Search

While the periodic equation-of-motion coupled-cluster (EOM-CC) method promises systematic improvement of electronic band gap calculations in solids, its practical application at the singles and doubles level (EOM-CCSD) is hindered by severe finite-size errors in feasible simulation cells. We present a hybrid approach combining EOM-CCSD with the computationally efficient $GW$ approximation to estimate thermodynamic limit band gaps for several insulators and semiconductors. Our method substantially reduces required cell sizes while maintaining accuracy. Comparisons with experimental gaps and self-consistent $GW$ calculations reveal that deviations in EOM-CCSD predictions correlate with reduced single excitation character of the excited many-electron states. Our work not only provides a computationally tractable approach to EOM-CC calculations in solids but also reveals fundamental insights into the role of single excitations in electronic-structure theory., Comment: 7 (+12 supplementary) pages, 3(+7) figures

Quantum-mechanical simulations can offer atomic-level insights into chemical processes on surfaces. This understanding is crucial for the rational design of new solid catalysts as well as materials to store energy and mitigate greenhouse gases. However, achieving the accuracy needed for reliable predictions has proven challenging. Density functional theory (DFT), the workhorse quantum-mechanical method, can often lead to inconsistent predictions, necessitating accurate methods from correlated wave-function theory (cWFT). However, the high computational demands and significant user intervention associated with cWFT have traditionally made it impractical to carry out for surfaces. In this work, we address this challenge, presenting an automated framework which leverages multilevel embedding approaches, to apply accurate cWFT methods to ionic surfaces with computational costs approaching DFT. With this framework, we have reproduced experimental adsorption enthalpies for a diverse set of 19 adsorbate-surface systems. Moreover, we resolve long-standing debates on the adsorption configuration of several systems, while offering valuable benchmarks to assess DFT. This framework is completely open-source, making it possible to now routinely apply cWFT to complex problems in ionic surface chemistry.

We present a study of the construction and spatial properties of localized Wannier orbitals in large supercells of insulating solids using plane waves as the underlying basis. The Pipek-Mezey (PM) functional in combination with intrinsic atomic orbitals (IAOs) as projectors is employed, resulting in so-called intrinsic bond orbitals (IBOs). Independent of the bonding type and band gap, a correlation between orbital spreads and geometric properties is observed. As a result, comparable sparsity patterns of the Hartree-Fock exchange matrix are found across all considered bulk 3D materials, exhibiting covalent bonds, ionc bonds, and a mixture of both bonding types. Recognizing the considerable computational effort required to construct localized Wannier orbitals for large periodic simulation cells, we address the performance and scaling of different solvers for the localization problem. This includes the Broyden-Fletcher-Goldfarb-Shanno (BFGS), Conjugate-Gradient (CG), Steepest Ascent (SA) as well as the Direct Inversion in the Iterative Subspace (DIIS) method. Each algorithm performs a Riemannian optimization under unitary matrix constraint, efficiently reaching the optimum in the "curved parameter space" on geodesics. We hereby complement the literature with an introduction to this topic along with key references. Furthermore, we observe that the construction of Wannier orbitals for supercells of metal oxides presents a significant challenge, requiring approximately one order of magnitude more iteration steps than other systems studied., Comment: 28 pages, 9 figures, 4 tables

We investigate the convergence of quasi-particle energies for periodic systems to the thermodynamic limit using increasingly large simulation cells corresponding to increasingly dense integration meshes in reciprocal space. The quasi-particle energies are computed at the level of equation-of-motion coupled-cluster theory for ionization (IP-EOM-CC) and electron attachment processes (EA-EOM-CC). By introducing an electronic correlation structure factor, the expected asymptotic convergence rates for systems with different dimensionality are formally derived. We rigorously test these derivations through numerical simulations for trans-Polyacetylene using IP/EA-EOM-CCSD and the G0W0@HF approximation, which confirm the predicted convergence behavior. Our findings provide a solid foundation for efficient schemes to correct finite-size errors in IP/EA-EOM-CCSD calculations., Comment: 16 pages, 9 figures

