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## Discovery and optimization of low-storage Runge-Kutta methods

Fonte: Monterey, California: Naval Postgraduate School
Publicador: Monterey, California: Naval Postgraduate School

Tipo: Tese de Doutorado

Português

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#low-storage Runge-Kutta (LSRK),stability region#Maxwell’s equations#half-explicit Runge-Kutta (HERK)#low-storage half-explicit runge-kutta (LSHERK)#differential algebraic equation (DAE)

Approved for public release; distribution is unlimited; Runge-Kutta (RK) methods are an important family of iterative methods for approximating the solutions of ordinary differential equations (ODEs) and differential algebraic equations (DAEs). It is common to use an RK method to discretize in time when solving time dependent partial differential equations (PDEs) with a method-of-lines approach as in Maxwell’s Equations. Different types of PDEs are discretized in such a way that could result in a high dimensional ODE or DAE.We use a low-storage RK (LSRK) method that stores two registers per ODE dimension, which limits the impact of increased storage requirements. Classical RK methods, however, have one storage variable per stage. In this thesis we compare the efficiency and accuracy of LSRK methods to RK methods. We then focus on optimizing the truncation error coefficients for LSRK to discover new methods. Reusing the tools from the optimization method, we discover new methods for low-storage half-explicit RK (LSHERK) methods for solving DAEs.; ; Captain, United States Army

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## A Dynamic Bi-orthogonal Field Equation Approach for Efficient Bayesian Calibration of Large-Scale Systems

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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This paper proposes a novel computationally efficient dynamic
bi-orthogonality based approach for calibration of a computer simulator with
high dimensional parametric and model structure uncertainty. The proposed
method is based on a decomposition of the solution into mean and a random field
using a generic Karhunnen-Loeve expansion. The random field is represented as a
convolution of separable Hilbert spaces in stochastic and spacial dimensions
that are spectrally represented using respective orthogonal bases. In
particular, the present paper investigates generalized polynomial chaos bases
for stochastic dimension and eigenfunction bases for spacial dimension. Dynamic
orthogonality is used to derive closed form equations for the time evolution of
mean, spacial and the stochastic fields. The resultant system of equations
consists of a partial differential equation (PDE) that define dynamic evolution
of the mean, a set of PDEs to define the time evolution of eigenfunction bases,
while a set of ordinary differential equations (ODEs) define dynamics of the
stochastic field. This system of dynamic evolution equations efficiently
propagates the prior parametric uncertainty to the system response. The
resulting bi-orthogonal expansion of the system response is used to reformulate
the Bayesian inference for efficient exploration of the posterior distribution.
Efficacy of the proposed method is investigated for calibration of a 2D
transient diffusion simulator with uncertain source location and diffusivity.
Computational efficiency of the method is demonstrated against a Monte Carlo
method and a generalized polynomial chaos approach.

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## Probabilistic Representation of Weak Solutions of Partial Differential Equations with Polynomial Growth Coefficients

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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In this paper we develop a new weak convergence and compact embedding method
to study the existence and uniqueness of the
$L_{\rho}^2({\mathbb{R}^{d}};{\mathbb{R}^{1}})\otimes
L_{\rho}^2({\mathbb{R}^{d}};{\mathbb{R}^{d}})$ valued solution of backward
stochastic differential equations with p-growth coefficients. Then we establish
the probabilistic representation of the weak solution of PDEs with p-growth
coefficients via corresponding BSDEs.

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## Symmetries of PDEs Systems in Solar Physics and Contact Geometry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 26/10/1999
Português

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One considers a special class of PDEs systems and one determines the
associated symmetry group. Particulary, for the Blair system, one finds the
symmetry group. A solutions of the Blair system gives a conformally flat
contact metric structure and also it defines a "force-free" model of solar
physics. By using the symmetry groups theory, one shows that the known
solutions are group-invariant solutions and one gives new solutions.; Comment: 16 pages

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## Backward Stochastic Differential Equations with Markov Chains and The Application: Homogenization of PDEs System

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 26/09/2010
Português

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Stemmed from the derivation of the optimal control to a stochastic
linear-quadratic control problem with Markov jumps, we study one kind of
backward stochastic differential equations (BSDEs) that the generator f is
affected by a Markovian switching. Then, the case that the Markov chain is
involved in a large state space is considered. Following the classical
approach, a hierarchical approach is adopted to reduce the complexity and a
singularly perturbed Markov chain is involved. We will study the asymptotic
property of BSDE with the singularly perturbed Markov chain. At last, as an
application of our theoretical result, we show the homogenization of one system
of partial differential equations (PDEs) with a singularly perturbed Markov
chain.

