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The secant equation traditionally constitutes the basis of quasi-Newton methods, as the updated Hessian approximations satisfy the equation on each iteration. Modified versions of the secant relation have recently been the focus of several papers with encouraging outcomes. This paper continues with that idea where a secant-like modification that utilises nonlinear quantities in constructing the Hessian (or its inverse) approximation updates is derived. The technique takes advantage of data readily computed from the two most recent steps. Thus, it offers a substitute to the secant equation to produce better Hessian approximations that result in accelerated convergence to the objective function minimiser. The reported results provide adequate evidence to suggest that the proposed method is promising and deserves attention.

The Newton polytope <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Subscript f"> <mml:semantics> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>f</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">P_f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a polynomial <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is well known to have a strong impact on its behavior. The Bernstein-Kouchnirenko Theorem asserts that even the number of simultaneous zeros in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis double-struck upper C Superscript asterisk Baseline right-parenthesis Superscript m"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi>m</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">(\mathbb {C}^*)^m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a system of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> polynomials depends on their Newton polytopes. In this article, we show that Newton polytopes also have a strong impact on the distribution of zeros and pointwise norms of polynomials, the basic theme being that Newton polytopes determine allowed and forbidden regions in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis double-struck upper C Superscript asterisk Baseline right-parenthesis Superscript m"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi>m</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">(\mathbb {C}^*)^m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for these distributions. Our results are statistical and asymptotic in the degree of the polynomials. We equip the space of polynomials of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="less-than-or-equal-to p"> <mml:semantics> <mml:mrow> <mml:mo> ≤ </mml:mo> <mml:mi>p</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\leq p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> complex variables with its usual SU <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis m plus 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(m+1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -invariant Gaussian probability measure and then consider the conditional measure induced on the subspace of polynomials with fixed Newton polytope <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We then determine the asymptotics of the conditional expectation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper E Subscript vertical-bar upper N upper P Baseline left-parenthesis upper Z Subscript f 1 comma ellipsis comma f Sub Subscript k Subscript Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">E</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>N</mml:mi> <mml:mi>P</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>Z</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>f</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {E}_{|N P}(Z_{f_1, \dots , f_k})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of simultaneous zeros of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:

&lt;p&gt;The focus for quasi-Newton methods is the quasi-Newton equation. A new quasi-Newton equation is derived for quadratic function. Then, based on this new quasi-Newton equation, a new quasi-Newton updating formulas are presented. Under appropriate conditions, it is shown that the proposed method is globally convergent. Finally, some numerical experiments are reported which verifies the effectiveness of the new method.&lt;/p&gt;

&lt;span id="docs-internal-guid-a04d8b24-7fff-eaad-9449-fe4b2527904b"&gt;&lt;span&gt;Quasi-Newton method is an efficient method for solving unconstrained optimization problems. Self-scaling is one of the common approaches in the modification of the quasi-Newton method. A large variety of self-scaling of quasi-Newton methods is very well known. In this paper, based on quadratic function we derive the new self-scaling of quasi-Newton method and study the convergence property. Numerical results on the collection of problems showed the self-scaling of quasi-Newton methods which improves overall numerical performance for BFGS method.&lt;/span&gt;&lt;/span&gt;

&lt;span&gt;&lt;span&gt;Quasi-Newton methods are a class of numerical methods for &lt;/span&gt;solving the problem of unconstrained optimization. To improve the overall efficiency of resulting algorithms, we use the quasi-Newton methods which is interesting for quasi-Newton equation. In this manuscript, we present a modified BFGS update formula based on the new quasi-Newton equation, which give a new search direction for solving unconstrained optimizations proplems. We analyse the convergence rate of quasi-Newton method under some mild condition. Numerical experiments are conducted to demonstrate the efficiency of new methods using some test problems. The results indicates that the proposed method is competitive compared to the BFGS methods as it yielded fewer iteration and fewer function evaluations.&lt;/span&gt;

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We show that optomechanical systems can test the Schr\"odinger-Newton equation of gravitational quantum mechanics due to Yang et al. Phys. Rev. Lett. 110, 170401 (2013). This equation is motivated by semiclassical gravity, a widely used theory of interacting gravitational and quantum fields. From the many-body Schr\"odinger-Newton equation follows an approximate equation for the center-of-mass dynamics of macroscopic objects. This predicts a distinctive double-peaked signature in the output optical quadrature power spectral density of certain optomechanical systems. Since the Schr\"odinger-Newton equation lacks free parameters, these will allow its experimental confirmation or refutation.