Many-electron correlation methods offer a systematic approach to predicting material properties with high precision. However, practically attaining accurate ground-state properties for bulk metals presents significant challenges. In this work, we propose a novel scheme to reach the thermodynamic limit of the total ground-state energy of metals using coupled cluster theory. We demonstrate that the coupling between long-range and short-range contributions to the correlation energy is sufficiently weak, enabling us to restrict long-range contributions to low-energy excitations in a controllable way. Leveraging this insight, we calculate the surface energy of aluminum and platinum (111), providing numerical evidence that coupled cluster theory is well-suited for modeling metallic materials, particularly in surface science. Notably, our results exhibit convergence with respect to finite-size effects, basis-set size, and coupled cluster expansion, yielding excellent agreement with experimental data. This paves the way for more efficient coupled cluster calculations for large systems and a broader utilization of the theory in realistic metallic models of materials., Comment: 22 pages, 5 figures

Quantum mechanical many-electron calculations can predict properties of atoms, molecules and even complex materials. The employed computational methods play a quintessential role in many scientifically and technologically relevant research fields. However, a question of paramount importance is whether approximations aimed at reducing the computational complexity for solving the many-electron Schr\"odinger equation, are accurate enough. Here, we investigate recently reported discrepancies of noncovalent interaction energies for large molecules predicted by two of the most widely-trusted many-electron theories: diffusion quantum Monte Carlo and coupled-cluster theory. We are able to unequivocally pin down the source of the puzzling discrepancies and present modifications to widely-used coupled-cluster methods needed for more accurate noncovalent interaction energies of large molecules on the hundred-atom scale. This enhances the reliability of predictions from quantum mechanical many-electron theories across a wide range of critical applications, including drug design, catalysis, and the innovation of new functional materials, such as those for renewable energy technologies., Comment: 14 pages, 3 figures, 2 tables

Matrix evolution equations occur in many applications, such as dynamical Lyapunov/Sylvester systems or Riccati equations in optimization and stochastic control, machine learning or data assimilation. In many cases, their tightest stability condition is coming from a linear term. Exponential time differencing (ETD) is known to produce highly stable numerical schemes by treating the linear term in an exact fashion. In particular, for stiff problems, ETD methods are a method of choice. We propose an extension of the class of ETD algorithms to matrix-valued dynamical equations. This allows us to produce highly efficient and stable integration schemes. We show their efficiency and applicability for a variety of real-world problems, from geophysical applications to dynamical problems in machine learning.

Recently, Josserand et al. proposed a stochastic nonlinear Schroedinger model for finite-time singularity-mediated turbulence [Phys. Rev. Fluids 5, 054607 (2020)]. Here, we use instanton calculus to quantify the effect of extreme fluctuations on the statistics of the energy dissipation rate. While the contribution of the instanton alone is insufficient, we obtain excellent agreement with direct simulations when including Gaussian fluctuations and the corresponding zero mode. Fluctuations are crucial to obtain the correct scaling when quasi-singular events govern the turbulence statistics., Comment: 5 pages, 3 figures, 5 pages appendix

We present a robust strategy to numerically sample the Coulomb potential in reciprocal space for periodic Born-von Karman cells of general shape. Our approach tackles two common issues of plane-wave based implementations of Coulomb integrals under periodic boundary conditions, the treatment of the singularity at the Brillouin-zone center, as well as quadrature errors, which can cause severe convergence problems in anisotropic cells, necessary for the calculation of low-dimensional systems. We apply our strategy to the Hartree-Fock (HF) and coupled cluster (CC) theory and discuss the consequences of different sampling strategies on the different theories. We show that sampling the Coulomb potential via the widely used probe-charge Ewald method is unsuitable for CC calculations in anisotropic cells. To demonstrate the applicability of our developed approach, we study two representative, low-dimensional use cases: the infinite carbon chain, for which we report the first periodic CCSD(T) potential energy surface, as well as a surface slab of lithium hydride, for which we demonstrate the impact of different sampling strategies for calculating surface energies. We find that our Coulomb sampling strategy serves as a vital solution, addressing the critical need for improved accuracy in plane-wave based CC calculations for low-dimensional systems., Comment: 7 pages, 3 figures

