Saddle-point Equations Research Articles

Linear Quadratic Differential Games: Saddle Point and Riccati Differential Equation

Zhang [SIAM J. Control Optim., 43 (2005), pp. 2157–2165] recently established the equivalence between the finiteness of the open loop value of a two-player zero-sum linear quadratic (LQ) game and the finiteness of its open loop lower and upper values. In this paper we complete and sharpen the results of Zhang for the finiteness of the lower value of the game by providing a set of necessary and sufficient conditions that emphasizes the feasibility condition: $(0,0)$ is a solution of the open loop lower value of the game for the zero initial state. Then we show that, under the assumption of an open loop saddle point in the time horizon $[0,T]$ for all initial states, there is an open loop saddle point in the time horizon $[s,T]$ for all initial times s, $0\le s<T$, and all initial states at time s. From this we get an optimality principle and adapt the invariant embedding approach to construct the decoupling symmetrical matrix function $P(s)$ and show that it is an $H^1(0,T)$ solution of the matrix Riccati differential equation. Thence an open loop saddle point in $[0,T]$ yields closed loop optimal strategies for both players. Furthermore, a necessary and sufficient set of conditions for the existence of an open loop saddle point in $[0,T]$ for all initial states is the convexity-concavity of the utility function and the existence of an $H^1(0,T)$ symmetrical solution to the matrix Riccati differential equation. As an illustration of the cases where the open loop lower/upper value of the game is $-\infty$/$+\infty$, we work out two informative examples of solutions to the Riccati differential equation with and without blow-up time.

Non-perturbative phenomena in the three-dimensional random field Ising model

A systematic approach to the calculations of the non-perturbative contributionsto the free energy in the ferromagnetic phase of the random field Ising modelis developed. It is demonstrated that such contributions appear due toinstanton-like excitations localized in space, which exist only in dimensionsD≤3.It is shown that away from the critical region such instanton solutions are described by the set ofthe mean-field saddle-point equations for the replica vector order parameter, and these equationscan be formally reduced to the only saddle-point equation of the pure system in dimensions(D−2). In themarginal case, D = 3, the corresponding non-analytic contribution is computed explicitly. The nature of thephase transition in the three-dimensional random field Ising model is discussed.

Infrared degrees of freedom of Yang-Mills theory in the Schrödinger representation

We set up a new calculational framework for the Yang-Mills vacuum transition amplitude in the Schroedinger representation. After integrating out hard-mode contributions perturbatively, we perform a gauge invariant gradient expansion of the ensuing soft mode action which renders a subsequent saddle point expansion for the vacuum overlap manageable. The standard "squeezed" approximation for the vacuum wave functional then allows for an essentially analytical treatment of physical amplitudes. Moreover, it leads to the identification of dominant and gauge invariant classes of gauge field orbits which play the role of gluonic infrared (IR) degrees of freedom. Those emerge as a rich variety of (mostly solitonic) solutions to the saddle point equations which are characterized by a common relative gauge orientation of the underlying gluon fields. We discuss their scale stability, guaranteed by a virial theorem, and other general properties including topological quantum numbers and action bounds. We then find important saddle point solutions explicitly and examine their physical impact. Some of them are related to tunneling solutions of the classical Yang-Mills equation, i.e. to instantons and merons, while others appear to play unprecedented roles. A remarkable new class of IR degrees of freedom comprises vortex and knot solutions of Faddeev-Niemi type, potentially related to glueballs.

Interferences of real trajectories and the emergence of quantum features in electron-atom scattering in a strong laser field

Using the example of electron-atom scattering in a strong laser field, it is shown that the oscillatory structure of the scattered electron spectrum can be explained as a consequence of the interference of the real electron trajectories in terms of Feynman's path integral. While in previous work on quantum-orbit theory the complex solutions of the saddle-point equations were considered, we show here that for the electron-atom scattering with much simpler real solutions a satisfactory agreement with the strong-field-approximation results can be achieved. Real solutions are applicable both for the direct (low-energy) and the rescattering (high-energy) plateau in the scattered electron spectrum. In between the plateaus and beyond the rescattering cutoff good results can be obtained using the complex (quantum) solutions and the uniform approximation. The interference of real solutions is related to the recent attosecond double-slit experiment in time.

