2,587 research outputs found
Linear Hamilton Jacobi Bellman Equations in High Dimensions
The Hamilton Jacobi Bellman Equation (HJB) provides the globally optimal
solution to large classes of control problems. Unfortunately, this generality
comes at a price, the calculation of such solutions is typically intractible
for systems with more than moderate state space size due to the curse of
dimensionality. This work combines recent results in the structure of the HJB,
and its reduction to a linear Partial Differential Equation (PDE), with methods
based on low rank tensor representations, known as a separated representations,
to address the curse of dimensionality. The result is an algorithm to solve
optimal control problems which scales linearly with the number of states in a
system, and is applicable to systems that are nonlinear with stochastic forcing
in finite-horizon, average cost, and first-exit settings. The method is
demonstrated on inverted pendulum, VTOL aircraft, and quadcopter models, with
system dimension two, six, and twelve respectively.Comment: 8 pages. Accepted to CDC 201
Solving rank structured Sylvester and Lyapunov equations
We consider the problem of efficiently solving Sylvester and Lyapunov
equations of medium and large scale, in case of rank-structured data, i.e.,
when the coefficient matrices and the right-hand side have low-rank
off-diagonal blocks. This comprises problems with banded data, recently studied
by Haber and Verhaegen in "Sparse solution of the Lyapunov equation for
large-scale interconnected systems", Automatica, 2016, and by Palitta and
Simoncini in "Numerical methods for large-scale Lyapunov equations with
symmetric banded data", SISC, 2018, which often arise in the discretization of
elliptic PDEs.
We show that, under suitable assumptions, the quasiseparable structure is
guaranteed to be numerically present in the solution, and explicit novel
estimates of the numerical rank of the off-diagonal blocks are provided.
Efficient solution schemes that rely on the technology of hierarchical
matrices are described, and several numerical experiments confirm the
applicability and efficiency of the approaches. We develop a MATLAB toolbox
that allows easy replication of the experiments and a ready-to-use interface
for the solvers. The performances of the different approaches are compared, and
we show that the new methods described are efficient on several classes of
relevant problems
Three Lectures: Nemd, Spam, and Shockwaves
We discuss three related subjects well suited to graduate research. The
first, Nonequilibrium molecular dynamics or "NEMD", makes possible the
simulation of atomistic systems driven by external fields, subject to dynamic
constraints, and thermostated so as to yield stationary nonequilibrium states.
The second subject, Smooth Particle Applied Mechanics or "SPAM", provides a
particle method, resembling molecular dynamics, but designed to solve continuum
problems. The numerical work is simplified because the SPAM particles obey
ordinary, rather than partial, differential equations. The interpolation method
used with SPAM is a powerful interpretive tool converting point particle
variables to twice-differentiable field variables. This interpolation method is
vital to the study and understanding of the third research topic we discuss,
strong shockwaves in dense fluids. Such shockwaves exhibit stationary
far-from-equilibrium states obtained with purely reversible Hamiltonian
mechanics. The SPAM interpolation method, applied to this molecular dynamics
problem, clearly demonstrates both the tensor character of kinetic temperature
and the time-delayed response of stress and heat flux to the strain rate and
temperature gradients. The dynamic Lyapunov instability of the shockwave
problem can be analyzed in a variety of ways, both with and without symmetry in
time. These three subjects suggest many topics suitable for graduate research
in nonlinear nonequilibrium problems.Comment: 40 pages, with 21 figures, as presented at the Granada Seminar on the
Foundations of Nonequilibrium Statistical Physics, 13-17 September, as three
lecture
Comparison of POD reduced order strategies for the nonlinear 2D Shallow Water Equations
This paper introduces tensorial calculus techniques in the framework of
Proper Orthogonal Decomposition (POD) to reduce the computational complexity of
the reduced nonlinear terms. The resulting method, named tensorial POD, can be
applied to polynomial nonlinearities of any degree . Such nonlinear terms
have an on-line complexity of , where is the
dimension of POD basis, and therefore is independent of full space dimension.
However it is efficient only for quadratic nonlinear terms since for higher
nonlinearities standard POD proves to be less time consuming once the POD basis
dimension is increased. Numerical experiments are carried out with a two
dimensional shallow water equation (SWE) test problem to compare the
performance of tensorial POD, standard POD, and POD/Discrete Empirical
Interpolation Method (DEIM). Numerical results show that tensorial POD
decreases by times the computational cost of the on-line stage of
standard POD for configurations using more than model variables. The
tensorial POD SWE model was only slower than the POD/DEIM SWE model
but the implementation effort is considerably increased. Tensorial calculus was
again employed to construct a new algorithm allowing POD/DEIM shallow water
equation model to compute its off-line stage faster than the standard and
tensorial POD approaches.Comment: 23 pages, 8 figures, 5 table
Neighborhoods of periodic orbits and the stationary distribution of a noisy chaotic system
The finest state space resolution that can be achieved in a physical
dynamical system is limited by the presence of noise. In the weak-noise
approximation the neighborhoods of deterministic periodic orbits can be
computed as distributions stationary under the action of a local Fokker-Planck
operator and its adjoint. We derive explicit formulae for widths of these
distributions in the case of chaotic dynamics, when the periodic orbits are
hyperbolic. The resulting neighborhoods form a basis for functions on the
attractor. The global stationary distribution, needed for calculation of
long-time expectation values of observables, can be expressed in this basis.Comment: 6 pages, 3 figure
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