Coupled-cluster theories can be used to compute ab initio electronic correlation energies of real materials with systematically improvable accuracy. However, the widely-used coupled cluster singles and doubles plus perturbative triples (CCSD(T)) method is only applicable to insulating materials. For zero-gap materials the truncation of the underlying many-body perturbation expansion leads to an infrared catastrophe. Here, we present a novel perturbative triples formalism that yields convergent correlation energies in metallic systems. Furthermore, the computed correlation energies for the three dimensional uniform electron gas at metallic densities are in good agreement with quantum Monte Carlo results. At the same time the newly proposed method retains all desirable properties of CCSD(T) such as its accuracy for insulating systems as well as its low computational cost compared to a full inclusion of the triples. This paves the way for ab initio calculations of real metals with chemical accuracy., Comment: 6 pages, 1 figure, 1 table plus a supplemental material of 14 pages, 4 figures and 1 table

Electron transfer with changing occupation in the 4f subshell poses a considerable challenge for quantitative predictions in quantum chemistry. Using the example of cerium oxide, we identify the main deficiencies of common parameter-dependent one-electron approaches, such as density functional theory (DFT) with a Hubbard correction, or hybrid functionals. As a response, we present the first benchmark of ab initio many-electron theory for electron transfer energies and lattice parameters under periodic boundary conditions. We show that the direct random phase approximation clearly outperforms all DFT variations. From this foundation, we, then, systematically improve even further. Periodic second-order M{\o}ller-Plesset perturbation theory meanwhile manages to recover standard hybrid functional values. Using these approaches to eliminate parameter bias allows for highly accurate benchmarks of strongly correlated materials, the reliable assessment of various density functionals, and functional fitting via machine-learning., Comment: 7 pages, 3 figures, 1 table

A first-principles study of the adsorption of a single water molecule on a layer of graphitic carbon nitride employing an embedding approach is presented. The embedding approach involves an algorithm to obtain localized Wannier orbitals of various types expanded in a plane-wave basis and intrinsic atomic orbital projectors. The localized occupied orbitals are employed in combination with unoccupied natural orbitals to perform many-electron perturbation theory calculations of local fragments. The fragments are comprised of a set of localized orbitals close to the adsorbed water molecule. Although the surface model contains more than 100 atoms in the simulation cell, the employed fragments are small enough to allow for calculations using high-level theories up to the coupled cluster ansatz with single, double and perturbative triple particle-hole excitation operators (CCSD(T)). To correct for the missing long-range correlation energy contributions to the adsorption energy, we embed CCSD(T) theory into the direct random phase approximation, yielding rapidly convergent adsorption energies with respect to the fragment size. Convergence of computed binding energies with respect to the virtual orbital basis set is achieved employing a number of recently developed techniques. Moreover, we discuss fragment size convergence for a range of approximate many-electron perturbation theories. The obtained benchmark results are compared to a number of density functional calculations., Comment: 12 pages, 7 figures

In case of an ocean oil spill, there are certain areas, e.g. shrimp farms, which are highly sensitive to small amounts of oil pollution while they are geographically far from the spill. We investigate the Lattice Boltzmann Method (LBM) and GNOME, a tool developed and used by NOAA, in terms of far-field impacts to sensitive areas. We present our simulation results of both models in limited scale (a sub area of Gulf of Mexico) under the same oil spill condition using real ocean current data from the Unified Wave Interface-Coupled Model (UWIN-CM). Our study shows that the kinetic theory based LBM model outperforms the stochastic particle based GNOME model in accuracy and computation time due to their fundamental difference in representation of oil pollution advection and diffusion mechanisms. We propose LBM as a viable alternative to the Lagrangian particle calculation component of GNOME in modeling the far-field impacts to highly sensitive areas.