Tail probability approximations for Student's t-statistics

In this paper, we derive saddlepoint approximations for Student's t-statistics for strongly nonlattice random variables without moment conditions. Under very mild conditions, we show that saddlepoint equations always have solutions.

Two-Dimensional Saddle Point Equation of Ginzburg–LandauHamiltonian for the Diluted Ising Model

The saddle point equation of Ginzburg–Landau Hamiltonian for thediluted Ising model is developed. The ground state is solvednumerically in two dimensions. The result is partly explained bythe coarse-grained approximation.

Uzawa Conjugate Gradient Domain Decomposition Methods for Coupled Stokes Flows

This paper deals with nonoverlapping domain decomposition methods for two coupled Stokes flows, based on the duality theory. By introducing a fictitious variable in the transmission condition and using saddle-point equations, the problem is restated as a linearly constrained maximization problem. According to whether constraints are uncoupled Stokes problems or uncoupled Poisson problems, two Uzawa-type domain decomposition algorithms are proposed. The results of some numerical experiments on a model problem are given.

On thea priori anda posteriori error analysis of a two-fold saddle-point approach for nonlinear incompressible elasticity

In this paper, we reconsider the a priori and a posteriori error analysis of a new mixed finite element method for nonlinear incompressible elasticity with mixed boundary conditions. The approach, being based only on the fact that the resulting variational formulation becomes a two-fold saddle-point operator equation, simplifies the analysis and improves the results provided recently in a previous work. Thus, a well-known generalization of the classical Babuska–Brezzi theory is applied to show the well-posedness of the continuous and discrete formulations, and to derive the corresponding a priori error estimate. In particular, enriched PEERS subspaces are required for the solvability and stability of the associated Galerkin scheme. In addition, we use the Ritz projection operator to obtain a new reliable and quasi-efficient a posteriori error estimate. Finally, several numerical results illustrating the good performance of the associated adaptive algorithm are presented. Copyright © 2006 John Wiley & Sons, Ltd.

Effect of second-rank random anisotropy on critical phenomena of a random-fieldO(N)spin model in the largeNlimit

We study the critical behavior of a random field O($N$) spin model with a second-rank random anisotropy term in spatial dimensions $4<d<6$, by means of the replica method and the 1/N expansion. We obtain a replica-symmetric solution of the saddle-point equation, and we find the phase transition obeying dimensional reduction. We study the stability of the replica-symmetric saddle point against the fluctuation induced by the second-rank random anisotropy. We show that the eigenvalue of the Hessian at the replica-symmetric saddle point is strictly positive. Therefore, this saddle point is stable and the dimensional reduction holds in the 1/N expansion. To check the consistency with the functional renormalization group method, we obtain all fixed points of the renormalization group in the large $N$ limit and discuss their stability. We find that the analytic fixed point yielding the dimensional reduction is practically singly unstable in a coupling constant space of the given model with large $N$. Thus, we conclude that the dimensional reduction holds for sufficiently large $N$.

Saddlepoint approximations for some models of circular data

In this article we provide saddlepoint approximations for some important models of circular data. The particularity of these saddlepoint approximations is that they do not require solving the saddlepoint equation iteratively, so their evaluation is immediate. We first give very accurate approximations to P -values, critical values and power functions for some optimal tests regarding the concentration parameter under wrapped symmetric α -stable and circular normal models. Then, we consider an approximation to the distribution of a projection of the two-dimensional Pearson random walk with exponential step sizes.

Numerical solution of the two-dimensional saddle point equation of Ginzburg-Landau Hamiltonian with random temperature

The two-dimensional saddle point equation of Ginzburg-Landau Hamiltonian with random temperature is studied. The random temperature is correlated in a finite range $l$. The distribution width of the random temperature is $\ensuremath{\Delta}$. The ground state of the saddle point equation is solved numerically. The average, fluctuation, and autocorrelation of the order parameter are obtained. For $\ensuremath{\Delta}⪢1∕{l}^{2}$, where $l$ is dimensionless, the correlation length is given by ${\ensuremath{\xi}}_{\ensuremath{\phi}}\ensuremath{\propto}l$. For $\ensuremath{\Delta}&lt;1∕{l}^{2}$, the correlation length ${\ensuremath{\xi}}_{\ensuremath{\phi}}\ensuremath{\propto}{\ensuremath{\Delta}}^{\ensuremath{-}1}$. The saddle point solutions for $\ensuremath{\Delta}&lt;1∕{l}^{2}$ are rescaled to that for $\ensuremath{\Delta}=1∕{l}^{2}$ by using a modified coarse-grained approximation.

