Trotter Formula Research Articles

Uniformly exponentially or polynomially stable approximations for second order evolution equations and some applications

In this paper, we consider the approximation of second order evolution equations. It is well known that the approximated system by finite element or finite difference is not uniformly exponentially or polynomially stable with respect to the discretization parameter, even if the continuous system has this property. Our goal is to damp the spurious high frequency modes by introducing numerical viscosity terms in the approximation scheme. With these viscosity terms, we show the exponential or polynomial decay of the discrete scheme when the continuous problem has such a decay and when the spectrum of the spatial operator associated with the undamped problem satisfies the generalized gap condition. By using the Trotter-Kato Theorem, we further show the convergence of the discrete solution to the continuous one. Some illustrative examples are also presented.

Stabilizations of the Trotter-Kato theorem and the Chernoff product formula

This paper concerns versions of the Trotter-Kato Theorem and the Chernoff Product Formula for C 0-semigroups in the absence of stability. Applications to \({\mathcal{A}}\)-stable rational approximations of semigroups are presented.

Sesquilinear quantum stochastic analysis in Banach space

A theory of quantum stochastic processes in Banach space is initiated. The processes considered here consist of Banach space valued sesquilinear maps. We establish an existence and uniqueness theorem for quantum stochastic differential equations in Banach modules, show that solutions in unital Banach algebras yield stochastic cocycles, give sufficient conditions for a stochastic cocycle to satisfy such an equation, and prove a stochastic Lie–Trotter product formula. The theory is used to extend, unify and refine standard quantum stochastic analysis through different choices of Banach space, of which there are three paradigm classes: spaces of bounded Hilbert space operators, operator mapping spaces and duals of operator space coalgebras. Our results provide the basis for a general theory of quantum stochastic processes in operator spaces, of which Lévy processes on compact quantum groups is a special case.

Erratum to: A Trotter product formula for gradient flows in metric spaces

d dt d2(u(t), y) ≤ φ(y)− φ(u(t)). The rescaled function v : [0, T∗/2] → X defined by v(t) = u(2t) satisfies the equations which have been stated for u. If the factor 1 2 is omitted, the theorem is correct. The proof is correct as stated and shows convergence to the correct equation, as can be seen, for example, in the last displayed formula of the proof on page 418. Further errors that need to be amended are given below: • On line 8 of page 413, [t j−1, t j ) should be replaced by [t j−1 h , t j h ). • The second line of the proof to Lemma 3.1 should be: “For the convenience of the reader, we provide a simple direct proof of (1) in the case that μ L d .”

On New Sum-Product--Type Estimates

New lower bounds involving sum, difference, product, and ratio sets of a set $A\subset {\mathbb C}$ are given. The estimates involving the sum set match, up to constants, the state-of-the-art estimates, proven by Solymosi for the reals and are obtained by generalizing his approach to the complex plane. The bounds involving the difference set improve the currently best known ones, also due to Solymosi, in both the real and complex cases by means of combining the Szemeredi--Trotter theorem with an arithmetic combinatorics technique.

Non-abelian Weyl commutation relations and the series product of quantum stochastic evolutions

We show that the series product, which serves as an algebraic rule for connecting state-based input-output systems, is intimately related to the Heisenberg group and the canonical commutation relations. The series product for quantum stochastic models then corresponds to a non-abelian generalization of the Weyl commutation relation. We show that the series product gives the general rule for combining the generators of quantum stochastic evolutions using a Lie-Trotter product formula.

Anisotropic magnetic molecular dynamics of cobalt nanowires

An investigation of thermally induced spin and lattice dynamics of a cobalt nanowire on a (111)Pt substrate is presented via magnetic molecular dynamics. This dynamical simulation model treats each atom as a particle supporting a classical spin. A coordinate dependent on both exchange and anisotropic functions ensures a minimal coupling between the spin and the lattice degrees of freedom to translate the magnetostrictive behavior of most magnetic materials. A spin-pair model of anisotropy is proposed to connect to the lattice thermodynamics. In order to solve linked spin-coordinate equations of motion, the efficiencies of algorithms based on Suzuki-Trotter decompositions are compared. The temperature dependence of the magnetic behavior of Co nanowires is investigated through thermal stochastic connections with mechanical and spin Langevin noises. From a magnetic Hamiltonian parametrized on ab initio calculations, the size dependence of the energy barriers and characteristic time scales of the magnetization relaxation are computed. In the superparamagnetic limit, it is shown that all spins in a nanowire evolve in a coherent rotation. When the size of the single nanowire increases, nucleations of domain walls let the activation energy be independent of the length of the wire.