We construct branched coverings such as matings and captures to describe the dynamics of every critically finite cubic Newton map. This gives a combinatorial model of the set of cubic Newton maps as the gluing of a subset of cubic polynomials with a part

Previous article Next article Extension of Davidon's Variable Metric Method to Maximization Under Linear Inequality and Equality ConstraintsDonald GoldfarbDonald Goldfarbhttps://doi.org/10.1137/0117067PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] M. J. Box, A comparison of several current optimization methods, and the use of transformations in constrained problems, Comput. J., 9 (1966), 67–77 MR0192645 0146.13304 CrossrefISIGoogle Scholar[2] Charles W. Carroll, The created response surface technique for optimizing nonlinear, restrained systems, Operations Res., 9 (1961), 169–185 MR0129020 0111.17004 CrossrefISIGoogle Scholar[3] R. Courant and , D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953xv+561 MR0065391 0051.28802 Google Scholar[4] W. C. Davidon, Variable metric method for minimization, Atomic Energy Commission Research Development Report, ANL-5990, 1959 Google Scholar[5] Jack B. 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Control, 4 (1966), 194–210 10.1137/0304019 MR0189832 0146.13303 LinkGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Exoskeleton kinematic design robustness: An assessment method to account for human variability4 November 2020 | Wearable Technologies, Vol. 1 Cross Ref FreePSI: an alignment-free approach to estimating exon-inclusion ratios without a reference transcriptome9 November 2017 | Nucleic Acids Research, Vol. 46, No. 2 Cross Ref A scaled three-term conjugate gradient method for unconstrained optimization13 December 2016 | Journal of Inequalities and Applications, Vol. 2016, No. 1 Cross Ref An active-set projected trust region algorithm for box constrained optimization problems28 November 2014 | Journal of Systems Science and Complexity, Vol. 28, No. 5 Cross Ref SOLVING A SPECIAL CLASS OF MULTIPLE OBJECTIVE LINEAR FRACTIONAL PROGRAMMING PROBLEMS9 October 2014 | The ANZIAM Journal, Vol. 56, No. 1 Cross Ref A New Procedure for Solving Integer 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Recherche opérationnelle, Vol. 8, No. V3 Cross Ref On the modification of 𝐿𝐷𝐿^{𝑇} factorizations1 January 1974 | Mathematics of Computation, Vol. 28, No. 128 Cross Ref A superlinearly convergent method for minimization problems with linear inequality constraintsMathematical Programming, Vol. 4, No. 1 Cross Ref A Kuhn–Tucker AlgorithmHilbert K. Schultz18 July 2006 | SIAM Journal on Control, Vol. 11, No. 3AbstractPDF (738 KB)Parameter Estimation in Spatial Interaction Modeling24 July 2016 | Environment and Planning A: Economy and Space, Vol. 5, No. 4 Cross Ref Experiment versus analysis: Computational techniques for the description of static material responseInternational Journal for Numerical Methods in Engineering, Vol. 5, No. 4 Cross Ref A variance algorithm for constrained minimization with linear constraintsJournal of Mathematical Analysis and Applications, Vol. 41, No. 2 Cross Ref Management scienceCommunications of the ACM, Vol. 15, No. 7 Cross Ref A New Tehoretical and Computational Approach to the Exact Solution of the Problem of the Optimal Control of Integrated (Hydro-Thermal) Power SystemsIFAC Proceedings Volumes, Vol. 5, No. 1 Cross Ref On the Convergence of the Conjugate Gradient Method for Singular Linear Operator EquationsW. J. Kammerer and M. Z. Nashed1 August 2006 | SIAM Journal on Numerical Analysis, Vol. 9, No. 1AbstractPDF (1764 KB)An efficient algorithm for minimizing barrier and penalty functionsMathematical Programming, Vol. 2, No. 1 Cross Ref An algorithm for solving linearly constrained optimization problemsMathematical Programming, Vol. 2, No. 1 Cross Ref Extension of techniques for Vol. 12, No. 1 Cross Ref Large optimization techniques to chemical and Journal of Chemical Engineering, Vol. No. 6 Cross Ref A method for structured nonlinear Programming, Vol. No. 1 Cross Ref GRADIENT AND GRADIENT OF of the American Water Resources Vol. No. 5 Cross Ref Some Algorithms Based on the of Feasible Cross Ref A Second Method for the Constrained Nonlinear Programming Problem Cross Ref for Feasible Direction Algorithms for Constrained Optimization Cross Ref of a for Cross Ref Optimization algorithms in 1979 Cross Ref Variable metric gradient projection method and replicator Cross Ref 17, Journal on Applied Mathematics History May July 2006 Society for Industrial and Applied & for Industrial and Applied Mathematics

