Spurious Growth Research Articles

Strong convergence of a class of adaptive numerical methods for SDEs with jumps

We develop adaptive time-stepping strategies for Itô-type stochastic differential equations (SDEs) with jump perturbations. Our approach builds on adaptive strategies for SDEs.Adaptive methods can ensure strong convergence of nonlinear SDEs with drift and diffusion coefficients that violate global Lipschitz bounds by adjusting the stepsize dynamically on each trajectory to prevent spurious growth that can lead to loss of convergence if it occurs with sufficiently high probability.In this article, we demonstrate the use of a jump-adapted mesh that incorporates jump times into the adaptive time-stepping strategy. We prove that any adaptive scheme satisfying a particular mean-square consistency bound for a nonlinear SDE in the non-jump case may be extended to a strongly convergent scheme in the Poisson jump case, where the jump and diffusion perturbations are mutually independent, and the jump coefficient satisfies a global Lipschitz condition.

Restricting the Spurious Growth of Knowledge Graphs by Using Ontology Graphs

Restricting the Spurious Growth of Knowledge Graphs by Using Ontology Graphs

A POD-RBF-FD scheme for simulating chemotaxis models on surfaces

The main aim of this paper is to develop a new framework of a meshless approximation for solving numerically three nonlinear partial differential equations in biology, i.e., the chemotaxis models defined on the smooth, closed manifolds embedded in R3. The radial basis function-generated finite difference scheme is considered to deal with the spatial variables, which depends only on the location of nodes and the value of normal vector at each point per spherical cap. The robust artificial hyperviscosity formulation is derived for each model, which has been used for preventing the spurious growth modes in the numerical solution. An implicit–explicit time discretization is employed to deal with the time variable. The resulting fully discrete scheme is solved via the biconjugate gradient stabilized method with a zero-fill incomplete lower upper preconditioner per time step, where a positivity-preserving filter is used to prevent the negative sign of the cell density variable. Besides, to reduce the used central processor unit (CPU) time, the proper orthogonal decomposition is considered for constructing a set of new orthogonal basis vectors based on the singular value decomposition. The developed numerical method is called a proper orthogonal decomposition-radial basis function-generated finite difference (POD-RBF-FD) scheme. Finally, the ability of the proposed method is investigated by simulation results showing the blowing-up, pattern formulation (perforated stripe) and aggregations of bacteria on some surfaces.

Simulating cold shear flows on a moving mesh

ABSTRACT Rotationally supported, cold, gaseous discs are ubiquitous in astrophysics and appear in a diverse set of systems, such as protoplanetary discs, accretion discs around black holes, or large spiral galaxies. Capturing the gas dynamics accurately in these systems is challenging in numerical simulations due to the low sound speed compared to the bulk velocity of the gas, the resolution limitations of full disc models, and the fact that numerical noise can easily source spurious growth of fluid instabilities if not suppressed sufficiently well, negatively interfering with real physical instabilities present in such discs (like the magnetorotational instability). Here, we implement the so-called shearing-box approximation in the moving-mesh code arepo in order to facilitate achieving high resolution in local regions of differentially rotating discs and to address these problems. While our new approach offers manifest translational invariance across the shearing-box boundaries and offers continuous local adaptivity, we demonstrate that the unstructured mesh of arepo introduces unwanted levels of ‘grid-noise’ in the default version of the code. We show that this can be rectified by high-order integrations of the flux over mesh boundaries. With our new techniques we obtain highly accurate results for shearing-box calculations of the magnetorotational instability that are superior to other Lagrangian techniques. These improvements are also of value for other applications of the code that feature strong shear flows.

Abnormal Silicon-Germanium (SiGe) Epitaxial Growth in FinFETs

In an advanced complementary metal-oxide-semiconductor (CMOS) technology node, small nano-scopic defects tend to have a significant impact on the yield and reliability of the final product. As the technology advances in fin field-effect-transistors (FinFETs), it has become increasingly challenging to control the extent of defects while also ensuring that the desired processing parameters are in place. The process of selective epitaxial growth of silicon-germanium (SiGe) on source/drain (S/D) regions is especially prone to defects due to the complexity of the process, thus giving a very narrow processing window to achieve the desired device, yield and reliability results. In this paper we evaluate a critical defect of “Abnormal epi” seen during the selective epitaxial growth of in-situ boron (B) doped SiGe on FinFETs. Abnormal epi here refers to abnormally large and spurious epitaxial growth defect occurring as random instances on a wafer die. These defects, depending on where in the layout they occur, can lead to catastrophic failure under higher test voltages due to the physical shorting of n and p FET devices. We also explore the process parameters which influence this defect including incoming surface conditions and propose mechanisms that lead to the defect. Further, an optimization of the processing parameters in the integration scheme to minimize the occurrence of these defects leads to yield gain and cost savings required for high volume manufacturing.