Computationally efficient and accurate quantum mechanical approximations to solve the many-electron Schr\"odinger equation are at the heart of computational materials science. In that respect the coupled cluster hierarchy of methods plays a central role in molecular quantum chemistry because of its systematic improvability and computational efficiency. In this hierarchy, coupled cluster singles and doubles (CCSD) is one of the most important steps in moving towards chemical accuracy and, in recent years, its scope has successfully been expanded to the study of insulating surfaces and solids. Here, we show that CCSD theory can also be applied to real metals. In so doing, we overcome the limitation of needing extremely large supercells to capture long range electronic correlation effects. An effective Hamiltonian can be found using the transition structure factor--a map of electronic excitations from the Hartree--Fock wavefunction--which has fewer finite size effects than conventional periodic boundary conditions. This not only paves the way of applying coupled cluster methods to real metals but also reduces the computational cost by two orders of magnitude compared to previous methods. Our applications to phases of lithium and silicon show a resounding success in reaching the thermodynamic limit, taking the first step towards a truly universal quantum chemical treatment of solids., Comment: 8 pages, 4 figures, 1 table

Freidlin-Wentzell theory of large deviations can be used to compute the likelihood of extreme or rare events in stochastic dynamical systems via the solution of an optimization problem. The approach gives exponential estimates that often need to be refined via calculation of a prefactor. Here it is shown how to perform these computations in practice. Specifically, sharp asymptotic estimates are derived for expectations, probabilities, and mean first passage times in a form that is geared towards numerical purposes: they require solving well-posed matrix Riccati equations involving the minimizer of the Freidlin-Wentzell action as input, either forward or backward in time with appropriate initial or final conditions tailored to the estimate at hand. The usefulness of our approach is illustrated on several examples. In particular, invariant measure probabilities and mean first passage times are calculated in models involving stochastic partial differential equations of reaction-advection-diffusion type.

We present an embedding approach to treat local electron correlation effects in periodic environments. In a single, consistent framework, our plane-wave based scheme embeds a local high-level correlation calculation (here Coupled Cluster Theory, CC), employing localized orbitals, into a low-level correlation calculation (here the direct Random Phase Approximation, RPA). This choice allows for an accurate and effcient treatment of long-range dispersion effects. Accelerated convergence with respect to the local fragment size can be observed if the low-level and high-level long-range dispersion are quantitatively similar, as is the case for CC in RPA. To demonstrate the capabilities of the introduced embedding approach, we calculate adsorption energies of molecules on a surface and in a chabazite crystal cage, as well as the formation energy of a lattice impurity in a solid at the level of highly accurate many-electron perturbation theories. The absorption energy of a methane molecule in a zeolite chabazite, for instance, is converged with an error well below 20 meV at the CC level. As our largest periodic benchmark system, we apply our scheme to the adsorption of a water molecule on titania in a supercell containing more than 1000 electrons., Comment: 6 pages, 2 figures

We present a method to approximate post-Hartree-Fock correlation energies by using approximate natural orbitals obtained by the random phase approximation (RPA). We demonstrate the method by applying it to the helium atom, the hydrogen and fluorine molecule, and to diamond as an example of a periodic system. For these benchmark systems, we show that RPA natural orbitals converge the MP2 correlation energy rapidly. Additionally, we calculated full configuration interaction energies for He and H$_2$, which are in excellent agreement with the literature and experimental values. We conclude that the proposed method may serve as a compromise to reach good approximations to correlation energies at moderate computational cost, and we expect the method to be especially useful for theoretical studies on surface chemistry by providing an efficient basis to correlated wave function based methods., Comment: 10 pages, 18 figures, 16 Equations; v2: fixed typo in metadata abstract, changed bibliography style; v3: added reference (4), added Acknowledgement