Often in nature different systems interact, like fluids and structures, heat and electricity, populations of species, etc. It is our aim in this thesis to find, describe and analyze solution methods to solve the equations resulting from the mathematical models describing those interacting systems. Even if powerful solvers often already exist for problems in a single physical domain (e.g. structural or fluid problems), the development of similar tools for multi-physics problems is still ongoing. When the interaction (or coupling) between the two systems is strong, many methods still fail or are computationally very expensive. Approaches for solving these multi-physics problems can be broadly put in two categories: monolithic or partitioned. While we are not claiming that the partitioned approach is panacea for all coupled problems, we will only focus our attention in this thesis on studying methods to solve (strongly) coupled problems with a partitioned approach in which each of the physical problems is solved with a specialized code that we consider to be a black box solver and of which the Jacobian is unknown. We also assume that calling these black boxes is the most expensive part of any algorithm, so that performance is judged by the number of times these are called. In 2005 Vierendeels presented a new coupling procedure for this partitioned approach in a fluid-structure interaction context, based on sensitivity analysis of the important displacement and pressure modes which are detected during the iteration process. This approach only uses input-output couples of the solvers (one for the fluid problem and one for the structural problem). In this thesis we will focus on establishing the properties of this method and show that it can be interpreted as a block quasi-Newton method with approximate Jacobians based on a least squares formulation. We also establish and investigate other algorithms that exploit the original idea but use a single approximate Jacobian. The main focus in this thesis lies on establishing the algebraic properties of the methods under investigation and not so much on the best implementation form.

Quasi-Newton methods are among the most practical and efficient iterative methods for solving unconstrained minimization problems. In this paper we give an overview of some of these methods with focus primarily on the Hessian approximation updates and modifications aimed at improving their performance.

The emergence of OpenAI's ChatGPT has put intense spotlight on Generative AI (Gen-AI) systems and their possible impacts on Academic integrity. This paper provides an overview of the current arguments around ChatGPT and Academic integrity and concludes that although these technologies are capable of revolutionising academia, the way ChatGPT and other generative AI systems are used could surely undermine academic integrity. However, to ensure that the risks to academic integrity are mitigated for greater maximisation, institutional and multi-stakeholder efforts are required.

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Regional relative sea level rise is exacerbating flooding hazards in the coastal zone. In addition to changes in the ocean, vertical land motion (VLM) is a driver of spatial variation in sea level change that can either diminish or enhance flood risk. Here, we apply state-of-the-art interferometric synthetic aperture radar and global navigation satellite system time series analysis to estimate velocities and corresponding uncertainties at 30-m resolution in the New York City metropolitan area, revealing VLM with unprecedented detail. We find broad subsidence of 1.6 mm/year, consistent with glacial isostatic adjustment to the melting of the former ice sheets, and previously undocumented hot spots of both subsidence and uplift that can be physically explained in some locations. Our results inform ongoing efforts to adapt to sea level rise and reveal points of VLM that motivate both future scientific investigations into surface geology and assessments of engineering projects.