Energy equipartition between stellar and dark matter particles in cosmological simulations results in spurious growth of galaxy sizes

ABSTRACT The impact of 2-body scattering on the innermost density profiles of dark matter haloes is well established. We use a suite of cosmological simulations and idealized numerical experiments to show that 2-body scattering is exacerbated in situations where there are two species of unequal mass. This is a consequence of mass segregation and reflects a flow of kinetic energy from the more to less massive particles. This has important implications for the interpretation of galaxy sizes in cosmological hydrodynamic simulations, which nearly always model stars with less massive particles than are used for the dark matter. We compare idealized models as well as simulations from the eagle project that differ only in the mass resolution of the dark matter component, but keep subgrid physics, baryonic mass resolution, and gravitational force softening fixed. If the dark matter particle mass exceeds the mass of stellar particles, then galaxy sizes – quantified by their projected half-mass radii, R50 – increase systematically with time until R50 exceeds a small fraction of the redshift-dependent mean interparticle separation, l (${\rm R_{50}} \gtrsim 0.05\times l$). Our conclusions should also apply to simulations that adopt different hydrodynamic solvers, subgrid physics, or adaptive softening, but in that case may need quantitative revision. Any simulation employing a stellar-to-dark matter particle mass ratio greater than unity will escalate spurious energy transfer from dark matter to baryons on small scales.

Stellarator microinstabilities and turbulence at low magnetic shear

Gyrokinetic simulations of drift waves in low-magnetic-shear stellarators reveal that simulation domains comprised of multiple turns can be required to properly resolve critical mode structures important in saturation dynamics. Marginally stable eigenmodes important in saturation of ion temperature gradient modes and trapped electron modes in the Helically Symmetric Experiment (HSX) stellarator are observed to have two scales, with the envelope scale determined by the properties of the local magnetic shear and an inner scale determined by the interplay between the local shear and magnetic field-line curvature. Properly resolving these modes removes spurious growth rates that arise for extended modes in zero-magnetic-shear approximations, enabling use of a zero-magnetic-shear technique with smaller simulation domains and attendant cost savings. Analysis of subdominant modes in trapped electron mode (TEM)-driven turbulence reveals that the extended marginally stable modes play an important role in the nonlinear dynamics, and suggests that the properties induced by low magnetic shear may be exploited to provide another route for turbulence saturation.

Hyperviscosity-based stabilization for radial basis function-finite difference (RBF-FD) discretizations of advection–diffusion equations

We present a novel hyperviscosity formulation for stabilizing RBF-FD discretizations of the advection–diffusion equation. The amount of hyperviscosity is determined quasi-analytically for commonly-used explicit, implicit, and implicit–explicit (IMEX) time integrators by using a simple 1D semi-discrete Von Neumann analysis. The analysis is applied to an analytical model of spurious growth in RBF-FD solutions that uses auxiliary differential operators mimicking the undesirable properties of RBF-FD differentiation matrices. The resulting hyperviscosity formulation is a generalization of existing ones in the literature, but is free of any tuning parameters and can be computed efficiently. To further improve robustness, we introduce a simple new scaling law for polynomial-augmented RBF-FD that relates the degree of polyharmonic spline (PHS) RBFs to the degree of the appended polynomial. When used in a novel ghost node formulation in conjunction with the recently-developed overlapped RBF-FD method, the resulting method is robust and free of stagnation errors. We validate the high-order convergence rates of our method on 2D and 3D test cases over a wide range of Peclet numbers (1–1000). We then use our method to solve a 3D coupled problem motivated by models of platelet aggregation and coagulation, again demonstrating high-order convergence rates.

