Adaptive time stepping in biomolecular dynamics

Adaptive time‐stepping methods based on the Monte Carlo Euler method for weak approximation of Itô stochastic differential equations are developed. The main result is new expansions of the computational error, with computable leading‐order term in a posteriori form, based on stochastic flows and discrete dual backward problems. The expansions lead to efficient and accurate computation of error estimates. Adaptive algorithms for either stochastic time steps or deterministic time steps are described. Numerical examples illustrate when stochastic and deterministic adaptive time steps are superior to constant time steps and when adaptive stochastic steps are superior to adaptive deterministic steps. Stochastic time steps use Brownian bridges and require more work for a given number of time steps. Deterministic time steps may yield more time steps but require less work; for example, in the limit of vanishing error tolerance, the ratio of the computational error and its computable estimate tends to 1 with negligible additional work to determine the adaptive deterministic time steps. © 2001 John Wiley & Sons, Inc.

The innovation presented is a method for adaptive time-stepping that allows clustering of time steps in portions of the cycle for which flow variables are rapidly changing, based on the concept of using a uniform step in a relevant dependent variable rather than a uniform step in the independent variable time. A user-defined function was developed to adapt the magnitude of the time step (adaptive time step) to a defined rate of change in inlet velocity. Quantitative comparison indicates that the new adaptive time stepping method significantly improves accuracy for simulations using an equivalent number of time steps per cycle.

Rate optimization of fractional flow reservoir model based on the continuous adjoint method

Implicit solvers present strong limitations when used on supercomputing facilities and in particular for adaptive mesh-refinement codes. We present a new method for implicit adaptive time-stepping on adaptive mesh refinement-grids. We implement it in the radiation hydrodynamics solver we designed for the RAMSES code for astrophysical purposes and, more particularly, for protostellar collapse. We briefly recall the radiation hydrodynamics equations and the adaptive time-stepping methodology used for hydrodynamical solvers. We then introduce the different types of boundary conditions (Dirichlet, Neumann, and Robin) that are used at the interface between levels and present our implementation of the new method in the RAMSES code. The method is tested against classical diffusion and radiation hydrodynamics tests, after which we present an application for protostellar collapse. We show that using Dirichlet boundary conditions at level interfaces is a good compromise between robustness and accuracy and that it can be used in structure formation calculations. The gain in computational time over our former unique time step method ranges from factors of 5 to 50 depending on the level of adaptive time-stepping and on the problem. We successfully compare the old and new methods for protostellar collapse calculations that involve highly nonlinear physics. We have developed a simple but robust method for adaptive time-stepping of implicit scheme on adaptive mesh-refinement grids. It can be applied to a wide variety of physical problems that involve diffusion processes.

Adaptive time-step unscented kalman filtering (ATS-UKF) based observer design for urea selective catalytic reduction (SCR) performance of diesel engines

Characterization of the hydrated state of a protein is crucial for understanding its structural stability and function. In the present study, we have investigated the 3D hydration structure of the protein BPTI (bovine pancreatic trypsin inhibitor) by molecular dynamics (MD) and the integral equation method in the three-dimensional reference interaction site model (3D-RISM) approach. Both methods have found a well-defined hydration layer around the protein and revealed the localization of BPTI buried water molecules corresponding to the X-ray crystallography data. Moreover, under 3D-RISM calculations, the obtained positions of waters bound firmly to the BPTI sites are in reasonable agreement with the experimental results mentioned above for the BPTI crystal form. The analysis of the 3D hydration structure (thickness of hydration shell and hydration numbers) was performed for the entire protein and its polar and non-polar parts using various cut-off distances taken from the literature as well as by a straightforward procedure proposed here for determining the thickness of the hydration layer. Using the thickness of the hydration shell from this procedure allows for calculating the total hydration number of biomolecules properly under both methods. Following this approach, we have obtained the thickness of the BPTI hydration layer of 3.6 Å with 369 water molecules in the case of MD simulation and 3.9 Å with 333 water molecules in the case of the 3D-RISM approach. The above procedure was also applied for a more detailed description of the BPTI hydration structure near the polar charged and uncharged radicals as well as non-polar radicals. The results presented for the BPTI as an example bring new knowledge to the understanding of protein hydration.