The Fermionic Minus-Sign Problem for the Ground State, Revisited with Higher Order Suzuki–Trotter Decompositions

A theorem by Yoshida which states that the discretization error for the Suzuki–Trotter (ST) decomposition is one order smaller than for the corresponding approximated Hamiltonian has long been overlooked with grave consequences for the accuracy of algorithms which use decomposition schemes. In our error analysis for the ground state energy of the Hubbard Hamiltonian by the projector quantum Monte Carlo method, we have used various orders of ST-decompositions, including for the first time in this field, pseudo-symplectic methods. We show that higher order does not necessarily imply better convergence, and identify the condition to obtain good convergence with higher order ST-decompositions. For the first order ST-decomposition, the ground state energy does not converge to the values for numerical diagonalization, in agreement with Yoshida's theorem, which may have caused confusion in connection with the “fermionic sign problem”. We show how the use of the sign of the fermion determinant is in fact a reweig...

Analysis and approximation of a reliable model

In this paper we consider the problem of approximating the dynamical system that models reliability of a system consisting of two machines separated by a finite storage buffer. The system is described as a distributed parameter system defined by a coupled partial and ordinary differential equations and formulated as an abstract Cauchy problem. To derive the dynamical solution and some instantaneous indexes of the model, we present a simple finite difference scheme and establish the convergence of this scheme by employing Trotter–Kato Theorem. Numerical results are given to illustrate the effectiveness of the scheme.

Decomposition of modified Landau-Lifshitz-Gilbert equation and corresponding analytic solutions

The Suzuki-Trotter decomposition in general allows one to divide the equation of motion of a dynamical system into smaller parts whose integration are easier than the original equation. In this study, we first rewrite by employing feasible approximations the modified Landau-Lifshitz-Gilbert equation for localized spins in a suitable form for simulations using the Suzuki-Trotter decomposition. Next we decompose the equation into parts and demonstrate that the parts are classified into three groups, each of which can be solved exactly. Since the modified Landau-Lifshitz-Gilbert equation from which we start is in rather a general form, simulations of spin dynamics in various systems accompanying only small numerical errors are possible.

Hierarchical fractional-step approximations and parallel kinetic Monte Carlo algorithms

We present a mathematical framework for constructing and analyzing parallel algorithms for lattice kinetic Monte Carlo (KMC) simulations. The resulting algorithms have the capacity to simulate a wide range of spatio-temporal scales in spatially distributed, non-equilibrium physiochemical processes with complex chemistry and transport micro-mechanisms. Rather than focusing on constructing exactly the stochastic trajectories, our approach relies on approximating the evolution of observables, such as density, coverage, correlations and so on. More specifically, we develop a spatial domain decomposition of the Markov operator (generator) that describes the evolution of all observables according to the kinetic Monte Carlo algorithm. This domain decomposition corresponds to a decomposition of the Markov generator into a hierarchy of operators and can be tailored to specific hierarchical parallel architectures such as multi-core processors or clusters of Graphical Processing Units (GPUs). Based on this operator decomposition, we formulate parallel Fractional step kinetic Monte Carlo algorithms by employing the Trotter Theorem and its randomized variants; these schemes, (a) are partially asynchronous on each fractional step time-window, and (b) are characterized by their communication schedule between processors.The proposed mathematical framework allows us to rigorously justify the numerical and statistical consistency of the proposed algorithms, showing the convergence of our approximating schemes to the original serial KMC. The approach also provides a systematic evaluation of different processor communicating schedules. We carry out a detailed benchmarking of the parallel KMC schemes using available exact solutions, for example, in Ising-type systems and we demonstrate the capabilities of the method to simulate complex spatially distributed reactions at very large scales on GPUs. Finally, we discuss work load balancing between processors and propose a re-balancing scheme based on probabilistic mass transport methods.