A One-Parameter Controlled Dissipative Unconditionally Stable Explicit Algorithm for Time History Analysis

A new family of one-step integration methods is presented herein. A free parameter is used to control the numerical properties and it can be considered as an indicator of numerical dissipation for the high frequency modes. This family method can have unconditional stability, explicit formulation and desired numerical damping, which implies that the low frequency modes can be accurately integrated while the spurious growth of high frequency modes can be suppressed or even eliminated. In addition, a zero damping ratio can be achieved. Since the unconditional stability and explicit formulation are integrated together for the proposed family method, it can drastically reduce the computational efforts when compared to the traditional integration methods.

A pseudodynamic testing algorithm for obtaining seismic responses of structures

A new family method is proposed for pseudodynamic tests. This family method can integrate favourable numerical properties together, such as the unconditional stability, explicit formulation, second-order accuracy and favourable numerical dissipation. Since the proposed family method can generally have an explicit formulation of each time step its pseudodynamic implementation involves no iteration procedure. Hence, the pseudodynamic tests can be easily conducted when compared to an implicit pseudodynamic method, where an iteration procedure must be involved for each time step. On the other hand, the properties of unconditional stability, second order accuracy and numerical dissipation indicate that the proposed pseudodynamic algorithm is promising for solving inertial problems, where the total response is dominated by low frequency modes and the high frequency responses are of no interest. The unconditional stability implies that there is no limitation on step size for the high frequency modes. Besides, the dominated low frequency modes can be reliably integrated by choosing an appropriate time step since the proposed family method can have a second order accuracy. Finally, the spurious growth of high frequency responses can be suppressed or filtered out by the desired numerical dissipation. As a result, the proposed pseudodynamic algorithm is very promising for a general pseudodynamic test or a substructure pseudodynamic test.

Tree demography dominates long-term growth trends inferred from tree rings.

Understanding responses of forests to increasing CO 2 and temperature is an important challenge, but no easy task. Tree rings are increasingly used to study such responses. In a recent study, van der Sleen et al. (2014) Nature Geoscience, 8, 4 used tree rings from 12 tropical tree species and find that despite increases in intrinsic water use efficiency, no growth stimulation is observed. This challenges the idea that increasing CO 2 would stimulate growth. Unfortunately, tree ring analysis can be plagued by biases, resulting in spurious growth trends. While their study evaluated several biases, it does not account for all. In particular, one bias may have seriously affected their results. Several of the species have recruitment patterns, which are not uniform, but clustered around one specific year. This results in spurious negative growth trends if growth rates are calculated in fixed size classes, as ‘fast‐growing’ trees reach the sampling diameter earlier compared to slow growers and thus fast growth rates tend to have earlier calendar dates. We assessed the effect of this ‘nonuniform age bias’ on observed growth trends and find that van der Sleen's conclusions of a lack of growth stimulation do not hold. Growth trends are – at least partially – driven by underlying recruitment or age distributions. Species with more clustered age distributions show more negative growth trends, and simulations to estimate the effect of species’ age distributions show growth trends close to those observed. Re‐evaluation of the growth data and correction for the bias result in significant positive growth trends of 1–2% per decade for the full period, and 3–7% since 1950. These observations, however, should be taken cautiously as multiple biases affect these trend estimates. In all, our results highlight that tree ring studies of long‐term growth trends can be strongly influenced by biases if demographic processes are not carefully accounted for.

Application of the logarithmic Hamiltonian algorithm to the circular restricted three-body problem with some post-Newtonian terms

An implementation of a fourth-order symplectic algorithm to the logarithmic Hamiltonian of the Newtonian circular restricted three-body problem in an inertial frame is detailed. The logarithmic Hamiltonian algorithm produces highly accurate results, comparable to the non-logarithmic one. Its numerical performance is independent of an orbital eccentricity. However, it is not when some post-Newtonian terms are included in this problem. Although the numerical accuracy becomes somewhat poorer as the orbital eccentricity gets larger, it is still much higher than that of the non-logarithmic Hamiltonian algorithm. As a result, the present code can drastically eliminate the overestimation of Lyapunov exponents and the spurious rapid growth of fast Lyapunov indicators for high-eccentricity orbits in the Newtonian or post-Newtonian circular restricted three-body problem.