Dynamical conductivity of disordered quantum Hall stripes

We present a detailed theory for finite-frequency conductivities Re$[\sigma_{\alpha\beta}(\omega)]$ of quantum Hall stripes, which form at Landau level $N\geq 2$ close to half filling, in the presence of weak Gaussian disorder. We use an effective elastic theory to describe the low-energy dynamics of the stripes with the dynamical matrix being determined through matching the density-density correlation function obtained in the microscopic time-dependent Hartree-Fock approximation. We then apply replicas and the Gaussian variational method to deal with the disorder. Within this method, a set of saddle point equations for the retarded self energies are obtained, which are solved numerically to get Re$[\sigma_{\alpha\beta}(\omega)]$. We find a quantum depinning transition as $\Delta\nu$, the fractional part of the filling factor, approaches a critical value $\Delta\nu_c$ from below. For $\Delta\nu<\Delta\nu_c$, the pinned state is realized in a replica symmetry breaking (RSB) solution, and the frequency-dependent conductivities in both the directions perpendicular and parallel to the stripes show resonant peaks. These peaks shift to zero frequency as $\Delta\nu\to \Delta\nu_c$. For $\Delta\nu\ge\Delta \nu_c$, we find a partial RSB (PRSB) solution in which there is RSB perpendicular to the stripes, but replica symmetry along the stripes, leading to free sliding along the stripe direction. The quantum depinning transition is in the Kosterlitz-Thouless universality class. The result is consistent with a previous renormalization group analysis.

Complexity in mean-field spin-glass models: Isingp-spin

The complexity of the Thouless-Anderson-Palmer (TAP) solutions of the Ising $p$-spin is investigated in the temperature regime where the equilibrium phase is one-step replica symmetry breaking. Two solutions of the resulting saddle point equations are found. One is supersymmetric (SUSY) and includes the equilibrium value of the free energy while the other is non-SUSY. The two solutions cross exactly at a value of the free energy where the replicon eigenvalue is zero; at low free energy the complexity is described by the SUSY solution while at high free energy it is described by the non-SUSY solution, the latter accounting for the total number of solutions. The relevant TAP solutions counted by the non-SUSY complexity share the same features of the corresponding solutions in the Sherrington-Kirkpatrick model; in particular their Hessian has a vanishing isolated eigenvalue. The TAP solutions corresponding to the SUSY complexity, instead, are well separated minima.

Effect of Second-Rank Random Anisotropy on Critical Phenomena of Random Field O(N) Spin Model

We study the critical behavior of a random field $\mathrm{O}(N)$ spin model with a second-rank random anisotropy term in spatial dimensions $4ldl6$, by means of the replica method and the $1∕N$ expansion. We obtain a replica-symmetric solution of the saddle-point equation, and we find the phase transition obeying dimensional reduction. We study the stability of the replica-symmetric saddle point against the fluctuation induced by the second-rank random anisotropy. We show that the eigenvalue of the Hessian at the replica-symmetric saddle point is strictly positive. Therefore, this saddle point is stable and the dimensional reduction holds in the $1∕N$ expansion. To check the consistency with the functional renormalization group method, we obtain all fixed points of the renormalization group in the large $N$ limit and discuss their stability. We find that the analytic fixed point yielding the dimensional reduction is practically singly unstable in a coupling constant space of the given model with large $N$. Thus, we conclude that the dimensional reduction holds for sufficiently large $N$.

An Efficient Joint Source-Channel Decoder with Dynamical Block Priors

An efficient joint source-channel (s/c) decoder based on the side information of the source and on the MN-Gallager algorithm over Galois fields is presented. The dynamical block priors (DBP) are derived either from a statistical mechanical approach via calculation of the entropy for the correlated sequences, or from the Markovian transition matrix. The Markovian joint s/c decoder has many advantages over the statistical mechanical approach. In particular, there is no need for the construction and the diagonalization of a qXq matrix and for a solution to saddle point equations in q dimensions. Using parametric estimation, an efficient joint s/c decoder with the lack of side information is discussed. Besides the variant joint s/c decoders presented, we also show that the available sets of autocorrelations consist of a convex volume, and its structure can be found using the Simplex algorithm.