Simple Proofs of Classical Theorems in Discrete Geometry via the Guth–Katz Polynomial Partitioning Technique

Recently Guth and Katz ( arXiv:1011.4105, 2010) invented, as a step in their nearly complete solution of Erdős’s distinct distances problem, a new method for partitioning finite point sets in ℝ d , based on the Stone–Tukey polynomial ham-sandwich theorem. We apply this method to obtain new and simple proofs of two well known results: the Szemeredi–Trotter theorem on incidences of points and lines, and the existence of spanning trees with low crossing numbers. Since we consider these proofs particularly suitable for teaching, we aim at self-contained, expository treatment. We also mention some generalizations and extensions, such as the Pach–Sharir bound on the number of incidences with algebraic curves of bounded degree.

Holographic entanglement entropy in Suzuki-Trotter decomposition of spin systems

In quantum spin chains at criticality, two types of scaling for the entanglement entropy exist: one comes from conformal field theory (CFT), and the other is for entanglement support of matrix product state (MPS) approximation. On the other hand, the quantum spin-chain models can be mapped onto two-dimensional (2D) classical ones by the Suzuki-Trotter decomposition. Motivated by the scaling and the mapping, we introduce information entropy for 2D classical spin configurations as well as a spectrum, and examine their basic properties in the Ising and the three-state Potts models on the square lattice. They are defined by the singular values of the reduced density matrix for a Monte Carlo snapshot. We find scaling relations of the entropy compatible with the CFT and the MPS results. Thus, we propose that the entropy is a kind of "holographic" entanglement entropy. At T(c), the spin configuration is fractal, and various sizes of ordered clusters coexist. Then, the singular values automatically decompose the original snapshot into a set of images with different length scales, respectively. This is the origin of the scaling. In contrast to the MPS scaling, long-range spin correlation can be described by only few singular values. Furthermore, the spectrum, which is a set of logarithms of the singular values, also seems to be a holographic entanglement spectrum. We find multiple gaps in the spectrum, and in contrast to the topological phases, the low-lying levels below the gap represent spontaneous symmetry breaking. These contrasts are strong evidence of the dual nature of the holography. Based on these observations, we discuss the amount of information contained in one snapshot.

Approximation for convex functionals on non-positively curved spaces and the Trotter–Kato product formula

Abstract. We study the validity of the Trotter–Kato product formula in the setting of gradient flows on CAT(0) metric spaces. We follow the strategy of the proof of the Hilbert space version of this theorem given by Kato–Masuda, but instead of the linear structure and inner product we have only geodesics and the CAT(0) inequality available. Thus we construct a counterpart of the approximation semigroups and their resolvents in CAT(0) spaces. We show that the convergence of the approximated resolvents to the resolvents of the sum functional implies convergence of the approximation semigroups to the semigroup associated to the sum functional. We also show that this resolvents convergence implies the product Trotter–Kato formula in CAT(0) spaces. These approximation theorems are of independent interest. A major difficulty compared to the linear theory is the lack of an appropriate notion of weak convergence on such spaces. This difficulty is successfully overcome with the aid of an ultra-filters technique. The main results of our investigation are various versions of the Trotter–Kato product formula in CAT(0) spaces.

A hyperbolic model of spatial evolutionary game theory

We present a one space dimensional model with finite speed of propagation for population dynamics, based both on thehyperbolic Cattaneo dynamics and the evolutionary game theory. We prove analytical properties of the model and globalestimates for solutions, by using a hyperbolic nonlinear Trotter product formula.

A simple counterexample related to the Lie–Trotter product formula

In this note a very simple example is given which shows that if the sum of two semigroup generators is itself a generator, the generated semigroup in general can not be given by the Lie–Trotter product formula.

On a Stochastic Trotter Formula with Application to Spontaneous Localization Models

We consider the relation between so called continuous localization models—i.e. non-linear stochastic Schrödinger evolutions—and the discrete GRW-model of wave function collapse. The former can be understood as scaling limit of the GRW process. The proof relies on a stochastic Trotter formula, which is of interest in its own right. Our Trotter formula also allows to complement results on existence theory of stochastic Schrödinger evolutions by Holevo and Mora/Rebolledo.

