Use Of Adjoint Equations Research Articles

The COSMO-Ru system of nonhydrostatic mesoscale short-range weather forecasting of the Hydrometcenter of Russia: The second stage of implementation and development

Described is the second stage of the work (2011-2014) on the implementation and development of the COSMO-Ru system of nonhydrostatic short-range weather forecasting (the first stage of the implementation and development of the COSMO-Ru system is described in [7, 8]). Demonstrated is how the research activities and ideas of G.I. Marchuk influenced modern methods for solving the systems of differential equations that describe atmospheric processes (in particular, the version of the Marchuk’s splitting method is used to find the solution of the finite-difference analog of the system of differential equations in the COSMO-Ru model); it is shown how he contributed to the development of the methods of assimilation of meteorological information associated with the use of adjoint equations. Given is a brief description of the COSMO model of the atmosphere and soil active layer, the COSMO-Ru system, and research activities on this system development.

Adjoint Equations in Stability Analysis

The objective of this article is to review some developments in the use of adjoint equations in hydrodynamic stability theory. Adjoint-based sensitivity analysis finds both analytical and numerical applications much beyond those originally imagined. It can be used to identify optimal perturbations, pinpoint the most receptive path to break down, select the most destabilizing base-flow defect in a nominally stable configuration, and map the structural sensitivity of an oscillator. We focus on two flow cases more closely: the noise-amplifying instability of a boundary layer and the global mode occurring in the wake of a cylinder. For both cases, the clever interpretation and use of direct and adjoint modes provide key insight into the process of the transition to turbulence.

Sensitivity analysis in models of spread of sexually transmitted infections with dynamic risk of infection

The study of the dynamics of sexually transmitted infections (STI) is important for development of efficient anti-epidemic measures. Traditionally, the models describing the spread of such infections are formulated in terms of the dynamics of the population of individuals infected and susceptible to infection in different risk groups. In general, suchmodels do not take into account the possibility of an individual changing its risk group during the observation period. However, for diseases lasting in time, such as HIV, such considerations may be necessary.In this paper we propose a general form of the model of sexually transmitted infections spreading with a dynamic risk of infection, present the results of analytic study of the sensitivity of the functionals of solutions to small perturbations of the model parameters with the use of adjoint equations, as well as computational results and comparison with regression analysis taking the example of an HIV infection spread model.

Point source identification in nonlinear advection–diffusion–reaction systems

We consider a problem of identification of point sources in time-dependent advection–diffusion systems with a nonlinear reaction term. The linear counterpart of the problem in question can be reduced to solving a system of nonlinear algebraic equations via the use of adjoint equations. We extend this approach by constructing an algorithm that solves the problem iteratively to account for the nonlinearity of the reaction term. We study the question of improving the quality of source identification by adding more measurements adaptively using the solution obtained previously with a smaller number of measurements.

Goal-oriented uncertainty propagation using stochastic adjoints

We propose a framework based on the use of adjoint equations to formulate an adaptive sampling strategy for uncertainty quantification for problems governed by algebraic or differential equations involving random parameters. The approach is non-intrusive and makes use of discrete sampling based on collocation on simplex elements in stochastic space. Adjoint or dual equations are introduced to estimate errors in statistical moments of random functionals resulting from the inexact reconstruction of the solution within the simplex elements. The approach is demonstrated to be accurate in estimating errors in statistical moments of interest and shown to exhibit super-convergence, in accordance with the underlying theoretical rates. Goal-oriented error indicators are then built using the adjoint solution and exploited to identify regions for adaptive sampling. The error-estimation and adaptive refinement strategy is applied to a range of problems including those governed by algebraic equations as well as scalar and systems of ordinary and partial differential equations. The strategy holds promise as a reliable method to set and achieve error tolerances for efficient aleatory uncertainty quantification in complex problems. Furthermore, the procedure can be combined with numerical error estimates in physical space so as to effectively manage a computational budget to achieve the best possible overall accuracy in the results.

Model Calibration in Option Pricing

We consider calibration problems for models of pricing derivatives which occur in mathematical finance. We discuss various approaches such as using stochastic differential equations or partial differential equations for the modeling process. We discuss the development in the past literature and give an outlook into modern approaches of modelling. Furthermore, we address important numerical issues in the valuation of options and likewise the calibration of these models. This leads to interesting problems in optimization, where, e.g., the use of adjoint equations or the choice of the parametrization for the model parameters play an important role.

