Uncertainty quantification and sensitivity analyses are a vital component for predictive modeling in the sciences and engineering. The adjoint approach to sensitivity analysis requires solving a primary system of equations and a mathematically related set of adjoint equations. The information contained in the equations can be combined to produce sensitivity information in a computationally efficient manner. In this work, sensitivity analyses are performed on systems described by flux-limited radiative diffusion using the adjoint approach. The sensitivities computed are shown to agree with standard perturbation theory, and can be obtained in significantly less computational time.
Verification, validation and uncertainty quantification (VVUQ) have become a common practice in thermal-hydraulics analysis. An important step in the uncertainty analysis is the sensitivity analysis of various uncertain input parameters. The common approach for computing the sensitivities, e.g. variance-based and regression-based methods, requires solving the governing equation multiple times, which is expensive in terms of computational cost. An alternative approach to compute the sensitivities is the adjoint method. The cost of solving an adjoint equation is comparable to the cost of solving…
The efficient method for computing the sensitivities is the adjoint method. The cost of solving an adjoint equation is comparable to the cost of solving the governing equation. Once the adjoint solution is obtained, the sensitivities to any number of parameters can be obtained with little effort. There are two methods to develop the adjoint equations: continuous method and discrete method. In the continuous method, the control theory is applied to the forward governing equation and produces an analytical partial differential equation for solving the adjoint variable; in the discrete method, th…
We develop the NILSAS algorithm, which performs adjoint sensitivity analysis of chaotic systems via computing the adjoint shadowing direction. NILSAS constrains its minimization to the adjoint unstable subspace, and can be implemented with little modification to existing adjoint solvers. The computational cost of NILSAS is independent of the number of parameters. We demonstrate NILSAS on the Lorenz 63 system and a weakly turbulent three-dimensional flow over a cylinder.
We introduce a technique to compute exact anelastic sensitivity kernels in the time domain using parsimonious disk storage. The method is based on a reordering of the time loop of time-domain forward/adjoint wave propagation solvers combined with the use of a memory buffer. It avoids instabilities that occur when time-reversing dissipative wave propagation simulations. The total number of required time steps is unchanged compared to usual acoustic or elastic approaches. The cost is reduced by a factor of 4/3 compared to the case in which anelasticity is partially accounted for by accommodating…
The friction coefficient and the base topography of a stationary and a dynamic ice sheet are perturbed in two models for the ice: the full Stokes equations and the shallow shelf approximation. The sensitivity to the perturbations of the velocity and the height at the surface is quantified by solving the adjoint equations of the stress and the height equations providing weights for the perturbed data. The adjoint equations are solved numerically and the sensitivity is computed in several examples in two dimensions. Comparisons are made with analytical solutions to simplified problems.
Sensitivity analysis plays an important role in searching for constitutive parameters (e.g. permeability) subsurface flow simulations. The mathematics behind is to solve a dynamic constrained optimization problem. Traditional methods like finite difference and forward sensitivity analysis require computational cost that increases linearly with the number of parameters times number of cost functions. Discrete adjoint sensitivity analysis (SA) is gaining popularity due to its computational efficiency. This algorithm requires a forward run followed by a backward run who involves integrating adjoi…
This paper describes a forward algorithm and an adjoint algorithm for computing sensitivity derivatives in chaotic dynamical systems, such as the Lorenz attractor. The algorithms compute the derivative of long time averaged "statistical" quantities to infinitesimal perturbations of the system parameters. The algorithms are demonstrated on the Lorenz attractor. We show that sensitivity derivatives of statistical quantities can be accurately estimated using a single, short trajectory (over a time interval of 20) on the Lorenz attractor.
The velocity field and the height at the surface of a dynamic ice sheet are observed. The ice sheets are modeled by the full Stokes equations and shallow shelf/shelfy stream approximations. Time dependence is introduced by a kinematic free surface equation which updates the surface elevation using the velocity solution. The sensitivity of the observed quantities at the ice surface to parameters in the models, for example the basal topography and friction coefficients, is analyzed by first deriving the time dependent adjoint equations. Using the adjoint solutions, the effect of a perturbation i…
In chaotic systems, such as turbulent flows, the solutions to tangent and adjoint equations exhibit an unbounded growth in their norms. This behavior renders the instantaneous tangent and adjoint solutions unusable for sensitivity analysis. The Lea-Allen-Haine ensemble sensitivity (ES) estimates provide a way of computing meaningful sensitivities in chaotic systems by utilizing tangent/adjoint solutions over short trajectories. In this paper, we analyze the feasibility of ES computations under optimistic mathematical assumptions on the flow dynamics. Furthermore, we estimate upper bounds on th…
Adjoint methods enable the accurate calculation of the sensitivities of a quantity of interest. The sensitivity is obtained by solving the adjoint system, which can be derived by continuous or discrete adjoint strategies. In acoustic wave propagation, continuous and discrete adjoint methods have been developed to compute the eigenvalue sensitivity to design parameters and passive devices (Aguilar, J. G. et al, 2017, J. Computational Physics, vol. 341, 163-181). In this short communication, it is shown that the continuous and discrete adjoint characteristic decompositions, and Riemann invariant…
In the context of adjoint-based optimization, nonlinear conservation laws pose significant problems regarding the existence and uniqueness of both direct and adjoint solutions, as well as the well-posedness of the problem for sensitivity analysis and gradient-based optimization algorithms. In this paper we will analyze the convergence of the adjoint equations to known exact solutions of the inviscid Burgers' equation for a variety of numerical schemes. The effect of the non-differentiability of the underlying approximate Riemann solver, complete vs. incomplete differentiation of the discrete s…
In this paper, we discuss selected adjoint approaches for the turbulent flow control. In particular, we focus on the application of adjoint solvers for the scope of noise reduction, in which flow solutions are obtained by large eddy and direct numerical simulations. Optimization results obtained with round and plane jet configurations are presented. The results indicate that using large control horizons poses a serious problem for the control of turbulent flows due to existence of very large sensitivity values with respect to control parameters. Typically these sensitivities grow in time and l…
The evolution of any complex dynamical system is described by its state derivative operators. However, the extraction of the exact N-order state derivative operators is often inaccurate and requires approximations. The open-source CFD code called BROADCAST discretises the compressible Navier-Stokes equations and then extracts the linearised Nderivative operators through Algorithmic Differentiation (AD) providing a toolbox for laminar flow dynamic analyses. Furthermore, the gradients through adjoint derivation are extracted either by transposition of the linearised operator or through the backw…
We describe how to apply adjoint sensitivity methods to backward Monte-Carlo schemes arising from simulations of particles passing through matter. Relying on this, we demonstrate derivative based techniques for solving inverse problems for such systems without approximations to underlying transport dynamics. We are implementing those algorithms for various scenarios within a general purpose differentiable programming C++17 library NOA (github.com/grinisrit/noa).
Interpretable Geometry Sensitivity for Inverse Design of Integrated Photonics
Department of Electrical and Computer Engineering, University of Michigan, Ann Arbor（密歇根大学）；College of Computing, Data Science, and Society, University of California, Berkeley（加州大学伯克利分校计算、数据与社会学院）