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## p-integrable solutions to multidimensional BSDEs and degenerate systems of PDEs with logarithmic nonlinearities

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 14/07/2010
Português

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560.90875%

We study multidimensional backward stochastic differential equations (BSDEs)
which cover the logarithmic nonlinearity u log u. More precisely, we establish
the existence and uniqueness as well as the stability of p-integrable solutions
(p > 1) to multidimensional BSDEs with a p-integrable terminal condition and a
super-linear growth generator in the both variables y and z. This is done with
a generator f(y, z) which can be neither locally monotone in the variable y nor
locally Lipschitz in the variable z. Moreover, it is not uniformly continuous.
As application, we establish the existence and uniqueness of Sobolev solutions
to possibly degenerate systems of semilinear parabolic PDEs with super-linear
growth generator and an p-integrable terminal data. Our result cover, for
instance, certain (systems of) PDEs arising in physics.; Comment: 35

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## Viscosity solutions of systems of PDEs with interconnected obstacles and Multi modes switching problems

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Optimization and Control#Computer Science - Systems and Control#60G40, 62P20, 91B99, 91B28, 35B37, 49L25

This paper deals with existence and uniqueness, in viscosity sense, of a
solution for a system of m variational partial differential inequalities with
inter-connected obstacles. A particular case of this system is the
deterministic version of the Verification Theorem of the Markovian optimal
m-states switching problem. The switching cost functions are arbitrary. This
problem is connected with the valuation of a power plant in the energy market.
The main tool is the notion of systems of reflected BSDEs with oblique
reflection.; Comment: 36 pages

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## Solutions of DEs and PDEs as Potential Maps Using First Order Lagrangians

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 10/07/2000
Português

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Using parametrized curves (Section 1) or parametrized sheets (Section 3), and
suitable metrics, we treat the jet bundle of order one as a semi-Riemann
manifold. This point of view allows the description of solutions of DEs as
pregeodesics (Section 1) and the solutions of PDEs as potential maps (Section
3), via Lagrangians of order one or via generalized Lorentz world-force laws.
Implicitly, we solved a problem rised first by Poincar\'e: find a suitable
geometric structure that converts the trajectories of a given vector field into
geodesics (see also [6] - [11]). Section 2 and Section 3 realize the passage
from the Lagrangian dynamics to the covariant Hamilton equations.; Comment: 18 pages

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## $k$-symplectic Pontryagin's Maximum Principle for some families of PDEs

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 25/10/2012
Português

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An optimal control problem associated with the dynamics of the orientation of
a bipolar molecule in the plane can be understood by means of tools in
differential geometry. For first time in the literature $k$-symplectic
formalism is used to provide the optimal control problems associated to some
families of partial differential equations with a geometric formulation. A
parallel between the classic formalism of optimal control theory with ordinary
differential equations and the one with particular families of partial
differential equations is established. This description allows us to state and
prove Pontryagin's Maximum Principle on $k$-symplectic formalism. We also
consider the unified Skinner-Rusk formalism for optimal control problems
governed by an implicit partial differential equation.; Comment: 21 pages

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## Control of Homodirectional and General Heterodirectional Linear Coupled Hyperbolic PDEs

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 28/04/2015
Português

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Research on stabilization of coupled hyperbolic PDEs has been dominated by
the focus on pairs of counter-convecting ("heterodirectional") transport PDEs
with distributed local coupling and with controls at one or both boundaries. A
recent extension allows stabilization using only one control for a system
containing an arbitrary number of coupled transport PDEs that convect at
different speeds against the direction of the PDE whose boundary is actuated.
In this paper we present a solution to the fully general case, in which the
number of PDEs in either direction is arbitrary, and where actuation is applied
on only one boundary (to all the PDEs that convect downstream from that
boundary). To solve this general problem, we solve, as a special case, the
problem of control of coupled "homodirectional" hyperbolic linear PDEs, where
multiple transport PDEs convect in the same direction with arbitrary local
coupling. Our approach is based on PDE backstepping and yields solutions to
stabilization, by both full-state and observer-based output feedback,
trajectory planning, and trajectory tracking problems.

