32,726 research outputs found
Kernel-based stochastic collocation for the random two-phase Navier-Stokes equations
In this work, we apply stochastic collocation methods with radial kernel
basis functions for an uncertainty quantification of the random incompressible
two-phase Navier-Stokes equations. Our approach is non-intrusive and we use the
existing fluid dynamics solver NaSt3DGPF to solve the incompressible two-phase
Navier-Stokes equation for each given realization. We are able to empirically
show that the resulting kernel-based stochastic collocation is highly
competitive in this setting and even outperforms some other standard methods
State estimation for linear stochastic differential equations with uncertain disturbances via BSDE approach
A backward stochastic differential equation (BSDE) is an Ito stochastic differential equation (SDE) for which a random terminal condition on the state has been specified. The paper deals with estimation problems for partly observed stochastic processes described by linear SDEs with uncertain disturbances. The disturbances and unknown initial states are supposed to be constrained by the inequality including mathematical expectation of the integral quadratic cost. We consider our equations as BSDEs, and construct at given instant the random information set of all possible states which are compatible with the measurements and the constraints. The center of this set represents the best estimation of the process' state. The evolutionary equations for the random information set and for the best estimation are given. Some examples and applications are considered. © 2012 American Institute of Physics
On the Interpretation of Delays in Delay Stochastic Simulation of Biological Systems
Delays in biological systems may be used to model events for which the
underlying dynamics cannot be precisely observed. Mathematical modeling of
biological systems with delays is usually based on Delay Differential Equations
(DDEs), a kind of differential equations in which the derivative of the unknown
function at a certain time is given in terms of the values of the function at
previous times. In the literature, delay stochastic simulation algorithms have
been proposed. These algorithms follow a "delay as duration" approach, namely
they are based on an interpretation of a delay as the elapsing time between the
start and the termination of a chemical reaction. This interpretation is not
suitable for some classes of biological systems in which species involved in a
delayed interaction can be involved at the same time in other interactions. We
show on a DDE model of tumor growth that the delay as duration approach for
stochastic simulation is not precise, and we propose a simulation algorithm
based on a ``purely delayed'' interpretation of delays which provides better
results on the considered model
Simultaneous Optimal Uncertainty Apportionment and Robust Design Optimization of Systems Governed by Ordinary Differential Equations
The inclusion of uncertainty in design is of paramount practical importance because all real-life systems are affected by it. Designs that ignore uncertainty often lead to poor robustness, suboptimal performance, and higher build costs. Treatment of small geometric uncertainty in the context of manufacturing tolerances is a well studied topic. Traditional sequential design methodologies have recently been replaced by concurrent optimal design methodologies where optimal system parameters are simultaneously determined along with optimally allocated tolerances; this allows to reduce manufacturing costs while increasing performance. However, the state of the art approaches remain limited in that they can only treat geometric related uncertainties restricted to be small in magnitude.
This work proposes a novel framework to perform robust design optimization concurrently with optimal uncertainty apportionment for dynamical systems governed by ordinary differential equations. The proposed framework considerably expands the capabilities of contemporary methods by enabling the treatment of both geometric and non-geometric uncertainties in a unified manner. Additionally, uncertainties are allowed to be large in magnitude and the governing constitutive relations may be highly nonlinear.
In the proposed framework, uncertainties are modeled using Generalized Polynomial Chaos and are solved quantitatively using a least-square collocation method. The computational efficiency of this approach allows statistical moments of the uncertain system to be explicitly included in the optimization-based design process. The framework formulates design problems as constrained multi-objective optimization problems, thus enabling the characterization of a Pareto optimal trade-off curve that is off-set from the traditional deterministic optimal trade-off curve. The Pareto off-set is shown to be a result of the additional statistical moment information formulated in the objective and constraint relations that account for the system uncertainties. Therefore, the Pareto trade-off curve from the new framework characterizes the entire family of systems within the probability space; consequently, designers are able to produce robust and optimally performing systems at an optimal manufacturing cost.
A kinematic tolerance analysis case-study is presented first to illustrate how the proposed methodology can be applied to treat geometric tolerances. A nonlinear vehicle suspension design problem, subject to parametric uncertainty, illustrates the capability of the new framework to produce an optimal design at an optimal manufacturing cost, accounting for the entire family of systems within the associated probability space. This case-study highlights the general nature of the new framework which is capable of optimally allocating uncertainties of multiple types and with large magnitudes in a single calculation
Atomic radius and charge parameter uncertainty in biomolecular solvation energy calculations
Atomic radii and charges are two major parameters used in implicit solvent
electrostatics and energy calculations. The optimization problem for charges
and radii is under-determined, leading to uncertainty in the values of these
parameters and in the results of solvation energy calculations using these
parameters. This paper presents a new method for quantifying this uncertainty
in implicit solvation calculations of small molecules using surrogate models
based on generalized polynomial chaos (gPC) expansions. There are relatively
few atom types used to specify radii parameters in implicit solvation
calculations; therefore, surrogate models for these low-dimensional spaces
could be constructed using least-squares fitting. However, there are many more
types of atomic charges; therefore, construction of surrogate models for the
charge parameter space requires compressed sensing combined with an iterative
rotation method to enhance problem sparsity. We demonstrate the application of
the method by presenting results for the uncertainties in small molecule
solvation energies based on these approaches. The method presented in this
paper is a promising approach for efficiently quantifying uncertainty in a wide
range of force field parameterization problems, including those beyond
continuum solvation calculations.The intent of this study is to provide a way
for developers of implicit solvent model parameter sets to understand the
sensitivity of their target properties (solvation energy) on underlying choices
for solute radius and charge parameters
- …