Saddle Point Structure Research Articles

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.

Asymmetric ImineN-Inversion in 3-Methyl-4-pyrimidinimine. Molecular Dipole Analysis of Solvation Effects

The thermal (E)/(Z)-isomerization of 3-methyl-4-pyrimidinimine, 3MePMI, has been studied in the gas phase at MP2/6-31G* and with the inclusion of medium effects using the polarizable continuum method, PCM(MP2/6-31G*), and the solvation model density method, SMD(MP2/6-31G*). For the free molecule and for 3MePMI in each of 14 solvents, the structures were determined of the (E)- and (Z)-isomers, of the transition state structure for isomerization ITS by asymmetric N-inversion, and of the second-order saddle point structure (SOSP) associated with in-plane N-inversion. The results predict a reduction of the (E)-isomer preference energy of 3MePMI, an increase of the deformation energy ΔE(def) = E(SOSP) - E(ITS), and an increase of the activation barrier E(act)(Z → E) with increasing solvent polarity. Electronic effects associated with N-inversion are analyzed using molecular orbital theory, results of population analysis, and electrostatic potential maps. The molecular dipole moments are superior parameters for the description of electronic relaxation in the imine basin during N-inversion. In particular, the analysis of dipole moments explains the compatibility of the increase of local CN polarity during N-inversion with the negative solvation effect on the activation barrier.

Ab initio characterization of the flexural [formula omitted] anion found in the reversible dehydrogenation

It was recently shown that the B 3 H 8 - anion can be produced during the reversible dehydrogenation of Mg(BH 4) 2 to Mg(B 3H 8) 2 in the solid under moderate conditions. In this letter we performed a global minimum search for the B 3 H 8 - anion and identified the global minimum structure and one low-lying saddle point structure, which explains the flexural behavior of this anion. Chemical bonding analysis performed by Adaptive Natural Density Partitioning method revealed that the B 3 H 8 - anion is a σ-aromatic species. Calculated VDEs for the global minimum structure can help to interpret future anionic photoelectron spectroscopic study of this anion.

First-principles calculations of impurity diffusion coefficients in dilute Mg alloys using the 8-frequency model

Diffusion in dilute Mg–X alloys, where X denotes Al, Zn, Sn and Ca impurities, was investigated with first-principles density functional theory in the local density approximation. Impurity diffusion coefficients were computed as a function of temperature using the 8-frequency model which provided the relevant impurity and solvent (Mg) jump frequencies and correlation factors. Minimum energy pathways for impurity diffusion and associated saddle point structures were computed with the climbing image nudged elastic band method. Vibrational properties were obtained with the supercell (direct) method for lattice dynamics. Calculated diffusion coefficients were compared with available experimental data. For diffusion between basal planes, we find D Mg–Ca > D Mg–Zn > D Mg–Sn > D Mg–Al, where D is the diffusion coefficient. For diffusion within a basal plane, the same trend holds except that D Mg–Zn overlaps with D Mg–Al at high temperatures and D Mg–Sn at low temperatures. These trends were explored with charge density contours in selected planes of each Mg–X alloy, the variation of the activation energy for diffusion with the atomic radius of each impurity and the electronic density of states. The theoretical methodology developed herein can be applied to impurity diffusion in other hexagonal materials.

Stability and preconditioning for a hybrid approximation on the sphere

This paper proposes a new preconditioning scheme for a linear system with a saddle-point structure arising from a hybrid approximation scheme on the sphere, an approximation scheme that combines (local) spherical radial basis functions and (global) spherical polynomials. In principle the resulting linear system can be preconditioned by the block-diagonal preconditioner of Murphy, Golub and Wathen. Making use of a recently derived inf–sup condition and the Brezzi stability and convergence theorem for this approximation scheme, we show that in this context the Schur complement in the above preconditioner is spectrally equivalent to a certain non-constant diagonal matrix. Numerical experiments with a non-uniform distribution of data points support the theoretically proved quality of the new preconditioner.

