30,367 research outputs found
General Doubly Stochastic Maximum Principle and Its Applications to Optimal Control of SPDEs
In this paper, we prove the necessary and sufficient maximum principles
(NSMPs in short) for the optimal control of systems described by a quasilinear
stochastic heat equation within convex control domains, which all the
coefficients contain control variables. For that, the optimal control problem
of fully coupled forward-backward doubly stochastic system is studied. We apply
our NSMPs to treat a kind of forward-backward doubly stochastic linear
quadratic optimal control problems and an example of optimal control of
stochastic partial differential equations (SPDEs in short) as well.Comment: arXiv admin note: text overlap with arXiv:1005.412
A Moment and Sum-of-Squares Extension of Dual Dynamic Programming with Application to Nonlinear Energy Storage Problems
We present a finite-horizon optimization algorithm that extends the
established concept of Dual Dynamic Programming (DDP) in two ways. First, in
contrast to the linear costs, dynamics, and constraints of standard DDP, we
consider problems in which all of these can be polynomial functions. Second, we
allow the state trajectory to be described by probability distributions rather
than point values, and return approximate value functions fitted to these. The
algorithm is in part an adaptation of sum-of-squares techniques used in the
approximate dynamic programming literature. It alternates between a forward
simulation through the horizon, in which the moments of the state distribution
are propagated through a succession of single-stage problems, and a backward
recursion, in which a new polynomial function is derived for each stage using
the moments of the state as fixed data. The value function approximation
returned for a given stage is the point-wise maximum of all polynomials derived
for that stage. This contrasts with the piecewise affine functions derived in
conventional DDP. We prove key convergence properties of the new algorithm, and
validate it in simulation on two case studies related to the optimal operation
of energy storage devices with nonlinear characteristics. The first is a small
borehole storage problem, for which multiple value function approximations can
be compared. The second is a larger problem, for which conventional discretized
dynamic programming is intractable.Comment: 33 pages, 9 figure
Reversible thermal diode and energy harvester with a superconducting quantum interference single-electron transistor
The density of states of proximitized normal nanowires interrupting
superconducting rings can be tuned by the magnetic flux piercing the loop.
Using these as the contacts of a single-electron transistor allows to control
the energetic mirror asymmetry of the conductor, this way introducing
rectification properties. In particular, we show that the system works as a
diode that rectifies both charge and heat currents and whose polarity can be
reversed by the magnetic field and a gate voltage. We emphasize the role of
dissipation at the island. The coupling to substrate phonons enhances the
effect and furthermore introduces a channel for phase tunable conversion of
heat exchanged with the environment into electrical current.Comment: 5 pages, 4 figures. Accepted versio
Unified Systems of FB-SPDEs/FB-SDEs with Jumps/Skew Reflections and Stochastic Differential Games
We study four systems and their interactions. First, we formulate a unified
system of coupled forward-backward stochastic partial differential equations
(FB-SPDEs) with Levy jumps, whose drift, diffusion, and jump coefficients may
involve partial differential operators. A solution to the FB-SPDEs is defined
by a 4-tuple general dimensional random vector-field process evolving in time
together with position parameters over a domain (e.g., a hyperbox or a
manifold). Under an infinite sequence of generalized local linear growth and
Lipschitz conditions, the well-posedness of an adapted 4-tuple strong solution
is proved over a suitably constructed topological space. Second, we consider a
unified system of FB-SDEs, a special form of the FB-SPDEs, however, with skew
boundary reflections. Under randomized linear growth and Lipschitz conditions
together with a general completely-S condition on reflections, we prove the
well-posedness of an adapted 6-tuple weak solution with boundary regulators to
the FB-SDEs by the Skorohod problem and an oscillation inequality.
Particularly, if the spectral radii in some sense for reflection matrices are
strictly less than the unity, an adapted 6-tuple strong solution is concerned.