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## Barrier Functionals for Output Functional Estimation of PDEs

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 09/12/2014
Português

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We propose a method for computing bounds on output functionals of a class of
time-dependent PDEs. To this end, we introduce barrier functionals for PDE
systems. By defining appropriate unsafe sets and optimization problems, we
formulate an output functional bound estimation approach based on barrier
functionals. In the case of polynomial data, sum of squares (SOS) programming
is used to construct the barrier functionals and thus to compute bounds on the
output functionals via semidefinite programs (SDPs). An example is given to
illustrate the results.; Comment: 8 pages, 1 figure, preprint submitted to 2015 American Control
Conference

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## The kinetic limit of a system of coagulating Brownian particles

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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We consider a random model of diffusion and coagulation. A large number of
small particles are randomly scattered at an initial time. Each particle has
some integer mass and moves in a Brownian motion whose diffusion rate is
determined by that mass. When any two particles are close, they are liable to
combine into a single particle that bears the mass of each of them. Choosing
the initial density of particles so that, if their size is very small, a
typical one is liable to interact with a unit order of other particles in a
unit of time, we determine the macroscopic evolution of the system, in any
dimension d \geq 3. The density of particles evolves according to the
Smoluchowski system of PDEs, indexed by the mass parameter, in which the
interaction term is a sum of products of densities. Central to the proof is
establishing the so-called Stosszahlensatz, which asserts that, at any given
time, the presence of particles of two distinct masses at any given point in
macroscopic space is asymptotically independent, as the size of the particles
is taken towards zero.; Comment: 58 pages. Theorem 1.1 and Proposition 1 rewritten to indicate how the
proved convergence to the Smoluchowski PDE is stronger when uniqueness of
this solution is known

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## On construction of symmetries and recursion operators from zero-curvature representations and the Darboux-Egoroff system

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Nonlinear Sciences - Exactly Solvable and Integrable Systems#Mathematical Physics#Mathematics - Differential Geometry#37K10, 37K35, 37K30

The Darboux-Egoroff system of PDEs with any number $n\ge 3$ of independent
variables plays an essential role in the problems of describing $n$-dimensional
flat diagonal metrics of Egoroff type and Frobenius manifolds. We construct a
recursion operator and its inverse for symmetries of the Darboux-Egoroff system
and describe some symmetries generated by these operators.
The constructed recursion operators are not pseudodifferential, but are
Backlund autotransformations for the linearized system whose solutions
correspond to symmetries of the Darboux-Egoroff system. For some other PDEs,
recursion operators of similar types were considered previously by
Papachristou, Guthrie, Marvan, Poboril, and Sergyeyev.
In the structure of the obtained third and fifth order symmetries of the
Darboux-Egoroff system, one finds the third and fifth order flows of an
$(n-1)$-component vector modified KdV hierarchy.
The constructed recursion operators generate also an infinite number of
nonlocal symmetries. In particular, we obtain a simple construction of nonlocal
symmetries that were studied by Buryak and Shadrin in the context of the
infinitesimal version of the Givental-van de Leur twisted loop group action on
the space of semisimple Frobenius manifolds.
We obtain these results by means of rather general methods...

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## On the Equality of Solutions of Max-Min and Min-Max Systems of Variational Inequalities with Interconnected Bilateral Obstacles

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 19/08/2014
Português

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In this paper we deal with the solutions of systems of PDEs with bilateral
inter-connected obstacles of min-max and max-min types. These systems arise
naturally in stochastic switching zero-sum game problems. We show that when the
switching costs of one side are regular, the solutions of the min-max and
max-min systems coincide. Furthermore, this solution is identified as the value
function of a zero-sum switching game.

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## The Maslov index in PDEs geometry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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It is proved that the Maslov index naturally arises in the framework of PDEs
geometry. The characterization of PDE solutions by means of Maslov index is
given. With this respect, Maslov index for Lagrangian submanifolds is given on
the ground of PDEs geometry. New formulas to calculate bordism groups of
$(n-1)$-dimensional compact sub-manifolds bording via $n$-dimensional
Lagrangian submanifolds of a fixed $2n$-dimensional symplectic manifold are
obtained too. As a by-product it is given a new proof of global smooth
solutions existence, defined on all $\mathbb{R}^3$, for the Navier-Stokes PDE.
Further, complementary results are given in Appendices concerning Navier-Stokes
PDE and Legendrian submanifolds of contact manifolds.; Comment: 40 pages, 2 figures

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## Second order quasilinear PDEs and conformal structures in projective space

