41,159 research outputs found
The Generalized Quasi-linearization Method for Reaction Diffusion Equations on an Unbounded Domain
AbstractThe method of generalized quasi-linearization has been well developed for ordinary differential equations. In this paper, we extend the method of generalized quasi-linearization to reaction diffusion equations on an unbounded domain. The iterates, which are solutions of linear equations starting from lower and upper solutions, converge uniformly and monotonically to the unique solution of the nonlinear reaction diffusion equation in an unbounded domain. Initially an existence theorem for the linear nonhomogeneous reaction diffusion equation in an unbounded domain has been proved under improved conditions. The quadratic convergence has been proved by using a comparison theorem of reaction diffusion equations with ordinary differential equations. This avoids the computational complexity of the quasi-linearization method, since the computation of Green's function at each stage of the iterates is avoided
Network Identification for Diffusively-Coupled Systems with Minimal Time Complexity
The theory of network identification, namely identifying the (weighted)
interaction topology among a known number of agents, has been widely developed
for linear agents. However, the theory for nonlinear agents using probing
inputs is less developed and relies on dynamics linearization. We use global
convergence properties of the network, which can be assured using passivity
theory, to present a network identification method for nonlinear agents. We do
so by linearizing the steady-state equations rather than the dynamics,
achieving a sub-cubic time algorithm for network identification. We also study
the problem of network identification from a complexity theory standpoint,
showing that the presented algorithms are optimal in terms of time complexity.
We also demonstrate the presented algorithm in two case studies.Comment: 12 pages, 3 figure
Gain Scheduling Control of Nonlinear Shock Motion Based on Equilibrium Manifold Linearization Model
AbstractThe equilibrium manifold linearization model of nonlinear shock motion is of higher accuracy and lower complexity over other models such as the small perturbation model and the piecewise-linear model. This paper analyzes the physical significance of the equilibrium manifold linearization model, and the self-feedback mechanism of shock motion is revealed. This helps to describe the stability and dynamics of shock motion. Based on the model, the paper puts forwards a gain scheduling control method for nonlinear shock motion. Simulation has shown the validity of the control scheme
Quantum algorithms for linear and non-linear fractional reaction-diffusion equations
High-dimensional fractional reaction-diffusion equations have numerous
applications in the fields of biology, chemistry, and physics, and exhibit a
range of rich phenomena. While classical algorithms have an exponential
complexity in the spatial dimension, a quantum computer can produce a quantum
state that encodes the solution with only polynomial complexity, provided that
suitable input access is available. In this work, we investigate efficient
quantum algorithms for linear and nonlinear fractional reaction-diffusion
equations with periodic boundary conditions. For linear equations, we analyze
and compare the complexity of various methods, including the second-order
Trotter formula, time-marching method, and truncated Dyson series method. We
also present a novel algorithm that combines the linear combination of
Hamiltonian simulation technique with the interaction picture formalism,
resulting in optimal scaling in the spatial dimension. For nonlinear equations,
we employ the Carleman linearization method and propose a block-encoding
version that is appropriate for the dense matrices that arise from the spatial
discretization of fractional reaction-diffusion equations
High-order regularized regression in Electrical Impedance Tomography
We present a novel approach for the inverse problem in electrical impedance
tomography based on regularized quadratic regression. Our contribution
introduces a new formulation for the forward model in the form of a nonlinear
integral transform, that maps changes in the electrical properties of a domain
to their respective variations in boundary data. Using perturbation theory the
transform is approximated to yield a high-order misfit unction which is then
used to derive a regularized inverse problem. In particular, we consider the
nonlinear problem to second-order accuracy, hence our approximation method
improves upon the local linearization of the forward mapping. The inverse
problem is approached using Newton's iterative algorithm and results from
simulated experiments are presented. With a moderate increase in computational
complexity, the method yields superior results compared to those of regularized
linear regression and can be implemented to address the nonlinear inverse
problem
Multiparameter spectral analysis for aeroelastic instability problems
This paper presents a novel application of multiparameter spectral theory to
the study of structural stability, with particular emphasis on aeroelastic
flutter. Methods of multiparameter analysis allow the development of new
solution algorithms for aeroelastic flutter problems; most significantly, a
direct solver for polynomial problems of arbitrary order and size, something
which has not before been achieved. Two major variants of this direct solver
are presented, and their computational characteristics are compared. Both are
effective for smaller problems arising in reduced-order modelling and
preliminary design optimization. Extensions and improvements to this new
conceptual framework and solution method are then discussed.Comment: 20 pages, 8 figure
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