Understanding the phase behavior of semiconductor materials is important for applications in solid state physics and nanoscience. Accurate experimental data is often difficult to obtain due to strong kinetic effects. In this work, we calculate the temperature-pressure phase diagram of lead selenide (PbSe) using the random phase approximation (RPA), an accurate wavefunction based many-body technique. We consider three crystalline phases, the low pressure B1 phase (NaCl-type), the intermediate B33 phase (CrB-type), and the high pressure B2 phase (CsCl-type). The electronic contributions to the free energy (at T=0K) are calculated in the Born-Oppenheimer approximation using the RPA, whereas phononic contributions are computed in the quasi-harmonic approximation using DFT and the PBEsol functional. At room temperature, we find transition pressures of 4.6 +/- 0.3 GPa for the B1-B33 transition and 18.7 +/- 0.3 GPa for the B33-B2 transition, in good agreement with experiments. In contrast to the interpretation of recent experiments, we observe a negative Clapeyron slope for both transitions. Gibbs free energy differences between competing structures have small gradients close to coexistence, consistent with pronounced hysteresis observed in experiments. The phase diagram presented in this work can serve as a reference for future studies of PbSe and should prove useful in the development of accurate and efficient force fields., Comment: 7 pages, 7 figures

Crystalline GeSb2Te4 (GST) is remarkable material, as it allows to continuously tune the electrical resistance by orders of magnitude without involving a phase transition or stoichiometric changes, just by altering the short-range order. While well-ordered specimen are metallic, increasing amounts of disorder can eventually lead to an insulating state with vanishing conductivity in the 0K limit, but a similar number of charge carriers. These observations make disordered GST one of the most promising candidates for the realization of a true Anderson insulator. While so far the low-temperature properties have mostly been studied in films of small grain size, here a sputter-deposition process is employed that enables preparation of a large variety of these GST states including metallic and truly insulating ones. By growing films of GST on mica substrates, biaxially textured samples with huge grain sizes are obtained. A series of these samples is employed for transport measurements, as their electron mean free path can be altered by a factor of 20. Yet, the mean free path always remains more than an order of magnitude smaller than the lateral grain size. This proves unequivocally that grain boundaries play a negligible role for electron scattering, while intragrain scattering, presumably by disordered vacancies, dominates. Most importantly, these findings underline that the Anderson insulating state as well as the system's evolution towards metallic conductivity are indeed intrinsic properties of the material.

Using the geometric minimum action method, we compute minimizers of the Freidlin-Wentzell functional for the dissipative linear and nonlinear Schroedinger equation. For the particular case of transitions between solitary waves of different amplitudes, we discuss the relationship of the minimizer of the PDE model to the minimizer of a finite-dimensional reduction.

We present an implementation and analysis of a stochastic high performance algorithm to calculate the correlation energy of three dimensional periodic systems in second-order M{\o}ller-Plesset perturbation theory (MP2). In particular we measure the scaling behavior of the sample variance and probe whether this stochastic approach is competitive if accuracies well below 1 meV per valence orbital are required, as it is necessary for calculations of adsorption, binding, or surface energies. The algorithm is based on the Laplace transformed MP2 (LTMP2) formulation in the plane wave basis. The time-dependent Hartree-Fock orbitals, appearing in the LTMP2 formulation, are stochastically rotated in the occupied and unoccupied Hilbert space. This avoids a full summation over all combinations of occupied and unoccupied orbitals, as inspired by the work of D. Neuhauser, E. Rabani, and R. Baer in J. Chem. Theory Comput. 9, 24 (2013). Additionally, correlated sampling is introduced, accelerating the statistical convergence significantly., Comment: 11 pages, 6 figures

Which density functional is the "best" for structure simulations of a particular material? A concise, first principles, approach to answer this question is presented. The random phase approximation (RPA)--- an accurate many body theory--- is used to evaluate various density functionals. To demonstrate and verify the method, we apply it to the hybrid perovskite MAPbI$_3$, a promising new solar cell material. The evaluation is done by first creating finite temperature ensembles for small supercells using RPA molecular dynamics, and then evaluating the variance between the RPA and various approximate density functionals for these ensembles. We find that, contrary to recent suggestions, van der Waals functionals do not improve the description of the material, whereas hybrid functionals and the SCAN (strongly constrained appropriately normed) density functional yield very good agreement with the RPA. Finally, our study shows that in the room temperature tetragonal phase of MAPbI$_3$, the molecules are preferentially parallel to the shorter lattice vectors but reorientation on ps timescales is still possible.