Third, we formulate a stochastic differential game (SDG) with general number of
players based on the FB-SDEs. By a solution to the FB-SPDEs, we get a solution
to the FB-SDEs under a given control rule and then obtain a Pareto optimal Nash
equilibrium policy process to the SDG. Fourth, we study the applications of the
FB-SPDEs/FB-SDEs in queueing systems and quantum statistics while we use them
to motivate the SDG.Comment: 58 pages, 6 figures, invited talks and plenary talks at a number of
conferences and workshop
Convergence and coupling for spin glasses and hard spheres
We discuss convergence and coupling of Markov chains, and present general
relations between the transfer matrices describing these two processes. We then
analyze a recently developed local-patch algorithm, which computes rigorous
upper bound for the coupling time of a Markov chain for non-trivial
statistical-mechanics models. Using the coupling from the past protocol, this
allows one to exactly sample the underlying equilibrium distribution. For spin
glasses in two and three spatial dimensions, the local-patch algorithm works at
lower temperatures than previous exact-sampling methods. We discuss variants of
the algorithm which might allow one to reach, in three dimensions, the
spin-glass transition temperature. The algorithm can be adapted to hard-sphere
models. For two-dimensional hard disks, the algorithm allows us to draw exact
samples at higher densities than previously possible.Comment: 22 pages, 19 figures, python cod
Symplectic Reconstruction of Data for Heat and Wave Equations
This report concerns the inverse problem of estimating a spacially dependent
coefficient of a partial differential equation from observations of the
solution at the boundary. Such a problem can be formulated as an optimal
control problem with the coefficient as the control variable and the solution
as state variable. The heat or the wave equation is here considered as state
equation. It is well known that such inverse problems are ill-posed and need to
be regularized. The powerful Hamilton-Jacobi theory is used to construct a
simple and general method where the first step is to analytically regularize
the Hamiltonian; next its Hamiltonian system, a system of nonlinear partial
differential equations, is solved with the Newton method and a sparse Jacobian
Optimal Estimation of Dynamically Evolving Diffusivities
The augmented, iterated Kalman smoother is applied to system identification
for inverse problems in evolutionary differential equations. In the augmented
smoother, the unknown, time-dependent coefficients are included in the state
vector, and have a stochastic component. At each step in the iteration, the
estimate of the time evolution of the coefficients is linear. We update the
slowly varying mean temperature and conductivity by averaging the estimates of
the Kalman smoother. Applications include the estimation of anomalous diffusion
coefficients in turbulent fluids and the plasma rotation velocity in plasma
tomography
Allostery and Kinetic Proofreading
Kinetic proofreading is an error correction mechanism present in the
processes of the central dogma and beyond, and typically requires the free
energy of nucleotide hydrolysis for its operation. Though the molecular players
of many biological proofreading schemes are known, our understanding of how
energy consumption is managed to promote fidelity remains incomplete. In our
work, we introduce an alternative conceptual scheme called 'the piston model of
proofreading' where enzyme activation through hydrolysis is replaced with
allosteric activation achieved through mechanical work performed by a piston on
regulatory ligands. Inspired by Feynman's ratchet and pawl mechanism, we
consider a mechanical engine designed to drive the piston actions powered by a
lowering weight, whose function is analogous to that of ATP synthase in cells.
Thanks to its mechanical design, the piston model allows us to tune the 'knobs'
of the driving engine and probe the graded changes and trade-offs between
speed, fidelity and energy dissipation. It provides an intuitive explanation of
the conditions necessary for optimal proofreading and reveals the unexpected
capability of allosteric molecules to beat the Hopfield limit of fidelity by
leveraging the diversity of states available to them. The framework that we
built for the piston model can also serve as a basis for additional studies of
driven biochemical systems
Minimum-Time Cavity Optomechanical Cooling
Optomechanical cooling is a prerequisite for many exotic applications
promised by modern quantum technology and it is crucial to achieve it in short
times, in order to minimize the undesirable effects of the environment. We
formulate cavity optomechanical cooling as a minimum-time optimal control
problem on anti-de Sitter space of appropriate dimension and use the Legendre
pseudospectral optimization method to find the minimum time and the
corresponding optimal control, for various values of the maximum coupling rate
between the cavity field and the mechanical resonator. The present framework
can also be applied to create optomechanical entanglement in minimum time and
to improve the efficiency of an optomechanical quantum heat engine
On the mean field games with common noise and the McKean-Vlasov SPDEs
We formulate the MFG limit for interacting agents with a common noise as
a single quasi-linear deterministic infinite-dimensional partial differential
second order backward equation. We prove that any its (regular enough) solution
provides an -Nash-equilibrium profile for the initial -player game. We
use the method of stochastic characteristics to provide the link with the basic
models of MFG with a major player. We develop two auxiliary theories of
independent interest: sensitivity and regularity analysis for the McKean-Vlasov
SPDEs and the -convergence rate for the propagation of chaos property of
interacting diffusions
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