The Monge–Ampère trajectory correction for semi-Lagrangian schemes

Requiring that numerical estimates of the flow trajectories comply with the fundamental Euler expansion formula that governs the evolution of a volume of fluid leads to a second-order nonlinear Monge–Ampère partial differential equation (MAE). In Cossette and Smolarkiewicz (2011) [15], a numerical algorithm based on solving the MAE with an inexact Newton–Krylov solver has been developed and used to correct standard estimates of the departure points of flow trajectories provided by a classical semi-Lagrangian scheme in the context of an incompressible fluid. Here we extend the theoretical analysis of the elemental rotational and deformational motions presented in Cossette and Smolarkiewicz (2011) [15]. In particular, closed-form analytic solutions are derived for both cases that serve to illustrate the mechanics of the enhanced trajectory scheme and to address the issues of existence and uniqueness. Scalar advection shows that the MA correction improves mass conservation substantially by suppressing anomalous fluid contraction. The impact of the MA correction on complex flows is studied in the framework of an ideal magneto-fluid in which the formation of current sheets leads to topological changes and reconnection of field lines. The use of the MA correction improves the numerical stability of the solutions and prevents trajectory intersections as well as the spurious growth of the magnetic energy. 2D and 3D examples are presented and the computational performance of the solver is documented.

Simple and efficient numerical methods for vortex sheet motion with surface tension

SUMMARYWe present two simple and efficient explicit methods for the vortex sheet with surface tension. The first one is the standard point vortex method, which has been known to be unstable in the presence of surface tension, due to spurious growth of waves of high modes. We show, for the first time, that the standard point vortex method is able to calculate the vortex sheet motion with surface tension by employing a Fourier filtering. The second method is a modification of the Pullin method using central differences for numerical differentiations. This method is more convenient to implement than other spectral methods and is free from the aliasing instability. We give a linear stability analysis for the numerical methods and show results for the long‐time evolution of the vortex sheet. We also propose a new redistribution procedure to control point clustering, by setting limits of minimum and maximum distances between neighboring points. This procedure is found to be very efficient for long‐time computations of the explicit methods. Copyright © 2013 John Wiley & Sons, Ltd.

THE FORMATION OF THE FIRST COSMIC STRUCTURES AND THE PHYSICS OF THEz∼ 20 UNIVERSE

We perform a suite of cosmological simulations in the LCDM paradigm of the formation of the first structures in the Universe prior to astrophysical reheating and reionization (15<~z<200). These are the first simulations initialized in a manner that self consistently account for the impact of pressure on the rate of growth of modes, temperature fluctuations in the gas, and the dark matter-baryon supersonic velocity difference. Even with these improvements, these are still difficult times to simulate accurately as the Jeans length of the cold intergalactic gas must be resolved while also capturing a representative sample of the Universe. Our simulations support the finding of recent studies that the dark matter-baryon velocity difference has a surprisingly large impact on the accretion of gas onto the first star-forming minihalos (with masses of ~10^6 Msun). In fact, the halo gas is often significantly downwind of such halos and with lower densities, which delays the formation of the first stars in most locations in the Universe. We also show that dynamical friction plays an important role in the nonlinear evolution of the differential velocity, acting to erase this velocity difference quickly in overdense gas as well as sourcing visually-apparent bow shocks and Mach cones throughout the Universe. We use simulations with both the GADGET and Enzo cosmological codes to test the robustness of these conclusions. We find that particle coupling in GADGET between the gas and dark matter particles can result in spurious growth that mimics nonlinear growth in the matter power spectrum. In a companion paper, we use the simulations presented here to make detailed estimates for the impact of the dark matter--baryon velocity differential on redshifted 21cm radiation. The initial conditions generator used in this study CICsASS can be publicly downloaded.

Disturbance energy norms: A critical analysis

The question of which norm should be used to characterize the fluctuating disturbance energy in studies on thermoacoustic instabilities remains a topic of continued debate. In this paper we formulate a strategy for developing mathematically consistent norms that do not support spurious growth of disturbance energy in the absence of ‘physical’ sources of energy. A critical analysis of various disturbance energy norms existing in literature is conducted and important conclusions regarding positive definitiveness and susceptibility to exhibiting unphysical growth are drawn. It is shown that the energy norm proposed by Cantrell and Hart [Interaction between sound and flow in acoustic cavities: mass, momentum and energy considerations, Journal of Acoustical Society of America 36 (1964) 697–706] and Myers' first order disturbance energy norm [Transport of energy by disturbances in arbitrary steady flow, Journal of Fluid Mechanics 226 (1991) 383–400] are positive definite if M0<1 and M0<1/γ, respectively, where M0 is the magnitude of the local mean flow Mach number, γ is the ratio of specific heats at constant pressure and volume and the stipulated conditions should be met at every point in the flow domain. Our analysis shows that the disturbance energy norm proposed by Chu [On the energy transfer to small disturbances in fluid flow (part I), Acta Mechanica 1 (1965) 215–234] does not exhibit unphysical growth. It is also shown that this property is not unique to Chu's disturbance energy norm and there exists a family of norms which satisfy this requirement. The analysis also leads to an interesting interpretation of the acoustic energy conservation principle of Cantrell and Hart. The potential held by various disturbance energy norms for exhibiting fictitious growth is quantified using tools from nonmodal stability theory. It is concluded that if the mean flow Mach number is small, Myers' norm is a suitable measure of the disturbance energy.