A priori and a posteriori error analyses of a velocity-pseudostress formulation for a class of quasi-Newtonian Stokes flows

In this paper we introduce and analyze new mixed finite element schemes for a class of nonlinear Stokes models arising in quasi-Newtonian fluids. The methods are based on a non-standard mixed approach in which the velocity, the pressure, and the pseudostress are the original unknowns. However, we use the incompressibility condition to eliminate the pressure, and set the velocity gradient as an auxiliary unknown, which yields a twofold saddle point operator equation as the resulting dual-mixed variational formulation. In addition, a suitable augmented version of the latter showing a single saddle point structure is also considered. We apply known results from nonlinear functional analysis to prove that the corresponding continuous and discrete schemes are well-posed. In particular, we show that Raviart–Thomas elements of order k ⩾ 0 for the pseudostress, and piecewise polynomials of degree k for the velocity and its gradient, ensure the well-posedness of the associated Galerkin schemes. Moreover, we prove that any finite element subspace of the square integrable tensors can be employed to approximate the velocity gradient in the case of the augmented formulation. Then, we derive a reliable and efficient residual-based a posteriori error estimator for each scheme. Finally, we provide several numerical results illustrating the good performance of the resulting mixed finite element methods, confirming the theoretical properties of the estimator, and showing the behaviour of the associated adaptive algorithms.

A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging

In this paper we study a first-order primal-dual algorithm for non-smooth convex optimization problems with known saddle-point structure. We prove convergence to a saddle-point with rate O(1/N) in finite dimensions for the complete class of problems. We further show accelerations of the proposed algorithm to yield improved rates on problems with some degree of smoothness. In particular we show that we can achieve O(1/N 2) convergence on problems, where the primal or the dual objective is uniformly convex, and we can show linear convergence, i.e. O(ω N ) for some ω∈(0,1), on smooth problems. The wide applicability of the proposed algorithm is demonstrated on several imaging problems such as image denoising, image deconvolution, image inpainting, motion estimation and multi-label image segmentation.

Finite element approximation of nematic liquid crystal flows using a saddle-point structure

In this work, we propose finite element schemes for the numerical approximation of nematic liquid crystal flows, based on a saddle-point formulation of the director vector sub-problem. It introduces a Lagrange multiplier that allows to enforce the sphere condition. In this setting, we can consider the limit problem (without penalty) and the penalized problem (using a Ginzburg–Landau penalty function) in a unified way. Further, the resulting schemes have a stable behavior with respect to the value of the penalty parameter, a key difference with respect to the existing schemes. Two different methods have been considered for the time integration. First, we have considered an implicit algorithm that is unconditionally stable and energy preserving. The linearization of the problem at every time step value can be performed using a quasi-Newton method that allows to decouple fluid velocity and director vector computations for every tangent problem. Then, we have designed a linear semi-implicit algorithm (i.e. it does not involve nonlinear iterations) and proved that it is unconditionally stable, verifying a discrete energy inequality. Finally, some numerical simulations are provided.

First-principles study of self-diffusion in hcp Mg and Zn

Self-diffusion coefficients for hcp Mg and Zn have been calculated from first-principles as a function of temperature, within the generalized gradient and local density approximations. The climbing image nudged elastic band (CI-NEB) method has been used to provide minimum energy pathways and associated saddle point structures. Vibrational properties have been calculated using the direct method to lattice dynamics. A good agreement between our calculated data and available experimental measurements has been obtained. For both Mg and Zn, LDA and GGA results form lower and upper bounds to the experimental data. Calculated results show that diffusion is faster in the basal plane in Mg and along the normal of the basal plane in Zn, in accordance with the available experimental data. Key differences in the diffusion anisotropy of both the metals have been examined in detail.

Theoretical study on the mechanism of double proton transfer in porphycene by path-integral molecular dynamics simulations

Full-dimensional path-integral molecular dynamics simulations were performed to determine whether the double proton transfer tautomerization of porphycene is a concerted or a stepwise process. We employed an on-the-fly direct dynamics technique at the semiempirical PM6 method whose parameters were determined so as that the relative energies of the stationary points approximately reproduce previously reported electronic structure calculations. It was found that double proton transfer occurs dominantly through the concerted pathway via the second-order saddle point structure and that contribution of the stepwise mechanism increases with a temperature increase. Nuclear quantum effects play essential roles in determining the proton transfer mechanism.

