9 research outputs found
Optimal solution of ordinary differential equations
AbstractWe survey some recent optimality results for the numerical solution of initial value problems for ODE. We assume that information used by an algorithm about a right-hand-side function is partial. Two settings of information-based complexity are considered: the worst case and asymptotic. Upper and lower bounds on the error are presented for three types of information: standard, linear, and nonlinear continuous. In both settings, minimum error algorithms are exhibited
Almost Optimal Solution of Initial-Value Problems by Randomized and Quantum Algorithms
We establish essentially optimal bounds on the complexity of initial-value
problems in the randomized and quantum settings. For this purpose we define a
sequence of new algorithms whose error/cost properties improve from step to
step. These algorithms yield new upper complexity bounds, which differ from
known lower bounds by only an arbitrarily small positive parameter in the
exponent, and a logarithmic factor. In both the randomized and quantum
settings, initial-value problems turn out to be essentially as difficult as
scalar integration.Comment: 16 pages, minor presentation change
On a randomized backward Euler method for nonlinear evolution equations with time-irregular coefficients
In this paper we introduce a randomized version of the backward Euler method,
that is applicable to stiff ordinary differential equations and nonlinear
evolution equations with time-irregular coefficients. In the finite-dimensional
case, we consider Carath\'eodory type functions satisfying a one-sided
Lipschitz condition. After investigating the well-posedness and the stability
properties of the randomized scheme, we prove the convergence to the exact
solution with a rate of in the root-mean-square norm assuming only that
the coefficient function is square integrable with respect to the temporal
parameter.
These results are then extended to the numerical solution of
infinite-dimensional evolution equations under monotonicity and Lipschitz
conditions. Here we consider a combination of the randomized backward Euler
scheme with a Galerkin finite element method. We obtain error estimates that
correspond to the regularity of the exact solution. The practicability of the
randomized scheme is also illustrated through several numerical experiments.Comment: 37 pages, 3 figure
Euler scheme for approximation of solution of nonlinear ODEs under inexact information
We investigate error of the Euler scheme in the case when the right-hand side
function of the underlying ODE satisfies nonstandard assumptions such as local
one-sided Lipschitz condition and local H\"older continuity. Moreover, we
assume two cases in regards to information availability: exact and noisy with
respect to the right-hand side function. Optimality analysis of the Euler
scheme is also provided. Finally, we present the results of some numerical
experiments.Comment: 18 pages, 9 figure
Complexity of randomized algorithms for underdamped Langevin dynamics
We establish an information complexity lower bound of randomized algorithms
for simulating underdamped Langevin dynamics. More specifically, we prove that
the worst strong error is of order , for
solving a family of -dimensional underdamped Langevin dynamics, by any
randomized algorithm with only queries to , the driving Brownian
motion and its weighted integration, respectively. The lower bound we establish
matches the upper bound for the randomized midpoint method recently proposed by
Shen and Lee [NIPS 2019], in terms of both parameters and .Comment: 27 pages; some revision (e.g., Sec 2.1), and new supplementary
materials in Appendice
Quantum Computing for Fusion Energy Science Applications
This is a review of recent research exploring and extending present-day
quantum computing capabilities for fusion energy science applications. We begin
with a brief tutorial on both ideal and open quantum dynamics, universal
quantum computation, and quantum algorithms. Then, we explore the topic of
using quantum computers to simulate both linear and nonlinear dynamics in
greater detail. Because quantum computers can only efficiently perform linear
operations on the quantum state, it is challenging to perform nonlinear
operations that are generically required to describe the nonlinear differential
equations of interest. In this work, we extend previous results on embedding
nonlinear systems within linear systems by explicitly deriving the connection
between the Koopman evolution operator, the Perron-Frobenius evolution
operator, and the Koopman-von Neumann evolution (KvN) operator. We also
explicitly derive the connection between the Koopman and Carleman approaches to
embedding. Extension of the KvN framework to the complex-analytic setting
relevant to Carleman embedding, and the proof that different choices of complex
analytic reproducing kernel Hilbert spaces depend on the choice of Hilbert
space metric are covered in the appendices. Finally, we conclude with a review
of recent quantum hardware implementations of algorithms on present-day quantum
hardware platforms that may one day be accelerated through Hamiltonian
simulation. We discuss the simulation of toy models of wave-particle
interactions through the simulation of quantum maps and of wave-wave
interactions important in nonlinear plasma dynamics.Comment: 42 pages; 12 figures; invited paper at the 2021-2022 International
Sherwood Fusion Theory Conferenc