Saddle Point Structure Research Articles

A Petrov–Galerkin reduced basis approximation of the Stokes equation in parameterized geometries

We present a Petrov–Galerkin reduced basis (RB) approximation for the parameterized Stokes equation. Our method, which relies on a reduced solution space and a parameter-dependent test space, is shown to be stable (in the sense of Babuška) and algebraically stable (a bound on the condition number of the online system can be established). Compared to other stable RB methods that can also be shown to be algebraically stable, our approach is among those with the smallest online time cost and it has general applicability to linear non-coercive problems without assuming a saddle-point structure.

Ab initio water pair potential with flexible monomers.

A potential energy surface for the water dimer with explicit dependence on monomer coordinates is presented. The surface was fitted to a set of previously published interaction energies computed on a grid of over a quarter million points in the 12-dimensional configurational space using symmetry-adapted perturbation theory and coupled-cluster methods. The present fit removes small errors in published fits, and its accuracy is critically evaluated. The minimum and saddle-point structures of the potential surface were found to be very close to predictions from direct ab initio optimizations. The computed second virial coefficients agreed well with experimental values. At low temperatures, the effects of monomer flexibility in the virial coefficients were found to be much smaller than the quantum effects.

Towards Textbook Efficiency for Parallel Multigrid

In this work, we extend Achi Brandt's notion of textbook multigrid efficiency (TME) to massively parallel algorithms. Using a finite element based geometric multigrid implementation, we recall the classical view on TME with experiments for scalar linear equations with constant and varying coefficients as well as linear systems with saddle-point structure. To extend the idea of TME to the parallel setting, we give a new characterization of a work unit (WU) in an architecture-aware fashion by taking into account performance modeling techniques. We illustrate our newly introduced parallel TME measure by large-scale computations, solving problems with up to 200 billion unknowns on a TOP-10 supercomputer.

Smooth roughness of exponential dichotomies, revisited

As a direct consequence of well-established proof techniques, we establish that the invariant projectors of exponential dichotomies for parameter-dependent nonautonomous difference equations are as smooth as their right-hand sides. For instance, this guarantees that the saddle-point structure in the vicinity of hyperbolic solutions inherits its differentiability properties from the particular given equation.

Moving Dirichlet boundary conditions

This paper develops a framework to include Dirichlet boundary conditions on a subset of the boundary which depends on time. In this model, the boundary conditions are weakly enforced with the help of a Lagrange multiplier method. In order to avoid that the ansatz space of the Lagrange multiplier depends on time, a bi-Lipschitz transformation, which maps a fixed interval onto the Dirichlet boundary, is introduced. An inf-sup condition as well as existence results are presented for a class of second order initial-boundary value problems. For the semi-discretization in space, a finite element scheme is presented which satisfies a discrete stability condition. Because of the saddle point structure of the underlying PDE, the resulting system is a DAE of index 3.

Local structural excitations in model glasses

Structural excitations of model Lennard-Jones glass systems are investigated using the activation-relaxation technique (ART$n$), which explores the potential energy landscape around a local minimum energy configuration by converging to a nearby saddle-point configuration. Performing ART$n$ results in a distribution of barrier energies that is single-peaked for well-relaxed samples. The present work characterizes such atomic-scale excitations in terms of their local structure and environment. It is found that, at zero applied stress, many of the identified events consist of chainlike excitations that can either be extended or ringlike in their geometry. The location and barrier energy of these saddle-point structures are found to correlate with the type of atom involved, and with spatial regions that have low Kelvin eigenshear moduli and are close to the excess free volume within the configuration. Such correlations are, however, weak and more generally the identified local structural excitations are seen to exist throughout the model glass sample. The work concludes with a discussion within the framework of $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ relaxation processes that are known to occur in the undercooled liquid regime.

Sampling saddle points on a free energy surface.

Many problems in biology, chemistry, and materials science require knowledge of saddle points on free energy surfaces. These saddle points act as transition states and are the bottlenecks for transitions of the system between different metastable states. For simple systems in which the free energy depends on a few variables, the free energy surface can be precomputed, and saddle points can then be found using existing techniques. For complex systems, where the free energy depends on many degrees of freedom, this is not feasible. In this paper, we develop an algorithm for finding the saddle points on a high-dimensional free energy surface "on-the-fly" without requiring a priori knowledge the free energy function itself. This is done by using the general strategy of the heterogeneous multi-scale method by applying a macro-scale solver, here the gentlest ascent dynamics algorithm, with the needed force and Hessian values computed on-the-fly using a micro-scale model such as molecular dynamics. The algorithm is capable of dealing with problems involving many coarse-grained variables. The utility of the algorithm is illustrated by studying the saddle points associated with (a) the isomerization transition of the alanine dipeptide using two coarse-grained variables, specifically the Ramachandran dihedral angles, and (b) the beta-hairpin structure of the alanine decamer using 20 coarse-grained variables, specifically the full set of Ramachandran angle pairs associated with each residue. For the alanine decamer, we obtain a detailed network showing the connectivity of the minima obtained and the saddle-point structures that connect them, which provides a way to visualize the gross features of the high-dimensional surface.

