Nonlinear Projection Operator Research Articles

Generalized nonlinear Langevin equation from quantum nonlinear projection operator

Generalized nonlinear Langevin equation from quantum nonlinear projection operator

A nodally bound-preserving finite element method

Abstract This work proposes a nonlinear finite element method whose nodal values preserve bounds known for the exact solution. The discrete problem involves a nonlinear projection operator mapping arbitrary nodal values into bound-preserving ones and seeks the numerical solution in the range of this projection. As the projection is not injective, a stabilisation based upon the complementary projection is added in order to restore well-posedness. Within the framework of elliptic problems, the discrete problem may be viewed as a reformulation of a discrete obstacle problem, incorporating the inequality constraints through Lipschitz projections. The derivation of the proposed method is exemplified for linear and nonlinear reaction-diffusion problems. Near-best approximation results in suitable norms are established. In particular, we prove that, in the linear case, the numerical solution is the best approximation in the energy norm among all nodally bound-preserving finite element functions. A series of numerical experiments for such problems showcase the good behaviour of the proposed bound-preserving finite element method.

Dynamic coarse-graining of linear and non-linear systems: Mori-Zwanzig formalism and beyond.

To investigate the impact of non-linear interactions on dynamic coarse graining, we study a simplified model system featuring a tracer particle in a complex environment. Using a projection operator formalism and computer simulations, we systematically derive generalized Langevin equations (GLEs) describing the dynamics of this particle. We compare different kinds of linear and non-linear coarse-graining procedures to understand how non-linearities enter reconstructed GLEs and how they influence the coarse-grained dynamics. For non-linear external potentials, we show analytically and numerically that the non-Gaussian parameter and the incoherent intermediate scattering function will not be correctly reproduced by the GLE if a linear projection is applied. This, however, can be overcome by using non-linear projection operators. We also study anharmonic coupling between the tracer and the environment and demonstrate that the reconstructed memory kernel develops an additional trap-dependent contribution. Our study highlights some open challenges and possible solutions in dynamic coarse graining.

Hilbert points in Hardy spaces

A Hilbert point in H p ( T d ) H^p(\mathbb {T}^d) , for d ≥ 1 d\geq 1 and 1 ≤ p ≤ ∞ 1\leq p \leq \infty , is a nontrivial function φ \varphi in H p ( T d ) H^p(\mathbb {T}^d) such that ‖ φ ‖ H p ( T d ) ≤ ‖ φ + f ‖ H p ( T d ) \| \varphi \|_{H^p(\mathbb {T}^d)} \leq \|\varphi + f\|_{H^p(\mathbb {T}^d)} whenever f f is in H p ( T d ) H^p(\mathbb {T}^d) and orthogonal to φ \varphi in the usual L 2 L^2 sense. When p ≠ 2 p\neq 2 , φ \varphi is a Hilbert point in H p ( T ) H^p(\mathbb {T}) if and only if φ \varphi is a nonzero multiple of an inner function. An inner function on T d \mathbb {T}^d is a Hilbert point in any of the spaces H p ( T d ) H^p(\mathbb {T}^d) , but there are other Hilbert points as well when d ≥ 2 d\geq 2 . The case of 1 1 -homogeneous polynomials is studied in depth and, as a byproduct, a new proof is given for the sharp Khinchin inequality for Steinhaus variables in the range 2 > p > ∞ 2>p>\infty . Briefly, the dynamics of a certain nonlinear projection operator is treated. This operator characterizes Hilbert points as its fixed points. An example is exhibited of a function φ \varphi that is a Hilbert point in H p ( T 3 ) H^p(\mathbb {T}^3) for p = 2 , 4 p=2, 4 , but not for any other p p ; this is verified rigorously for p > 4 p>4 but only numerically for 1 ≤ p > 4 1\leq p>4 .

Backward Stackelberg Differential Game with Constraints: A Mixed Terminal-Perturbation and Linear-Quadratic Approach

We discuss an open-loop backward Stackelberg differential game involving single leader and single follower. Unlike most Stackelberg game literature, the state to be controlled is characterized by a backward stochastic differential equation (BSDE) for which the terminal- instead initial-condition is specified as a priori; the decisions of leader consist of a static terminal-perturbation and a dynamic linear-quadratic control. In addition, the terminal control is subject to (convex-closed) pointwise and (affine) expectation constraints. Both constraints are arising from real applications such as mathematical finance. For information pattern: the leader announces both terminal and open-loop dynamic decisions at the initial time while takes account the best response of follower. Then, two interrelated optimization problems are sequentially solved by the follower (a backward linear-quadratic (BLQ) problem) and the leader (a mixed terminal-perturbation and backward-forward LQ (BFLQ) problem). Our open-loop Stackelberg equilibrium is represented by some coupled backward-forward stochastic differential equations (BFSDEs) with mixed initial-terminal conditions. Our BFSDEs also involve nonlinear projection operator (due to pointwise constraint) combining with a Karush-Kuhn-Tucker (KKT) system (due to expectation constraint) via Lagrange multiplier. The global solvability of such BFSDEs is also discussed in some nontrivial cases. Our results are applied to one financial example.