We present a low-complexity algorithm to calculate the correlation energy of periodic systems in second-order M\o ller-Plesset perturbation theory (MP2). In contrast to previous approximation-free MP2 codes, our implementation possesses a quartic scaling, $\mathcal O (N^4)$, with respect to the system size $N$ and offers an almost ideal parallelization efficiency. The general issue that the correlation energy converges slowly with the number of basis functions is eased by an internal basis set extrapolation. The key concept to reduce the scaling is to eliminate all summations over virtual orbitals which can be elegantly achieved in the Laplace transformed MP2 (LTMP2) formulation using plane wave basis sets and Fast Fourier transforms. Analogously, this approach could allow to calculate second order screened exchange (SOSEX) as well as particle-hole ladder diagrams with a similar low complexity. Hence, the presented method can be considered as a step towards systematically improved correlation energies., Comment: 12 pages, 10 figures

We discuss that in the random phase approximation (RPA) the first derivative of the energy with respect to the Green's function is the self-energy in the GW approximation. This relationship allows us to derive compact equations for the RPA interatomic forces. We also show that position dependent overlap operators are elegantly incorporated in the present framework. The RPA force equations have been implemented in the projector augmented wave formalism, and we present illustrative applications, including ab initio molecular dynamics simulations, the calculation of phonon dispersion relations for diamond and graphite, as well as structural relaxations for water on boron nitride. The present derivation establishes a concise framework for forces within perturbative approaches and is also applicable to more involved approximations for the correlation energy., Comment: 5 pages, 3 figures

Small random perturbations may have a dramatic impact on the long time evolution of dynamical systems, and large deviation theory is often the right theoretical framework to understand these effects. At the core of the theory lies the minimization of an action functional, which in many cases of interest has to be computed by numerical means. Here we review the theoretical and computational aspects behind these calculations, and propose an algorithm that simplifies the geometric minimum action method to minimize the action in the space of arc-length parametrized curves. We then illustrate this algorithm's capabilities by applying it to various examples from material sciences, fluid dynamics, atmosphere/ocean sciences, and reaction kinetics. In terms of models, these examples involve stochastic (ordinary or partial) differential equations with multiplicative or degenerate noise, Markov jump processes, and systems with fast and slow degrees of freedom, which all violate detailed balance, so that simpler computational methods are not applicable.

A precise characterization of structures occurring in turbulent fluid flows at high Reynolds numbers is one of the last open problems of classical physics. In this review we discuss recent developments related to the application of instanton methods to turbulence. Instantons are saddle point configurations of the underlying path integrals. They are equivalent to minimizers of the related Freidlin-Wentzell action and known to be able to characterize rare events in such systems. While there is an impressive body of work concerning their analytical description, this review focuses on the question on how to compute these minimizers numerically. In a short introduction we present the relevant mathematical and physical background before we discuss the stochastic Burgers equation in detail. We present algorithms to compute instantons numerically by an efficient solution of the corresponding Euler-Lagrange equations. A second focus is the discussion of a recently developed numerical filtering technique that allows to extract instantons from direct numerical simulations. In the following we present modifications of the algorithms to make them efficient when applied to two- or three-dimensional fluid dynamical problems. We illustrate these ideas using the two-dimensional Burgers equation and the three-dimensional Navier-Stokes equations.