A first-principles study of self-diffusion coefficients of fcc Ni

Self-diffusion coefficients for fcc Ni are obtained as a function of temperature by first-principles calculations based on density functional theory within the local density approximation. To provide the minimum energy pathway and the associated saddle point structures of an elementary atomic jump, the nudged elastic band method is employed. Two magnetic settings, ferromagnetic and non-magnetic, and two vibrational contribution calculation methods, the phonon supercell approach and the Debye model, create four calculation settings for the self-diffusion coefficient in nickel. The results from these four methods are compared to each other and presented with the known experimental data. The use of the Debye model in lieu of the phonon supercell approach is shown to be a viable and computationally time saving alternative for the finite temperature thermodynamic properties. Consistent with other observations in the literature, the use of the phonon supercell approach for the calculation of the finite temperature thermodynamic properties within the LDA results in an underestimation of the diffusion coefficient with respect to experimental data. The calculated Ni self-diffusion coefficients for all four conditions in the present work are compared to a statistical consensus analysis previously performed on all known experimental self-diffusion data. With the exception of the NM phonon setting, the other three conditions fall within the 95% confidence interval for the consensus analysis.

Weighted Toeplitz Regularized Least Squares Computation for Image Restoration

The main aim of this paper is to develop a fast algorithm for solving weighted Toeplitz regularized least squares problems arising from image restoration. Based on augmented system formulation, we develop new Hermitian and skew-Hermitian splitting (HSS) preconditioners for solving such linear systems. The advantage of the proposed preconditioner is that the blurring matrix, weighting matrix, and regularization matrix can be decoupled such that the resulting preconditioner is not expensive to use. We show that for a preconditioned system that is derived from a saddle point structure of size $(m+n)\times(m+n)$, the preconditioned matrix has an eigenvalue at 1 with multiplicity $n$ and the other $m$ eigenvalues of the form $1 - \lambda$ with $|\lambda| < 1$. We also study how to choose the HSS parameter to minimize the magnitude of $\lambda$, and therefore the Krylov subspace method applied to solving the preconditioned system converges very quickly. Experimental results for image restoration problems are re...

First-principles study of temperature-dependent diffusion coefficients for helium in α-Ti

The temperature-dependent diffusion coefficients of interstitial helium atom in α-Ti are predicted using the transition state theory. The microscopic parameters in the pre-factor and activation energy of the impurity diffusion coefficients are obtained from first-principles total energy and phonon calculations including the full coupling between the vibrational modes of the diffusing atom and the host lattice. The climbing image nudged elastic band method is used to search for the minimum energy pathways and associated saddle point structures. It is demonstrated that the diffusion coefficients within the xy plane (Dxy) is always higher than that along the z axis (Dz), showing remarkable anisotropy. Also, it is found that the formation of helium dimer centered at the octahedral site reduces the total energy and confines the diffusion of helium atoms.

Quantum Thermochemistry: Multistructural Method with Torsional Anharmonicity Based on a Coupled Torsional Potential.

We present a new approximation for calculating partition functions and thermodynamic functions by the multistructural method with torsional anharmonicity (MS-T). The new approximation is based on a reference potential with torsional barriers obtained from a calculation that includes local torsional coupling. By comparing to a fully coupled classical rotational-torsional partition function evaluated as a numerical phase space integral, the method is shown to provide improved accuracy in the classical limit. Quantum effects, which are most important at low temperatures, are included based on the harmonic approximation (which can be upgraded to a quasiharmonic approximation, that is, harmonic formulas with effective frequencies). Calculations were performed for six molecules (ethanol, 1-butanol, hexane, isohexane, heptane, and isoheptane), one radical (1-pentyl radical), and the saddle point structures of a hydrogen abstraction reaction (hydroxyl plus ethanol) to illustrate the difference between the new coupled-potential MS-T approximation and the original uncoupled-potential MS-T approximation. The new method improves the agreement with experimental results of calcuated thermodynamic functions for 1-butanol, hexane, isohexane, and heptane.

Unconventional Peroxy Chemistry in Alcohol Oxidation: The Water Elimination Pathway.