Anomalies in the response of V, Nb, and Ta to tensile and shear loading:Ab initiodensity functional theory calculations

Ab initio density functional calculations of the response of V, Nb, and Ta to tensile and shear loading have been performed. We find that the behavior of all three metals of the V group under large strains differs considerably from that reported for the other body-centered cubic (bcc) transition metals (Mo, W, and Fe). Under unaxial $⟨100⟩$ tensile loading, V and Nb undergo a bifurcation from a tetragonal to an orthorhombic deformation path, associated with a shear instability, before reaching the stress maximum. The bifurcation strongly reduces the ideal tensile strengths to 11.5 GPa for V and 12.5 GPa for Nb. For Ta the bifurcation point coincides with the stress maximum of 13.6 GPa along the tetragonal path so that the metal fails under tensile strain by shear and not by cleavage. The stress-strain curves calculated for the ${110}⟨111⟩$ and ${211}⟨111⟩$ slip systems are strongly asymmetric; the ideal shear strengths for both slip systems are 6.5 (5.5) GPa for V and 7.8 (6.0) GPa for Nb. Hence the ideal shear strengths are only half as large as for the bcc metals Mo and W, but much larger than expected on the basis for the low shear moduli of V and Nb. Ta shows a behavior intermediate between these two groups of metals, with a calculated ideal shear strength of 7.1 (6.5) GPa, in excellent agreement with the experimental estimate from nanoindentation experiments. The saddle-point structure determining the shear strength for both slip systems is a special body-centered-tetragonal structure. This structure is identical to the saddle-point structure identified on the orthorhombic deformation path under uniaxial tension. Hence also V and Nb will fail under uniaxial tension not by cleavage, but by shear.

Preconditioning Stochastic Galerkin Saddle Point Systems

Mixed finite element discretizations of deterministic second-order elliptic PDEs lead to saddle point systems for which the study of iterative solvers and preconditioners is mature. Galerkin approximation of solutions of stochastic second-order elliptic PDEs, which couple standard mixed finite element discretizations in physical space with global polynomial approximation on a probability space, also give rise to linear systems with familiar saddle point structure. For stochastically nonlinear problems, the solution of such systems presents a serious computational challenge. The blocks are sums of Kronecker products of pairs of matrices associated with two distinct discretizations, and the systems are large, reflecting the curse of dimensionality inherent in most stochastic approximation schemes. Moreover, for the problems considered herein, the leading blocks of the saddle point matrices are block-dense, and the cost of a matrix vector product is nontrivial. We implement a stochastic Galerkin discretization for the steady-state diffusion problem written as a mixed first-order system. The diffusion coefficient is assumed to be a lognormal random field, approximated via a nonlinear function of a finite number of Gaussian random variables. We study the resulting saddle point systems and investigate the efficiency of block-diagonal preconditioners of Schur complement and augmented type for use with the minimal residual method (MINRES). By introducing so-called Kronecker product preconditioners, we improve the robustness of cheap, mean-based preconditioners with respect to the statistical properties of the stochastically nonlinear diffusion coefficients.

The photoelectron spectrum of the ethoxide anion: Conical intersections, the spin-orbit interaction, and sequence bands

The negative ion photoelectron spectrum of the ethoxide anion (ethoxide-h(5)) and that of its fully deuterated analog, ethoxide-d(5), are calculated using the multimode vibronic coupling approach. A two state quasidiabatic Hamiltonian H(d) is constructed which includes all terms through second order in the full 18 dimensional internal coordinate space. H(d) is centered at the ab initio determined minimum energy crossing (MEX) point on the symmetry-allowed (2)A(")-(2)A(') accidental seam of conical intersection and determined from ab initio energy gradients and derivative couplings. It reproduces the local topography of the (2)A(")-(2)A(') MEX, in addition to accurately representing the geometries, energetics, and harmonic frequencies of equilibrium and saddle point structures located on the ground electronic state potential energy surface in the vicinity of the MEX. Spin-orbit effects are included. The results for ethoxy-h(5) are compared to photoelectron and slow electron velocity-map imaging (SEVI) spectra. By comparing the measured and predicted photoelectron spectrum, the accuracy of the electronic structure treatment is inferred. The existence of sequence bands in the SEVI spectrum is established.