The status of experimental tests of general relativity and of theoretical frameworks for analyzing them is reviewed. Einstein's equivalence principle (EEP) is well supported by experiments such as the Eötvös experiment, tests of special relativity, and the gravitational redshift experiment. Ongoing tests of EEP and of the inverse square law are searching for new interactions arising from unification or quantum gravity. Tests of general relativity at the post-Newtonian level have reached high precision, including the light deflection, the Shapiro time delay, the perihelion advance of Mercury, and the Nordtvedt effect in lunar motion. Gravitational wave damping has been detected in an amount that agrees with general relativity to better than half a percent using the Hulse-Taylor binary pulsar, and other binary pulsar systems have yielded other tests, especially of strong-field effects. When direct observation of gravitational radiation from astrophysical sources begins, new tests of general relativity will be possible.

Abstract A large range of sophisticated brain image analysis tools have been developed by the neuroscience community, greatly advancing the field of human brain mapping. Here we introduce the Computational Anatomy Toolbox (CAT) – a powerful suite of tools for brain morphometric analyses with an intuitive graphical user interface, but also usable as a shell script. CAT is suitable for beginners, casual users, experts, and developers alike providing a comprehensive set of analysis options, workflows, and integrated pipelines. The available analysis streams – illustrated on an example dataset – allow for voxel-based, surface-based, as well as region-based morphometric analyses. Notably, CAT incorporates multiple quality control options and covers the entire analysis workflow, including the preprocessing of cross-sectional and longitudinal data, statistical analysis, and the visualization of results. The overarching aim of this article is to provide a complete description and evaluation of CAT, while offering a citable standard for the neuroscience community.

The atomic simulation environment (ASE) is a software package written in the Python programming language with the aim of setting up, steering, and analyzing atomistic simulations. In ASE, tasks are fully scripted in Python. The powerful syntax of Python combined with the NumPy array library make it possible to perform very complex simulation tasks. For example, a sequence of calculations may be performed with the use of a simple 'for-loop' construction. Calculations of energy, forces, stresses and other quantities are performed through interfaces to many external electronic structure codes or force fields using a uniform interface. On top of this calculator interface, ASE provides modules for performing many standard simulation tasks such as structure optimization, molecular dynamics, handling of constraints and performing nudged elastic band calculations.

Seven knees were studied to determine the contact area and pressure distribution of the tibiofemoral joint, under various loads and at 0 degrees flexion, using the casting method and special sensor sheets. At a load of 1000N (Newton) the contact area of the knee was 11.5 x 10(2) mm2 with menisci and 5.2 x 10(2) mm2 without menisci, and the menisci occupied 70% of the total contact area. Peak pressure at 1000N was 3MPa (Mega Pascal) with the menisci and 6MPa without them. The high pressure areas were located on the lateral meniscus as well as on the uncovered part of the articular cartilage of the lateral compartment, and on the uncovered cartilage in the medial compartment. After removal of the menisci the contact area decreased to below one half that of the intact knee and the contact pressure considerably increased. These facts imply that the menisci have load bearing and load spreading functions. The contact areas were also measured in two osteoarthrotic knees and they were significantly larger than those in normal knees. In these arthrotic knees the menisci seemed to play a less significant role in transmission of weight than in the normal knees.

Viewing the brain as a complex computer of simple neurons cannot account for consciousness nor essential features of cognition. Single cell organisms with no synapses perform purposeful intelligent functions using their cytoskeletal microtubules. A new paradigm is needed to view the brain as a scale-invariant hierarchy extending both upward from the level of neurons to larger and larger neuronal networks, but also downward, inward, to deeper, faster quantum and classical processes in cytoskeletal microtubules inside neurons. Evidence shows self-similar patterns of conductive resonances repeating in terahertz, gigahertz, megahertz, kilohertz and hertz frequency ranges in microtubules. These conductive resonances apparently originate in terahertz quantum dipole oscillations and optical interactions among pi electron resonance clouds of aromatic amino acid rings of tryptophan, phenylalanine and tyrosine within each tubulin, the component subunit of microtubules, and the brain’s most abundant protein. Evidence from cultured neuronal networks also now shows that gigahertz and megahertz oscillations in dendritic-somatic microtubules regulate specific firings of distal axonal branches, causally modulating membrane and synaptic activities. The brain should be viewed as a scale-invariant hierarchy, with quantum and classical processes critical to consciousness and cognition originating in microtubules inside neurons.