Symplectic integrators with adaptive time steps

In recent decades, there have been many attempts to construct symplectic integrators with variable time steps, with rather disappointing results. In this paper, we identify the causes for this lack of performance, and find that they fall into two categories. In the first, the time step is considered a function of time alone, Δ = Δ(t). In this case, backward error analysis shows that while the algorithms remain symplectic, parametric instabilities may arise because of resonance between oscillations of Δ(t) and the orbital motion. In the second category the time step is a function of phase space variables Δ = Δ(q, p). In this case, the system of equations to be solved is analyzed by introducing a new time variable τ with dt = Δ(q, p) dτ. The transformed equations are no longer in Hamiltonian form, and thus do not benefit from integration methods which would be symplectic for Hamiltonian systems. We analyze two methods for integrating the transformed equations which do, however, preserve the structure of the original equations. The first is an extended phase space method, which has been successfully used in previous studies of adaptive time step symplectic integrators. The second, novel, method is based on a non-canonical mixed-variable generating function. Numerical trials for both of these methods show good results, without parametric instabilities or spurious growth or damping. It is then shown how to adapt the time step to an error estimate found by backward error analysis, in order to optimize the time-stepping scheme. Numerical results are obtained using this formulation and compared with other time-stepping schemes for the extended phase space symplectic method.

Numerical dissipation for pseudodynamic testing of nonlinear systems

It is well known that for linear elastic systems favorable numerical dissipation can suppress the spurious growth of high frequency modes while low frequency modes are almost unaffected. Thus, it is interesting to analytically explore the performance of numerical dissipation in the solution of a nonlinear system. In this paper, a novel technique is introduced to evaluate the numerical and error propagation properties of the γ‐function explicit method for nonlinear systems. It is analytically and numerically verified that this method has favorable numerical dissipation not only for linear elastic but also nonlinear systems. In fact, the suppression of high frequency modes for nonlinear systems is analytically proved. The importance of the spectral radius for a nonlinear system is also addressed. The rate of suppression of the spurious growth of high frequency responses will be accelerated due to stiffness hardening while it is decelerated due to stiffness softening.

A Family of Explicit Dissipative Algorithms for Pseudodynamic Testing

AbstractThe α-function method is a family of second-order explicit methods with controlled numerical dissipation. Thus, it is very promising for the pseudodynamic testing of a system where high frequency responses are of no interest. This is because that favorable numerical dissipation can suppress the spurious growth of high frequency responses, which might arise from numerical and/or experimental errors during a test. Furthermore, the implementation of an explicit method for the pseudodynamic testing is much simpler than for an implicit method. The superiority of using this method in performing a pseudodynamic test was verified both analytically and experimentally. In fact, results of error propagation analysis reveal that the spurious growth of high frequency responses can be suppressed and less error propagation is identified when compared to the Newmark explicit method. Actual tests were conducted pseudodynamically to confirm all the analytical results. It is also illustrated that although the high frequency response is insignificant to the total response it may be significantly amplified and propagated and finally destroys the pseudodynamic test results.

The discrete energy method in numerical relativity: towards long-term stability

The energy method can be used to identify well-posed initial boundary value problems for quasi-linear, symmetric hyperbolic partial differential equations with maximally dissipative boundary conditions. A similar analysis of the discrete system can be used to construct stable finite difference equations for these problems at the linear level. In this paper we apply these techniques to some test problems commonly used in numerical relativity and observe that while we obtain convergent schemes, fast growing modes, or ‘artificial instabilities’, contaminate the solution. We find that these growing modes can partially arise from the lack of a Leibnitz rule for discrete derivatives and discuss ways to limit this spurious growth.