach (that avoids having to calculate gradients) is to employ the direct search method (also known as compass search or pattern search) [21,22]. Furthermore, because it does not require gradient information, direct search is robust even when the objective function lacks smoothness or continuity.…”

Section: Parameter Optimizationmentioning

confidence: 99%

Interpolating sparse scattered data using flow information

Streletz

Gebbie

Kreylos

et al. 2016

Journal of Computational Science

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a b s t r a c tScattered data interpolation and approximation techniques allow for the reconstruction of a scalar field based upon a finite number of scattered samples of the field. In general, the fidelity of the reconstruction with respect to the original scalar field tends to deteriorate as the number of samples decreases. For the situation of very sparse sampling, the results may not be acceptable at all. However, if it is known that the scalar field of interest is correlated with a known flow field -as is the case when the scalar field represents the value of an oceanographic tracer that propagates under the influence of the ocean's flow -then this knowledge can be exploited to enhance the scattered data reconstruction method. One way to exploit flow field information is to use it to construct a modified notion of distance between points. Replacing the standard Euclidean distance metric with a flow-field-aware notion of distance provides a method for extending standard scattered data interpolation methods into flow-based methods that produce superior results for very sparse data. The resulting reconstructions typically have lower root-mean-square errors than reconstructions that do not use the flow information, and qualitatively they often appear physically more realistic.

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Optimization by Direct Search: New Perspectives on Some Classical and Modern Methods

Cited by 1,387 publications

References 183 publications

On NUFFT-based gridding for non-Cartesian MRI

On NUFFT-based gridding for non-Cartesian MRI

Using pattern search methods for surface structure determination of nanomaterials

Interpolating sparse scattered data using flow information

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