Position-constrained containment for second-order discrete-time multi-agent systems

In this paper, we study a containment control problem for second-order multi-agent systems in the presence of convex position constraints and switching graphs. A distributed algorithm with a nonlinear projection operator is proposed to guarantee that each agent eventually converges into the convex hull spanned by some given multiple static points, while the position of each agent is kept lying in a given closed set. It is proved that all agents ultimately converge into the convex hull while each agent remains in its constraint set as long as there exists a directed path from the static points to the agents in the union of the graphs in each bounded time interval. Finally, a numerical example is given to illustrate the obtained theoretical results.

Handling Gaussian blur without deconvolution

The paper presents a new theory of invariants to Gaussian blur. Unlike earlier methods, the blur kernel may be arbitrary oriented, scaled and elongated. Such blurring is a semi-group action in the image space, where the orbits are classes of blur-equivalent images. We propose a non-linear projection operator which extracts blur-insensitive component of the image. The invariants are then formally defined as moments of this component but can be computed directly from the blurred image without an explicit construction of the projections. Image description by the new invariants does not require any prior knowledge of the blur kernel parameters and does not include any deconvolution. The invariance property could be extended also to linear transformation of the image coordinates and combined affine-blur invariants can be constructed. Experimental comparison to three other blur-invariant methods is given. Potential applications of the new invariants are in blur/position invariant image recognition and in robust template matching.

Iteration Methods with Multigrid in Energy for Eigenvalue Neutron Diffusion Problems

In this paper we present nonlinear multilevel methods with multiple grids in energy for solving the k-eigenvalue problem for multigroup neutron diffusion equations. We develop multigrid-in-energy algorithms based on a nonlinear projection operator and several advanced prolongation operators. The evaluation of the eigenvalue is performed in the space with smallest dimensionality by solving the effective one-group diffusion problem. We consider two-dimensional Cartesian geometry. The multilevel methods are formulated in discrete form for the second-order finite volume discretization of the diffusion equation. The homogenization in energy is based on a spatially consistent discretization of the group diffusion equations on coarse grids in energy. We present numerical results of model reactor-physics problems with 44 energy groups. They demonstrate performance and main properties of the proposed iterative methods with multigrid in energy.

A semigroup approach to an integro-differential equation modeling slow erosion

The paper is concerned with a scalar conservation law with nonlocal flux, providing a model for granular flow with slow erosion and deposition. While the solution u=u(t,x) can have jumps, the inverse function x=x(t,u) is always Lipschitz continuous; its derivative has bounded variation and satisfies a balance law with measure-valued sources. Using a backward Euler approximation scheme combined with a nonlinear projection operator, we construct a continuous semigroup whose trajectories are the unique entropy weak solutions to this balance law. Going back to the original variables, this yields the global well-posedness of the Cauchy problem for the granular flow model.

A PROJECTION OPERATOR APPROACH TO THE OPTIMIZATION OF TRAJECTORY FUNCTIONALS

We develop a Newton method for the optimization of trajectory functionals. Through the use of a trajectory tracking nonlinear projection operator, the dynamically constrained optimization problem is converted into an unconstrained problem, making many aspects of the algorithm rather transparent. Examples: first and second order optimality conditions, search direction and step length calculations, update rule—all developed from an unconstrained point of view. Quasi-Newton methods are easily developed as well, allowing straightforward globalization of the Newton method. As all operations are set in an appropriate Banach space, properties such as solution regularity are retained so that implementation decisions (level of discretation, etc.) are based on approximating the solution rather than the problem. Convergence in Banach space is shown to be quadratic as is usual for Newton methods.

Towards a microscopic theory of nonisothermal stochastic processes

Using a nonlinear projection operator technique, the Markovian stochastic dynamics of a classical Brownian particle is derived in the case that the effective temperature is not constant. After recapitulating the general formalism and its application to isothermal Brownian motion, an isolated ‘particle plus environment’ system is analyzed in the microcanomical ensemble. The ensuing generalized Kramers equation is shown to involve a microcanonical potential of mean force different from the conditional free energy, and the conditional entropy emerges as the relevant equilibrium potential. The Smoluchowski limit is presented, and the effective spatial diffusion coefficient is calculated. Upon coupling the system to a heat bath, the microcanonical theory is seen to correspond to the case of small heat conductance. The final result for the stochastic process in the position, momentum, and energy variables is shown to be consistent with both isothermal Kramers theory and nonlinear equilibrium thermodynamics.

Method of projection operators in the theory of irreversible processes

The Zwanzig-Nakajima projection-operator method is extended to the case of time-dependent nonlinear projection operators. A method employing these operators is proposed for constructing the non-Markovian kinetic equation for a single-particle, time-dependent distribution function. An important assumption is made in the derivation of the equation regarding the factorization of the initial nonequilibrium distribution of the multi-particle system. An approximate kinetic equation is obtained for a slightly nonequilibrium system which asymptotically approaches the equilibrium canonical distribution at a fixed temperature. The effects of the self-consistent Vlasov field and the non-Markovian Fokker-Planck collision operator with a microscopic parameter appear in this equation. This is the first derivation of such an equation.