The Isomerization Barrier in Cyanocyclobutadienes: An ab Initio Multireference Average Quadratic Coupled Cluster Study

The energy profiles of the isomerization of mono, di-, and tetracyano-substituted cyclobutadienes (CBDs) are computed at the multireference average quadratic coupled cluster/complete active space self-consistent field level of theory. It was found that the energy barrier heights for the automerization reaction are 2.6 (tetracyano-CBD), 5.1 (1,3-dicyano-CBD), and 6.4 (cyano-CBD) kcal mol(-1), implying that they are lowered relative to that in the parent CBD (6.4 kcal mol(-1)), the monosubstituted derivative being an exception. Since the free CBD shuttles between two equivalent structures even at low temperature of 10 K, it follows that bond-stretch isomerism does not take place in cyanocyclobutadienes. Instead, these compounds exhibit rapid fluxional interconversion at room temperature between two bond-stretch isomers by the double bond flipping mechanism. The reason behind the decrease in the barrier heights is identified as a slightly enhanced resonance effect at the saddle points separating two (equivalent) bond-stretch isomers, compared to that in the equilibrium structures, predominantly due to the diradical character of the former. It is also shown that the energy gap between the singlet ground state saddle point structure and the first triplet equilibrium geometry decreases upon multiple substitution by the cyano groups. The splitting of the S and T energy is small being within the range of 6.5-8.2 kcal mol(-1).

An 𝐸-based mixed formulation for a time-dependent eddy current problem

In this paper, we analyze a mixed form of a time-dependent eddy current problem formulated in terms of the electric fieldE\boldsymbol {E}. We show that this formulation admits a well-posed saddle point structure when the constraints satisfied by the primary unknown in the dielectric material are handled by means of a Lagrange multiplier. We use Nédélec edge elements and standard nodal finite elements to define a semi-discrete Galerkin scheme. Furthermore, we introduce the corresponding backward-Euler fully-discrete formulation and prove error estimates.

Duality, magnetic space group and their applications to quantum phases and phase transitions on bipartite lattices in several experimental systems

By using a dual vortex method, we study phases such as superfluid, solids, supersolids and quantum phase transitions in a unified scheme in extended boson Hubbard models at and slightly away from half filling on bipartite optical lattices such as honeycomb and square lattice. We also map out its global phase diagram at T = 0 of chemical potential versus the ratio of kinetic energy over the interaction. We stress the importance of the self-consistence condition on the saddle point structure of the dual gauge fields in the translational symmetry breaking insulating sides, especially in the charge density wave side. We find that in the translational symmetry breaking side, different kinds of supersolids are generic possible states slightly away from half filling. We propose a new kind of supersolid: valence bond supersolid (VB-SS). In this VB-SS, the density fluctuation at any site is very large indicating its superfluid nature, but the boson kinetic energies on bonds between two sites are given and break the lattice translational symmetries indicating its valence bound nature. We show that 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 a logarithmic correction. Comparisons with previous quantum Monte Carlo (QMC) simulations on a square lattice are made. Implications on possible future QMC simulations in both bipartite lattices are given. All these phases and phase transitions can be potentially realized in ultra-cold atoms loaded on optical bipartite lattices. Then we apply our results to investigate the reentrant “superfluid” in a narrow region of coverages in the second layer of 4He adsorbed on graphite and the low temperature phase diagram of hydrogen physisorbed on krypton-preplated graphite (H 2/Kr/graphite) near half filling. We suggest that 4He and H 2 lattice supersolids maybe responsible for the experimental signals in the two systems. Finally, we suggest Cooper supersolid is repressible for the phase diagram of La 2− x Ba x CuO 4 near x = 1 / 8 .