Predictive simulation for designing efficient engines requires detailed modeling of combustion chemistry, for which the possibility of unknown pathways is a continual concern. Here, we characterize a low-lying water elimination pathway from key hydroperoxyalkyl (QOOH) radicals derived from alcohols. The corresponding saddle-point structure involves the interaction of radical and zwitterionic electronic states. This interaction presents extreme difficulties for electronic structure characterizations, but we demonstrate that these properties of this saddle point can be well captured by M06-2X and CCSD(T) methods. Experimental evidence for the existence and relevance of this pathway is shown in recently reported data on the low-temperature oxidation of isopentanol and isobutanol. In these systems, water elimination is a major pathway, and is likely ubiquitous in low-temperature alcohol oxidation. These findings will substantially alter current alcohol oxidation mechanisms. Moreover, the methods described will be useful for the more general phenomenon of interacting radical and zwitterionic states.

Analysis of the Minimal Residual Method Applied to Ill Posed Optimality Systems

We analyze the performance of the minimal residual (MINRES) method applied to linear Karush--Kuhn--Tucker systems arising in connection with inverse problems. Such optimality systems typically have a saddle point structure and have unique solutions for all $\alpha>0$, where $\alpha$ is the parameter employed in the Tikhonov regularization. Unfortunately, the associated spectral condition number is very large for small values of $\alpha$, which strongly indicates that their numerical treatment is difficult. Our main result shows that a broad range of linear ill posed optimality systems can be solved efficiently with the MINRES method. This result is obtained by carefully analyzing the spectrum of the associated saddle point operator: Except for a few isolated eigenvalues, the spectrum consists of three bounded intervals. Krylov subspace methods handle such problems very well. For severely ill posed cases, techniques based on Chebyshev polynomials are applied to prove that the number of iterations needed by...

Quantum phases, supersolids and quantum phase transitions of interacting bosons in frustrated lattices

By using the dual vortex method (DVM), we develop systematically a simple and effective scheme to use the vortex degree of freedoms on dual lattices to characterize the symmetry breaking patterns of the boson insulating states in the direct lattices. Then we apply our scheme to study quantum phases and phase transitions in an extended boson Hubbard model slightly away from 1/3 (2/3) filling on frustrated lattices such as triangular and Kagome lattice. In a triangular lattice at 1/3, we find a X-CDW, a stripe CDW phase which was found previously by a density operator formalism (DOF). Most importantly, we also find a new CDW-VB phase which has both local CDW and local VB orders, in sharp contrast to a bubble CDW phase found previously by the DOF. In the Kagome lattice at 1/3, we find a VBS phase and a 6-fold CDW phase. Most importantly, we also identify a CDW-VB phase which has both local CDW and local VB orders which was found in previous QMC simulations. We also study several other phases which are not found by the DVM. By analyzing carefully the saddle point structures of the dual gauge fields in the translational symmetry breaking sides and pushing the effective actions slightly away from the commensurate filling f=1/3(2/3), we classified all the possible types of supersolids and analyze their stability conditions. In a triangular lattice, there are X-CDW supersolid, stripe CDW supersolid, but absence of any valence bond supersolid (VB-SS). There are also a new kind of supersolid: CDW-VB supersolid. In a Kagome lattice, there are 6-fold CDW supersolid, stripe CDW supersolid, but absence of any valence bond supersolid (VB-SS). There are also a new kind of supersolid: CDW-VB supersolid. We show that independent of the types of the SS, the quantum phase transitions from solids to supersolids driven by a chemical potential are in the same universality class as that from a Mott insulator to a superfluid, therefore have exact exponents z=2, ν=1/2, η=0 (with logarithmic corrections). Excitation spectra of all these insulating phases and supersolid phases are also studied. Implications on QMC simulations with both nearest neighbor and next nearest neighbor interactions in both lattices are given. Some possible intrinsic problems of the DOF in identifying the insulating phases are also pointed out.

Dynamics, transition states, and timing of bond formation in Diels–Alder reactions

The time-resolved mechanisms for eight Diels–Alder reactions have been studied by quasiclassical trajectories at 298 K, with energies and derivatives computed by UB3LYP/6-31G(d). Three of these reactions were also simulated at high temperature to compare with experimental results. The reaction trajectories require 50–150 fs on average to transverse the region near the saddle point where bonding changes occur. Even with symmetrical reactants, the trajectories invariably involve unequal bond formation in the transition state. Nevertheless, the time gap between formation of the two new bonds is shorter than a C─C vibrational period. At 298 K, most Diels–Alder reactions are concerted and stereospecific, but at high temperatures (approximately 1,000 K) a small fraction of trajectories lead to diradicals. The simulations illustrate and affirm the bottleneck property of the transition state and the close connection between dynamics and the conventional analysis based on saddle point structure.