Error analysis of molecular dynamics and fractal time approximants from a combinatorial perspective

Trotter's theorem forms the theoretical basis of most modern molecular dynamics. In essence this theorem states that a time displacement operator (a Lie operator) constructed by exponentiating a sum of noncommuting operators can be approximated by a product of single operators provided the time interval is "very small." In theory "very small" implies infinitesimally small (at which point the approximate product becomes exact), while in practical analysis a finite time interval is divided into several small subintervals or steps. It follows, therefore, that the larger the number of steps the better the approximation to the exact time displacement operator. The question therefore arises: How many steps are sufficient? For bounded operators, standard theorems are available to provide the answer. In this paper we show that a very simple combinatorial formula can be derived which allows the computation of the global differences (as a function of the number of steps) between the Taylor coefficients of the exact time displacement operator and an approximate one constructed by using a finite number of steps. The formula holds for both bounded and nonbounded operators and shows, quantitatively, what is qualitatively expected-that the error decreases with increasing number of steps. Furthermore, the formula applies irrespective of the complexity of the system, boundary conditions, or the thermodynamic ensemble employed for averaging the initial conditions. The analysis yields explicit expressions for the Taylor coefficients which are then used to compute the errors. In the case of the algorithmically based practical numerical simulations in which fixed, albeit small, steps are repeatedly applied, the rise in the number of steps does not reduce the size of the steps but increases the total time of interest. The combinatorial formula shows that, here, the errors diverge. Furthermore, this work can be used to supplement other efforts such as the use of shadow Hamiltonians where the truncation of the series expansion of the latter will produce errors in the higher order propagator moments.

Nonlinear analysis of a fractional reaction diffusion model for tumour invasion

Mathematical models in general and Reaction Diffusion Models in particular have been rigorously studied and applied in different forms to explain situation in biomedical and allied sciences including the complex tumour microenvironment. They have been proven to be really significant in cancer research. The not so extensively known fractional reaction-diffusion model is presented in this research as a plausible model for the invasion of tumour and its consequences in the human organ in which it resides. In this model we make use of fractional derivatives as a replacement for the normal derivatives to express the diffusion of the tumour cells, normal cells and hydrogen ion concentration. The nonlinear analysis of this model predicts that a death situation always arise if some of the parameters are of a certain magnitude. In this situation, the tumour cells population increases (with an eventual bound after death) by eating up all the normal cells of their host organ. Effectively, the model predicts that the relative interaction between the tumor and normal cells population gives rise to a bifurcation parameter for which a Hopf bifurcation occurs between the period of attack and death. Also the death situation using this model comes faster than the predictions of other models for tumour invasion. This research may therefore give a useful indication of the possible timing of the drugs and dosages required for the treatment or control of tumour progression in human. Key words: Tumour, fractional reaction diffusion equation, trotter product formula, hopf bifurcation, vasculature, oxygenation, Hypoxia.

Feynman formulas for the diffusion of particles with position-dependent mass

In the present paper, Feynman formulas are obtained for Schrodinger semigroups generated by self-adjoint operators which are perturbations of self-adjoint extensions of the second-order Hamiltonian operator −Δg,0/2+V (throughout the paper, the coefficient −1/2 at Δg,0 is omitted to simplify the formulas) which describe the diffusion of a quasiparticle with position-dependent mass varying jump-like on a line. Every extension of this kind is defined by some invertible operator and is characterized by matching conditions at a jump point. The Schrodinger semigroups generated by self-adjoint Laplace operators and defined by the corresponding boundary conditions define solutions of initial-boundary value problems. In turn, the term “Feynman formulas” is applied (in the present case) to an explicit representation of the Schrodinger semigroup \( e^{t\hat H^T } \) in the form of a limit of integrals of finite multiplicity over Cartesian powers of some configuration space. In essence, the Feynman-Kac formula is a “probabilistic interpretation” of the Feynman formulas. Namely, the multiple integrals in the Feynman formulas approximate integrals against some measures on the space of trajectories (functions defined on an interval of the real line and ranging in the configuration space). Thus, the Feynman formulas enable one to evaluate integrals over spaces of trajectories. A crucial role in the proof of the Feynman formulas is played by the Chernoff theorem, which is a generalization of the famous Trotter formula. The result proved in the present paper is a demonstration of a part of the results recently announced by O. G. Smolyanov and H. von Weizacker (“Feynman Formulas Generated by Self-Adjoint Extensions of the Laplacian,” Dokl. Ross. Akad. Nauk 426 (2), 162–165 (2009) [Doklady Mathematics, 2009 79 (3), 335–338 (2009)]). The formulations of the results in question are inessentially modified here.