<p>Natural and induced seismicity is widely investigated, and extensive knowledge has been acquired in the last years, but exact earthquake mechanisms remain elusive and poorly understood. The high impact of earthquakes on human society emphasizes that a deeper understanding of earthquake processes must be a priority in order to improve seismic hazard assessment and mitigate associated risks. Pervasive fluid flow is a key process significantly influencing rock physics and mechanics, and thus has a crucial impact on natural and induced earthquakes. Seismo-hydro-thermo-mechanical (SHTM) modelling is an important nascent branch of geodynamic modelling, which investigates evolution of coupled fluid-solid systems under conditions of both slow tectonic and fast seismic deformation rates. Here, we present a new fully coupled two-dimensional seismo-hydro-mechanical numerical code, with a poro-visco-elasto-plastic rheology, based on fully staggered finite differences with marker-in-cell technique, adaptive time stepping and global Picard iterations. The presented numerical code combines inertial mechanical deformation with pervasive fluid flow. The adaptive time stepping allows the resolution of co-seismic and interseismic phases with time steps ranging from milliseconds to years.</p><p>First, we demonstrate how fluid-bearing subducting rocks are intrinsically seismic and how seismic events in the form of highly localized ruptures spontaneously nucleate along the subduction interface. Nucleation and propagation of such events are driven by rapid fluid pressurization caused by visco-plastic compaction, counterbalanced by a nearly simultaneous and equivalent poroelastic decompaction inside the propagating and rupturing fault. Successive post and interseismic fluid pressure release, generated by poroelastic compaction along the fault, allows strength recovery of the megathrust. The model reproduces the broad range of seismic events present at the subduction interface, including slower events, regular earthquakes, and earthquakes that rupture the entire megathrust and reach velocities on the order of m/s, without employing slip rate dependent frictional properties.</p><p>Next, we show how our approach can be successfully adapted for fluid injection setups to model induced seismicity phenomena. The numerical code successfully modelled fluid induced seismic events and the resulting seismic wave propagation.  Preliminary results show that faults can form spontaneously and grow aseismically at the injection site thereby creating favorable conditions for the development of broad induced seismicity region where different faults can be activated seismically at different time.</p><p>Finally, we outline in short future SHTM modeling directions that should account for fracture-induced dilatation and dynamic permeability variations, thermal effects and three-dimensionality.</p>

Adaptive time-step control for modal methods to integrate the neutron diffusion equation

Time-averaging and exponential integrators for non-homogeneous linear IVPs and BVPs

Adaptive second-order Crank-Nicolson time-stepping methods using the recent scalar auxiliary variable (SAV) approach are developed for the time-fractional Molecular Beam Epitaxial models with Caputo's derivative. Based on the piecewise linear interpolation, the Caputo's fractional derivative is approximated by a novel second-order formula, which is naturally suitable for a general class of nonuniform meshes and essentially preserves the positive semi-definite property of integral kernel. The resulting Crank-Nicolson SAV time-stepping schemes are unconditional energy stable on nonuniform time meshes, and are computationally efficient in multiscale time simulations when combined with adaptive time steps, such as are appropriate for accurately resolving the intrinsically initial singularity of solution and for efficiently capturing fast dynamics away from the initial time. Numerical examples are presented to show the effectiveness of our methods.

In molecular dynamics (MD) calculations, reduction in calculation time per MD loop is essential. A multiple time-step (MTS) integration algorithm, the RESPA (Tuckerman and Berne, J. Chem. Phys. 1992, 97, 1990-2001), enables reductions in calculation time by decreasing the frequency of time-consuming long-range interaction calculations. However, the RESPA MTS algorithm involves uncertainties in evaluating the atomic interaction-based pressure (i.e., atomic pressure) of systems with and without holonomic constraints. It is not clear which intermediate forces and constraint forces in the MTS integration procedure should be used to calculate the atomic pressure. In this article, we propose a series of equations to evaluate the atomic pressure in the RESPA MTS integration procedure on the basis of its equivalence to the Velocity-Verlet integration procedure with a single time step (STS). The equations guarantee time-reversibility even for the system with holonomic constrants. Furthermore, we generalize the equations to both (i) arbitrary number of inner time steps and (ii) arbitrary number of force components (RESPA levels). The atomic pressure calculated by our equations with the MTS integration shows excellent agreement with the reference value with the STS, whereas pressures calculated using the conventional ad hoc equations deviated from it. Our equations can be extended straightforwardly to the MTS integration algorithm for the isothermal NVT and isothermal-isobaric NPT ensembles. © 2017 Wiley Periodicals, Inc.