PINNING AND SLIDING OF QUANTUM HALL STRIPES AND BUBBLES

We report on the calculation of the frequency-dependent conductivity of the quantum Hall stripes and bubble crystals that form in high Landau level. We use the replica and Gaussian variational methods (GVM) with a dynamical matrix obtained from the time-dependent Hartree-Fock approximation. In the stripe state, we go beyond the semiclassical approximation of the saddle point equations obtained with the GVM and demonstrate the existence of a quantum depinning transition as a function of filling factor. Below a critical filling factor, the pinned state is described by a replica symmetry breaking (RSB) solution that gives resonant peaks in the frequency-dependent conductivity in both directions, parallel and perpendicular to the stripes orientation. These peaks shift to zero frequency as the critical filling is approached. Above the critical filling, we find a depinned stripe state described by a partial replica symmetry breaking solution in which there is free sliding only along the stripe direction. The transition has a Kosterlitz-Thouless character and includes a jump in the low-frequency exponent of the dynamical conductivity. In the bubble crystals a semiclassical approximation yields a pinning peak frequency and resonance width that generally decrease with increasing filling factor, in accordance with recent microwave absorption experiments.

Evolution of Néel order and localized spin moment in the doped two-dimensional Hubbard model

We investigate the effect of hole hopping on Néel order upon doping in the two-dimensional Hubbard model. The semiclassical staggered moment is computed in the doped region by solving saddle point equations based on a path-integral formalism. Quantum fluctuations in the localized spin system are taken into account by the Schwinger boson mean field theory. We argue that hopping of doped holes is rather ineffective in suppressing Néel order compared to rapid suppression of Néel order in high-temperature superconductors. The quantum disordered phase is defined as the phase in which the non-zero semiclassical staggered moments are less than the critical value ∼0.20. In the strong coupling limit, the quantum disordered phase is described by a system of spinless fermions and bosons without gauge field interaction.

Numerical solution of the one-dimensional saddle point equation of the Ginzburg–Landau Hamiltonian with random temperature

Numerical solution of the one-dimensional saddle point equation of the Ginzburg–Landau Hamiltonian with random temperature

Numerical solution of the one-dimensional saddle point equation of the Ginzburg–Landau Hamiltonian with random temperature

The one-dimensional saddle point equation of the Ginzburg–Landau Hamiltonian with random temperature is studied with a numerical method. The random temperature is correlated with a finite range l. The distribution width of the random temperature is � . The ground state of the saddle point equation is solved. The average, fluctuation and auto-correlation of the order parameter are obtained. It is found that the auto-correlation function behaves like ∼exp −x 2 ξ 2 φ .F or� � 1 /l 2 , where l is dimensionless, the correlation length is given by ξφ ∝ l .F or �< 1 /l 2 ,a st = 0, the correlation length ξφ ∝ l� −α , where α = 0.65. All the saddle point solutions for �< 1 /l 2 can be mapped to that for � = 1 /l 2 by using a coarse-grained approximation.

Functional renormalization group at largeNfor disordered elastic systems, and relation to replica symmetry breaking

We study the replica field theory which describes the pinning of elastic manifolds of arbitrary internal dimension d in a random potential, with the aim of bridging the gap between mean field and renormalization theory. The full effective action is computed exactly in the limit of large embedding space dimension N. The second cumulant of the renormalized disorder obeys a closed self-consistent equation. It is used to derive a Functional Renormalization Group (FRG) equation valid in any dimension d, which correctly matches the Balents-Fisher result to first order in epsilon=4-d. We analyze in detail the solutions of the large-N FRG for both long-range and short-range disorder, at zero and finite temperature. We find consistent agreement with the results of Mezard Parisi (MP) from the Gaussian variational method (GVM) in the case where full replica symmetry breaking (RSB) holds there. We prove that the cusplike non-analyticity in the large N FRG appears at a finite scale, corresponding to the instability of the replica symmetric solution of MP. We show that the FRG exactly reproduces, for any disorder correlator and with no need to invoke Parisi's spontaneous RSB, the non-trivial result of the GVM for small overlap. A formula is found yielding the complete RSB solution for all overlaps. Since our saddle-point equations for the effective action contain both the MP equations and the FRG, it can be used to describe the crossover from FRG to RSB. A qualitative analysis of this crossover is given, as well as a comparison with previous attempts to relate FRG to GVM. Finally, we discuss applications to other problems and new perspectives.