Instanton calculations are performed in the context of stationary Burgers turbulence to estimate the tails of the probability density function (PDF) of velocity gradients. These results are then compared to those obtained from massive direct numerical simulations (DNS) of the randomly forced Burgers equation. The instanton predictions are shown to agree with the DNS in a wide range of regimes, including those that are far from the limiting cases previously considered in the literature. These results settle the controversy of the relevance of the instanton approach for the prediction of the velocity gradient PDF tail exponents. They also demonstrate the usefulness of the instanton formalism in Burgers turbulence, and suggest that this approach may be applicable in other contexts, such as 2D and 3D turbulence in compressible and incompressible flows.

This work considers the problem of quickest detection of signals in a coupled system of $N$ sensors, which receive continuous sequential observations from the environment. It is assumed that the signals, which are modeled by general It\^{o} processes, are coupled across sensors, but that their onset times may differ from sensor to sensor. Two main cases are considered; in the first one signal strengths are the same across sensors while in the second one they differ by a constant. The objective is the optimal detection of the first time at which any sensor in the system receives a signal. The problem is formulated as a stochastic optimization problem in which an extended minimal Kullback-Leibler divergence criterion is used as a measure of detection delay, with a constraint on the mean time to the first false alarm. The case in which the sensors employ cumulative sum (CUSUM) strategies is considered, and it is proved that the minimum of $N$ CUSUMs is asymptotically optimal as the mean time to the first false alarm increases without bound. In particular, in the case of equal signal strengths across sensors, it is seen that the difference in detection delay of the $N$-CUSUM stopping rule and the unknown optimal stopping scheme tends to a constant related to the number of sensors as the mean time to the first false alarm increases without bound. Alternatively, in the case of unequal signal strengths, it is seen that this difference tends to zero., Comment: 29 pages. SIAM Journal on Control and Optimization, forthcoming

We present a mathematical framework for mapping second-order logic relations onto a simple state vector algebra. Using this algebra, basic theorems of set theory can be proven in an algorithmic way, hence by an expert system. We illustrate the use of the algebra with simple examples and show that, in principle, all theorems of basic set theory can be recovered in an elementary way. The developed technique can be used for an automated theorem proving in the 1st and 2nd order logic., Comment: This paper has been withdrawn by the author due to numerous errors found

We consider the two-dimensional advection-diffusion equation on a bounded domain subject to either Dirichlet or von Neumann boundary conditions and study both time-independent and time-periodic cases involving Liouville integrable Hamiltonians that satisfy conditions conducive to applying the averaging principle. Transformation to action-angle coordinates permits averaging in time and angle, leading to an underlying eigenvalue equation that allows for separation of the angle and action coordinates. The result is a one-dimensional second-order equation involving an anti-symmetric imaginary potential. For radial flows on a disk or an annulus, we rigorously apply existing complex-plane WKBJ methods to study the spectral properties in the semi-classical limit for vanishing diffusivity. In this limit, the spectrum is found to be a complicated set consisting of lines related to Stokes graphs. Eigenvalues in the neighborhood of these graphs exhibit nonlinear scaling with respect to diffusivity leading to convection-enhanced rates of dissipation (relaxation, mixing) for initial data which are mean-free in the angle coordinate. These branches coexist with a diffusive branch of eigenvalues that scale linearly with diffusivity and contain the principal eigenvalue (no dissipation enhancement)., Comment: 28 pages, 7 figures

A method is presented to compute minimizers (instantons) of action functionals using arclength parametrization of Hamilton's equations. This method can be interpreted as a local variant of the geometric minimum action method (gMAM) introduced to compute minimizers of the Freidlin-Wentzell action functional that arises in the context of large deviation theory for stochastic differential equations. The method is particularly well-suited to calculate expectations dominated by noise-induced excursions from deterministically stable fixpoints. Its simplicity and computational efficiency are illustrated here using several examples: a finite-dimensional stochastic dynamical system (an Ornstein-Uhlenbeck model) and two models based on stochastic partial differential equations: the $\phi^4$-model and the stochastically driven Burgers equation., Comment: 15 pages, 6 figures