High-order arbitrary Lagrangian–Eulerian discontinuous Galerkin methods for the incompressible Navier–Stokes equations

Adaptive time-stepping with high-order embedded Runge-Kutta pairs and rejection sampling provides efficient approaches for solving differential equations. While many such methods exist for solving deterministic systems, little progress has been made for stochastic variants. One challenge in developing adaptive methods for stochastic differential equations (SDEs) is the construction of embedded schemes with direct error estimates. We present a new class of embedded stochastic Runge-Kutta (SRK) methods with strong order 1.5 which have a natural embedding of strong order 1.0 methods. This allows for the derivation of an error estimate which requires no additional function evaluations. Next we derive a general method to reject the time steps without losing information about the future Brownian path termed Rejection Sampling with Memory (RSwM). This method utilizes a stack data structure to do rejection sampling, costing only a few floating point calculations. We show numerically that the methods generate statistically-correct and tolerance-controlled solutions. Lastly, we show that this form of adaptivity can be applied to systems of equations, and demonstrate that it solves a stiff biological model 12.28x faster than common fixed timestep algorithms. Our approach only requires the solution to a bridging problem and thus lends itself to natural generalizations beyond SDEs.

SUMMARYWe present a 2-D thermomechanical computational framework for simulating earthquake sequences in a non-linear visco-elasto-plastic compressible medium. The method is developed for a plane-strain problem and incorporates an invariant formulation of the classical rate- and state-dependent friction law and an adaptive time-stepping, which allows the time step to vary by many orders of magnitude during a simulation. Long-term tectonic convergence is imposed by displacing a boundary at a constant rate, whereas temperature-dependent viscosity is solved using a rapidly converging Newton–Raphson scheme. The 2-D volume is discretized using finite differences on a fully staggered grid and marker-in-cell techniques. An adaptive free-surface approximation is used to modulate the air viscosity with the time step, which allows stresses to vanish on the free surface during the propagation of fast slipping events. We present a set of increasingly complex models in which we investigate how inertia, radiation damping, thermally activated non-linear rheology and off-megathrust splay-fault events affect sequences of seismic and aseismic slip on a simplified subduction megathrust. The new method provides a unique computational framework to analyse earthquake sequences and to connect forearc deformation with the dynamic properties of the megathrust, thus providing a physical link between observations spanning from slow interseismic strain accumulation to localized coseismic slip of individual earthquakes and post-seismic viscoelastic relaxation.

In this paper is presented a method to capture the motion of the human body from multi image video sequences without using markers. The process is composed of five steps: acquisition of video sequences, calibration of the system, surface measurement of the human body for each frame, 3-D surface tracking and tracking of key points. The image acquisition system is currently composed of three synchronized progressive scan CCD cameras and a frame grabber which acquires a sequence of triplet images. Self calibration methods are applied to gain exterior orientation of the cameras, the parameters of internal orientation and the parameters modeling the lens distortion. From the video sequences, two kinds of 3-D information are extracted: a three-dimensional surface measurement of the visible parts of the body for each triplet and 3-D trajectories of points on the body. The approach for surface measurement is based on multi-image matching, using the adaptive least squares method. A full automatic matching process determines a dense set of corresponding points in the triplets. The 3-D coordinates of the matched points are then computed by forward ray intersection using the orientation and calibration data of the cameras. The tracking process is also based on least squares matching techniques. Its basic idea is to track triplets of corresponding points in the three images through the sequence and compute their 3-D trajectories. The spatial correspondences between the three images at the same time and the temporal correspondences between subsequent frames are determined with a least squares matching algorithm. The results of the tracking process are the coordinates of a point in the three images through the sequence, thus the 3-D trajectory is determined by computing the 3-D coordinates of the point at each time step by forward ray intersection. Velocities and accelerations are also computed. The advantage of this tracking process is twofold: it can track natural points, without using markers; and it can track local surfaces on the human body. In the last case, the tracking process is applied to all the points matched in the region of interest. The result can be seen as a vector field of trajectories (position, velocity and acceleration). The last step of the process is the definition of selected key points of the human body. A key point is a 3-D region defined in the vector field of trajectories, whose size can vary and whose position is defined by its center of gravity. The key points are tracked in a simple way: the position at the next time step is established by the mean value of the displacement of all the trajectories inside its region. The tracked key points lead to a final result comparable to the conventional motion capture systems: 3-D trajectories of key points which can be afterwards analyzed and used for animation or medical purposes.