The Characteristic Invariant Manifolds of the Adjoint System

AbstractWe consider a scalar function depending on a numerical solution of an initial value problem, and its second-derivative (Hessian) matrix for the initial value. The need to extract the information of the Hessian or to solve a linear system having the Hessian as a coefficient matrix arises in many research fields such as optimization, Bayesian estimation, and uncertainty quantification. From the perspective of memory efficiency, these tasks often employ a Krylov subspace method that does not need to hold the Hessian matrix explicitly and only requires computing the multiplication of the Hessian and a given vector. One of the ways to obtain an approximation of such Hessian-vector multiplication is to integrate the so-called second-order adjoint system numerically. However, the error in the approximation could be significant even if the numerical integration to the second-order adjoint system is sufficiently accurate. This paper presents a novel algorithm that computes the intended Hessian-vector multiplication exactly and efficiently. For this aim, we give a new concise derivation of the second-order adjoint system and show that the intended multiplication can be computed exactly by applying a particular numerical method to the second-order adjoint system. In the discussion, symplectic partitioned Runge–Kutta methods play an essential role.

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Using Global Invariant Manifolds to Understand Metastability in the Burgers Equation With Small Viscosity

SIAM Journal on Applied Dynamical Systems ◽

10.1137/08073651x ◽

2009 ◽

Vol 8 (3) ◽

pp. 1043-1065 ◽

Cited By ~ 9

Author(s):

Margaret Beck ◽

C. Eugene Wayne

Keyword(s):

Invariant Manifolds ◽

Burgers Equation ◽

Small Viscosity ◽

Global Invariant

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Fast and slow invariant manifolds in chemical kinetics

Computers & Mathematics with Applications ◽

10.1016/j.camwa.2013.01.040 ◽

2013 ◽

Vol 65 (10) ◽

pp. 1502-1515 ◽

Cited By ~ 11

Author(s):

V. Bykov ◽

V. Gol’dshtein

Keyword(s):

Chemical Kinetics ◽

Invariant Manifolds

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Local Mass Conservation and Velocity Splitting in PV-Based Balanced Models. Part II: Numerical Results

Journal of the Atmospheric Sciences ◽

10.1175/jas3934.1 ◽

2007 ◽

Vol 64 (6) ◽

pp. 1794-1810 ◽

Cited By ~ 7

Author(s):

Ali R. Mohebalhojeh ◽

Michael E. McIntyre

Keyword(s):

Potential Vorticity ◽

Wave Speed ◽

Invariant Manifolds ◽

Mass Conservation ◽

Flow Speed ◽

Local Mass ◽

Potential Vorticity Inversion ◽

Divergence Equation ◽

Local Mass Conservation ◽

Derivatives Of

The effects of enforcing local mass conservation on the accuracy of non-Hamiltonian potential-vorticity- based balanced models (PBMs) are examined numerically for a set of chaotic shallow-water f-plane vortical flows in a doubly periodic square domain. The flows are spawned by an unstable jet and all have domain-maximum Froude and Rossby numbers Fr ∼0.5 and Ro ∼1, far from the usual asymptotic limits Ro → 0, Fr → 0, with Fr defined in the standard way as flow speed over gravity wave speed. The PBMs considered are the plain and hyperbalance PBMs defined in Part I. More precisely, they are the plain-δδ, plain-γγ, and plain-δγ PBMs and the corresponding hyperbalance PBMs, of various orders, where “order” is related to the number of time derivatives of the divergence equation used in defining balance and potential-vorticity inversion. For brevity the corresponding hyperbalance PBMs are called the hyper-δδ, hyper-γγ, and hyper-δγ PBMs, respectively. As proved in Part I, except for the leading-order plain-γγ each plain PBM violates local mass conservation. Each hyperbalance PBM results from enforcing local mass conservation on the corresponding plain PBM. The process of thus deriving a hyperbalance PBM from a plain PBM is referred to for brevity as plain-to-hyper conversion. The question is whether such conversion degrades the accuracy, as conjectured by McIntyre and Norton. Cumulative accuracy is tested by running each PBM alongside a suitably initialized primitive equation (PE) model for up to 30 days, corresponding to many vortex rotations. The accuracy is sensitively measured by the smallness of the ratio ϵ = ||QPBM − QPE||2/||QPE||2, where QPBM and QPE denote the potential vorticity fields of the PBM and the PEs, respectively, and || ||2 is the L2 norm. At 30 days the most accurate PBMs have ϵ ≈ 10−2 with PV fields hardly distinguishable visually from those of the PEs, even down to tiny details. Most accurate is defined by minimizing ϵ over all orders and truncation types δδ, γγ, and δγ. Contrary to McIntyre and Norton’s conjecture, the minimal ϵ values did not differ systematically or significantly between plain and hyperbalance PBMs. The smallness of ϵ suggests that the slow manifolds defined by the balance relations of the most accurate PBMs, both plain and hyperbalance, are astonishingly close to being invariant manifolds of the PEs, at least throughout those parts of phase space for which Ro ≲ 1 and Fr ≲ 0.5. As another way of quantifying the departures from such invariance, that is, of quantifying the fuzziness of the PEs’ slow quasimanifold, initialization experiments starting at days 1, 2, . . . 10 were carried out in which attention was focused on the amplitudes of inertia–gravity waves representing the imbalance arising in 1-day PE runs. With balance defined by the most accurate PBMs, and imbalance by departures therefrom, the results of the initialization experiments suggest a negative correlation between early imbalance and late cumulative error ϵ. In such near-optimal conditions the imbalance seems to be acting like weak background noise producing an effect analogous to so-called stochastic resonance, in that a slight increase in noise level brings PE behavior closer to the balanced behavior defined by the most accurate PBMs when measured cumulatively over 30 days.

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GEOMETRY OF HOMOCLINIC CONNECTIONS IN A PLANAR CIRCULAR RESTRICTED THREE-BODY PROBLEM

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127407017744 ◽

2007 ◽

Vol 17 (04) ◽

pp. 1151-1169 ◽

Cited By ~ 16

Author(s):

MARIAN GIDEA ◽

JOSEP J. MASDEMONT

Keyword(s):

Symbolic Dynamics ◽

Body Problem ◽

Invariant Manifolds ◽

Libration Point ◽

Homoclinic Orbits ◽

Restricted Three Body Problem ◽

Three Body Problem ◽

Lyapunov Orbits ◽

Homoclinic Connections ◽

Three Body

The stable and unstable invariant manifolds associated with Lyapunov orbits about the libration point L1between the primaries in the planar circular restricted three-body problem with equal masses are considered. The behavior of the intersections of these invariant manifolds for values of the energy between that of L1and the other collinear libration points L2, L3is studied using symbolic dynamics. Homoclinic orbits are classified according to the number of turns about the primaries.

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