The present work provides a critical assessment of numerical solutions of the space-fractional diffusion-advection equation, which is of high significance for applications in various natural sciences. In view of the fact that, in contrast to the case of normal (Gaussian) diffusion, no standard methods and corresponding numerical codes for anomalous diffusion problems have been established yet, it is of importance to critically assess the accuracy and performance of existing approaches. Three numerical methods, namely a finite-difference method, the so-called matrix transfer technique, and a Monte-Carlo method based on the solution of stochastic differential equations, are analyzed and compared by applying them to three selected test problems for which analytical or semi-analytical solutions were known or are newly derived. The accuracy and performance differences are critically discussed with the result that the use of stochastic differential equations appears to be advantageous., Comment: 20 pages, 4 figures

We study the propagation of ultra-short short solitons in a cubic nonlinear medium modeled by nonlinear Maxwell's equations with stochastic variations of media. We consider three cases: variations of (a) the dispersion, (b) the phase velocity, (c) the nonlinear coefficient. Using a modified multi-scale expansion for stochastic systems, we derive new stochastic generalizations of the short pulse equation that approximate the solutions of stochastic nonlinear Maxwell's equations. Numerical simulations show that soliton solutions of the short pulse equation propagate stably in stochastic nonlinear Maxwell's equations and that the generalized stochastic short pulse equations approximate the solutions to the stochastic Maxwell's equations over the distances under consideration. This holds for both a pathwise comparison of the stochastic equations as well as for a comparison of the resulting probability densities.

We address the question whether one can identify instantons in direct numerical simulations of the stochastically driven Burgers equation. For this purpose, we first solve the instanton equations using the Chernykh-Stepanov method [Phys. Rev. E 64, 026306 (2001)]. These results are then compared to direct numerical simulations by introducing a filtering technique to extract prescribed rare events from massive data sets of realizations. Using this approach we can extract the entire time history of the instanton evolution which allows us to identify the different phases predicted by the direct method of Chernykh and Stepanov with remarkable agreement.

Using renormalization group techniques, we derive an extended short- pulse equation as approximation to a nonlinear wave equation. We investigate the new equation numerically and show that the new equation captures efficiently higher- order effects on pulse propagation in cubic nonlinear media. We illustrate our findings using one- and two-soliton solutions of the first-order short-pulse equation as initial conditions in the nonlinear wave equation.

We present a new method to transform an expanded class of non-selfadjoint advection-diffusion operators into self-adjoint operators. The transform is based on a combination of a point transform and Lie transform in conjunction with an asymptotic expansion in terms of the diffusivity. We illustrate the method in the context of simple shear flow where the expansion is exact and all transformation steps can be performed explicitly.

This work considers the problem of quickest detection of signals in a coupled system of N sensors, which receive continuous sequential observations from the environment. It is assumed that the signals, which are modeled a general Ito processes, are coupled across sensors, but that their onset times may differ from sensor to sensor. The objective is the optimal detection of the first time at which any sensor in the system receives a signal. The problem is formulated as a stochastic optimization problem in which an extended average Kullback- Leibler divergence criterion is used as a measure of detection delay, with a constraint on the mean time between false alarms. The case in which the sensors employ cumulative sum (CUSUM) strategies is considered, and it is proved that the minimum of N CUSUMs is asymptotically optimal as the mean time between false alarms increases without bound., Comment: 6 pages, 48th IEEE Conference on Decision and Control, Shanghai 2009 December 16 - 18

We study the effect of advection and small diffusion on passive tracers. The advecting velocity field is assumed to have mean zero and to possess time-periodic stream lines. Using a canonical transform to action-angle variables followed by a Lie-transform, we derive an averaged equation describing the effective motion of the tracers. An estimate for the time validity of the first-order approximation is established. For particular cases of a regularized vortical flow we present explicit formulas for the coefficients of the averaged equation both at first and at second order. Numerical simulations indicate that the validity of the above first-order estimate extends to the second order.