Exploring and exploiting the structure of saddle points in Gaussian scale space

When an image is filtered with a Gaussian of width σ and σ is considered as an extra dimension, the image is extended to a Gaussian scale-space ( GSS ) image. In earlier work it was shown that the GSS -image contains an intensity-based hierarchical structure that can be represented as a binary ordered rooted tree. Key elements in the construction of the tree are iso-intensity manifolds and scale-space saddles. A scale-space saddle is a critical point in scale space. When it connects two different parts of an iso-intensity manifold, it is called “dividing”, otherwise it is called “void”. Each dividing scale-space saddle is connected to an extremum in the original image via a curve in scale space containing critical points. Using the nesting of the iso-intensity manifolds in the GSS -image and the dividing scale-space saddles, each extremum is connected to another extremum. In the tree structure, the dividing scale-space saddles form the connecting elements in the hierarchy: they are the nodes of the tree. The extrema of the image form the leaves, while the critical curves are represented as the edges. To identify the dividing scale-space saddles, a global investigation of the scale-space saddles and the iso-intensity manifolds through them is needed. In this paper an overview of the situations that can occur is given. In each case it is shown how to distinguish between void and dividing scale-space saddles. Furthermore, examples are given, and the difference between selecting the dividing and the void scale-space saddles is shown. Also relevant geometric properties of GSS images are discussed, as well as their implications for algorithms used for the tree extraction. As main result, it is not necessary to search through the whole GSS image to find regions related to each relevant scale-space saddle. This yields a considerable reduction in complexity and computation time, as shown in two examples.

A Unified Perspective on the Hydrogen Atom Transfer and Proton-Coupled Electron Transfer Mechanisms in Terms of Topographic Features of the Ground and Excited Potential Energy Surfaces As Exemplified by the Reaction between Phenol and Radicals

The relation between the hydrogen atom transfer (HAT) and proton-coupled electron transfer (PCET) mechanisms is discussed and is illustrated by multiconfigurational electronic structure calculations on the ArOH + R(*) --> ArO(*) + RH reactions. The key topographic features of the Born-Oppenheimer potential energy surfaces that determine the predominant reaction mechanism are the conical intersection seam of the two lowest states and reaction saddle points located on the shoulders of this seam. The saddle point corresponds to a crossing of two interacting valence bond states corresponding to the reactant and product bonding patterns, and the conical intersection corresponds to the noninteracting intersection of the same two diabatic states. The locations of mechanistically relevant conical intersection structures and relevant saddle point structures are presented for the reactions between phenol and the N- and O-centered radicals, (*)NH2 and (*)OOCH3. Points on the conical intersection of the ground doublet D0 and first excited doublet D1 states are found to be in close geometric and energetic proximity to the reaction saddle points. In such systems, either the HAT mechanism or both the HAT mechanism and the proton-coupled electron transfer (PCET) mechanism can take place, depending on the relative energetic accessibility of the reaction saddle points and the D0/D1 conical intersection seams. The discussion shows how the two mechanisms are related and how they blend into each other along intermediate reaction paths. The recognition that the saddle point governing the HAT mechanism is on the shoulder of the conical intersection governing the PCET mechanism is used to provide a unified view of the competition between the two mechanisms (and the blending of the two mechanisms) in terms of the prominent and connected features of the potential energy surface, namely the saddle point and the conical intersection. The character of the dual mechanism may be understood in terms of the dominant valence bond configurations of the intersecting states, which are zero-order approximations to the diabatic states.

Confinement, screening and the center onS3×S1

We compute the one-loop effective potential for the Polyakov loop on S3 × S1 for an asymptotically free gauge theory of arbitrary group G and a generic matter content. We apply this result to study the phase structures of G2, SO(N) and Sp(2N) gauge theories which turn out to be in qualitative agreement with the results of lattice calculations. On S3 × S1, at zero temperature, the Polyakov loop is zero for kinematical reasons. For small but non-zero temperature, the Polyakov loop is still zero if the gauge theory has an unbroken center, while it acquires a small vacuum expectation value for gauge theories whose center is trivial or explicitly broken by the presence of dynamical matter fields. At high temperatures, the saddle point structure of the effective potential is different from the low temperature case suggesting that the theory is in the deconfined phase. At finite N, the Polyakov loop is non-zero in the high temperature phase only in theories with no unbroken center symmetry, consistent with screening of the external charge introduced by the Polyakov loop and Gauss law on a compact space.

Coupled time‐domain analysis with boundary and finite elements

AbstractA methodology for the combination of boundary and finite element discretizations for the numerical analysis of time‐dependent problems is presented. The interface conditions arising from the partitioning of the problem are incorporated in a weak form by means of Lagrange multiplier fields and, therefore, allow for nonconform interface discretizations. The resulting system matrices have the same saddle point structure as in the FETI method. Possible applications of the proposed method are the dynamic analysis of soil‐structure interaction and similar wave propagation phenomena in unbounded media. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)