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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We investigate second order quasilinear equations of the form f_{ij}
u_{x_ix_j}=0 where u is a function of n independent variables x_1, ..., x_n,
and the coefficients f_{ij} are functions of the first order derivatives
p^1=u_{x_1}, >..., p^n=u_{x_n} only. We demonstrate that the natural
equivalence group of the problem is isomorphic to SL(n+1, R), which acts by
projective transformations on the space P^n with coordinates p^1, ..., p^n. The
coefficient matrix f_{ij} defines on P^n a conformal structure f_{ij} dp^idp^j.
In this paper we concentrate on the case n=3, although some results hold in any
dimension. The necessary and sufficient conditions for the integrability of
such equations by the method of hydrodynamic reductions are derived. These
conditions constitute a complicated over-determined system of PDEs for the
coefficients f_{ij}, which is in involution. We prove that the moduli space of
integrable equations is 20-dimensional. Based on these results, we show that
any equation satisfying the integrability conditions is necessarily
conservative, and possesses a dispersionless Lax pair. Reformulated in
differential-geometric terms, the integrability conditions imply that the
conformal structure f_{ij} dp^idp^j is conformally flat...

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## On the Lagrangian formulation of multidimensionally consistent systems

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 11/08/2010
Português

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#Nonlinear Sciences - Exactly Solvable and Integrable Systems#Mathematical Physics#Mathematics - Symplectic Geometry

Multidimensional consistency has emerged as a key integrability property for
partial difference equations (P$\Delta$Es) defined on the "space-time" lattice.
It has led, among other major insights, to a classification of scalar
affine-linear quadrilateral P$\Delta$Es possessing this property, leading to
the so-called ABS list. Recently, a new variational principle has been proposed
that describes the multidimensional consistency in terms of discrete Lagrangian
multi-forms. This description is based on a fundamental and highly nontrivial
property of Lagrangians for those integrable lattice equations, namely the fact
that on the solutions of the corresponding P$\Delta$E the Lagrange forms are
closed, i.e. they obey a {\it closure relation}. Here we extend those results
to the continuous case: it is known that associated with the integrable
P$\Delta$Es there exist systems of PDEs, in fact differential equations with
regard to the parameters of the lattice as independent variables, which equally
possess the property of multidimensional consistency. In this paper we
establish a universal Lagrange structure for affine-linear quad-lattices
alongside a universal Lagrange multi-form structure for the corresponding
continuous PDEs, and we show that the Lagrange forms possess the closure
property.; Comment: 22 pages

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## Uniqueness of Viscosity Solutions for Optimal Multi-Modes Switching Problem with Risk of default

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 06/02/2012
Português

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In this paper we study the optimal m-states switching problem in finite
horizon as well as infinite horizon with risk of default. We allow the
switching cost functionals and cost of default to be of polynomial growth and
arbitrary. We show uniqueness of a solution for a system of m variational
partial differential inequalities with inter-connected obstacles. This system
is the deterministic version of the Verification Theorem of the Markovian
optimal m-states switching problem with risk of default. This problem is
connected with the valuation of a power plant in the energy market.; Comment: 25 pages; Real options, Backward stochastic differential equations,
Snell envelope, Stopping times, Switching, Viscosity solution of PDEs,
Variational inequalities. arXiv admin note: text overlap with arXiv:0805.1306
and arXiv:0904.0707

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## New Method for Solving Large Classes of Nonlinear Systems of PDEs

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 09/10/2006
Português

Relevância na Pesquisa

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The essentials of a new method in solving very large classes of nonlinear
systems of PDEs, possibly associated with initial and/or boundary value
problems, are presented. The PDEs can be defined by continuous, not necessarily
smooth expressions, and the solutions obtained cab be assimilated with usual
measurable functions, or even with Hausdorff continuous ones. The respective
result sets aside completely, and with a large nonlinear margin, the celebrated
1957 impossibility of Hans Lewy regarding the nonexistence of solution in
distributions of large classes of linear smooth coefficient PDEs.

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## Open intersection numbers and the wave function of the KdV hierarchy

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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Recently R. Pandharipande, J. Solomon and R. Tessler initiated a study of the
intersection theory on the moduli space of Riemann surfaces with boundary. They
conjectured that the generating series of the intersection numbers is a
specific solution of a system of PDEs, that they called the open KdV equations.
In this paper we show that the open KdV equations are closely related to the
equations for the wave function of the KdV hierarchy. This allows us to give an
explicit formula for the specific solution in terms of Witten's generating
series of the intersection numbers on the moduli space of stable curves.; Comment: Revised version: 15 pages, to be published in the Moscow Mathematical
Journal

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