Investigation of attachment saddle point structure of 3-D steady separation in laminar juncture flow using PIV

An experimental study of laminar horseshoe vortex upstream of a cylinder/flat plate juncture has been conducted using PIV in symmetric plane to verify the existence of attachment saddle point topology. The experimental results confirm the existence of attachment saddle point structure, which is different from conventional separation saddle point flow topology. In one, two and three primary vortices condition, upstream streamlines near surface do not rise up or separate from the plate surface but attach to plate surface. The outmost vortex initiates from the spatial singular point and not from surface singular point. Especially, the skin-friction-lines convergence corresponds to spatial attached flow but not separated flow as in case of conventional separation saddle point topology, alternatively, the spatial attached flow correspond to convergent but not divergent skin-friction-lines, as happens in conventional attachment flows. The results indicate that attachment saddle point structure is a kind of separation in which vorticity separates from spatial layer of boundary layer but not from the plate surface. The study also reveals three types of attachment saddle point topologies, these three topologies all satisfy the topological, singular point index rule.

Differentiation of ferrocene D5d and D5h conformers using IR spectroscopy

Fingerprinting in the infrared (IR) spectral region of 450–500 cm−1 of ferrocene is discovered to be a sensitive way to differentiate the eclipsed (D5h) and staggered (D5d) conformers of ferrocene. The present study simulates accurate IR spectra for ferrocene using density functional theory (DFT) based on the B3LYP/m6-31G(d) model without any scaling and manipulation. The agreement with an early experiment and other theory is excellent. It is found that in the vacuum, the eclipsed conformer represents the true minimum structure of ferrocene, whereas the staggered conformer represents the saddle point structure, in agreement with a number of other theoretical and experimental studies. The study further reveals that the sandwich complexes are formed by stacking two Cp rings with an Fe atom in the middle and there are no conventionally localised Fe–C bonds in ferrocene. The vibrational frequency splitting of Δυ = 17 cm−1 in the IR region of 450–500 cm−1 (expt. Δυ = 16 cm−1) becomes the fingerprint of the eclipsed conformer in contrast with the staggered conformer of ferrocene. In addition, the present study suggests that the earlier IR spectral measurement of Lippincott and Nelson (1958) of ferrocene was not D5d alone but was likely a mixture of both eclipsed and staggered ferrocene. Finally, the study highlights that Fe-related properties of ferrocene hold the key to reveal one ferrocene conformer from the other.

A New Class of Stabilized WEB-Spline-Based Mesh-Free Finite Elements for the Approximation of Maxwell Equations

The objective of this article is to discuss the existence and the uniqueness of a weighted extended B-spline-(WEB-spline) based discrete solution for the Maxwell equations in low frequency limit. The domain is composed of insulating and conducting regions. This problem has saddle point structure where the electric field in insulating region is the Lagrange multiplier that forces curl-free constraint on the magnetic field.

Spin waves in the block checkerboard antiferromagnetic phase

Motivated by the discovery of a new family of 122 iron-based superconductors, we present the theoretical results on the ground state phase diagram, spin wave, and dynamic structure factor obtained from the extended J1−J2 Heisenberg model. In the reasonable physical parameter region of K2Fe4Se5, we find that the block checkerboard antiferromagnetic order phase is stable. There are two acoustic spin wave branches and six optical spin wave branches in the block checkerboard antiferromagnetic phase, which have analytic expressions at the high-symmetry points. To further compare the experimental data on neutron scattering, we investigate the saddlepoint structure of the magnetic excitation spectrum and the inelastic neutron scattering pattern based on linear spin wave theory.

On the emergence of competitive equilibrium growth cycles

A basic discrete-time heterogeneous capital goods competitive environment is considered, its potential for displaying steady growth solutions analyzed and the properties of the latter characterized. A first composite good may be used for consumption or investment on a one-to-one basis, while a second good is only used for accumulation, solely capital inputs being part of the production process. This framework is first considered from the allocative standpoint through the derivation of the frontier of the production possibility set. Having defined the perfect foresight competitive equilibrium that also describes a Pareto optimum over time, attention is then given to the potential for unbounded steady growth solutions. Under interiority, summability, and expansivity restrictions, there is a unique optimal steady growth rate. For unitary depreciation rates of both capital goods, locally there exists a unique convergent sequence to this steady growth solution that exhibits a saddlepoint structure. However, as soon as one of the depreciation rates of the capital goods is non-unitary and the profit share accruing to the first capital stock is greater in the second pure accumulation industry than in the first composite good industry, the steady growth solution shows a loss of stability, and competitive equilibrium growth cycles emerge through the occurrence of a flip bifurcation in its neighborhood. This is the first optimal cycles result based upon a production set that does not explicitly incorporate any exogenously determined primary labor input in its definition.