Solution of the nonlinear elasticity imaging inverse problem: The incompressible case

We have recently developed and tested an efficient algorithm for solving the nonlinear inverse elasticity problem for a compressible hyperelastic material. The data for this problem are the quasi-static deformation fields within the solid measured at two distinct overall strain levels. The main ingredients of our algorithm are a gradient based quasi-Newton minimization strategy, the use of adjoint equations and a novel strategy for continuation in the material parameters. In this paper we present several extensions to this algorithm. First, we extend it to incompressible media thereby extending its applicability to tissues which are nearly incompressible under slow deformation. We achieve this by solving the forward problem using a residual-based, stabilized, mixed finite element formulation which circumvents the Ladyzenskaya–Babuska–Brezzi condition. Second, we demonstrate how the recovery of the spatial distribution of the nonlinear parameter can be improved either by preconditioning the system of equations for the material parameters, or by splitting the problem into two distinct steps. Finally, we present a new strain energy density function with an exponential stress–strain behavior that yields a deviatoric stress tensor, thereby simplifying the interpretation of pressure when compared with other exponential functions. We test the overall approach by solving for the spatial distribution of material parameters from noisy, synthetic deformation fields.

Control of sudden releases in channel flow

We present a method for the detection and real-time control of chemical releases in channel flow. Sensor arrays capable of detecting a broad menu of chemical agents are required at strategic locations of the channel. The sensors detect the instantaneous, spatially distributed concentration of the chemical agent and transmit the associated information to a predictive control model. The model provides optimal operation scenarios for computer controlled bleed valves mounted on the channel walls and connected to a common manifold. Control and elimination of the chemical cloud are achieved by optimal blowing and suction of ambient fluid. Gradient information is obtained by use of adjoint equations, so optimization of the control actions is achieved with the highest possible efficiency. The control is optimized over a finite prediction horizon and instructions are sent to the valve manifold. Next, the sensor arrays detect all changes effected by the control and report them to the control model, which advances the process over the next control horizon. Non-reflective boundary conditions for the adjoint equations are derived by a characteristic analysis, which minimizes spurious information in the computation of sensitivities.

Testing a method for analyzing the effect of parameter change in climate driven ecological systems

In this study, we consider a herbivore-predator metapopulation model, consisting of two patches. Only the herbivores are allowed to migrate between the patches. Furthermore, the intrinsic growth rates of the herbivores depend on climatic fluctuations. Our objective is the conservation of the herbivores. We are interested in finding the parameter perturbations that, when applied, will reduce the extinction risk of one of the subpopulations the most. The risk of extinction is measured in terms of the fifth percentiles of the subpopulations. This is the value below which the subpopulation is found 5 out of 100 times in a series of values taken at fixed time intervals. We make use of the short-term behavior of the model by formulating the tangent linear equations. In the neighborhood of a short interval of a reference orbit the linear error growth can then be calculated. The parameter perturbation that causes the largest error growth, the so-called first singular vector, can be computed with the use of adjoint equations. The adjoint system acts as a backward integration. It turns out that at certain moments in time such a singular vector has a direction for which a parameter perturbation is likely effective in changing the dynamics in a long model simulation. The selection is based on a specific local behavior of the error growth. This adjoint method is compared with a method in which parameter perturbations are randomly chosen. Here we carry out a test for a model with only few parameters. However, it can be applied to models with a very large number of parameters, making the adjoint method an interesting alternative for the random method as then the required number of runs cannot be realized within a realistic computing time. A same objection holds for the use of a systematic method of finding an optimum, e.g. by computing the gradient in the fifth percentile with respect to the parameter values.

The Use of Adjoint Equations in Numerical Weather Prediction

ABSTRACT The adjoint equations allow computation of the sensitivity of one output parameter of a model to all input parameters. After a brief introduction of this technique, its main applications to numerical weather prediction are described. The most trivial application of the adjoint model is to investigate the sensitivity to initial conditions or model parameters. The fact that the adjoint method allows computation of the gradient of a cost function with respect to some parameters in an efficient way makes the minimization of a cost function using descent algorithms possible. This may be applied to estimation problems like variational assimilation. Another use of the adjoint model is the evaluation of the covariances of forecast error in the Kalman filtering context. Finally, the estimation of the singular vectors of a linearized model using its adjoint is relevant for predictability studies.

Important literature on the use of adjoint, variational methods and the Kalman filter in meteorology

The use of adjoint equations is proving to be invaluable in many areas of meteorological research. Unlike a forecast model which describes the evolution of meteorological fields forward in time, the adjoint equations describe the evolution of sensitivity (to initial, boundary and parametric conditions) backward in time. Essentially, by utilizing this sensitivity information, many types of problems can be solved more efficiently than in the past, including variational data assimilation, parameter fitting, optimal instability and sensitivity analysis in general. For this reason, the adjoints of various models and their applications have been appearing more and more frequently in meteorological research. This paper is a bibliography in chronological order of published works in meteorology dealing with adjoints which have appeared prior to this issue of Tellus. Also included are meteorological works regarding variational methods (even without adjoints) and Kalman filtering in data assimilation, plus some references outside meteorology. These additional works are included here because the main thrust for adjoint application within meteorology is currently concentrated in the development of next-generation data assimilation systems. DOI: 10.1034/j.1600-0870.1993.t01-4-00002.x

The use of adjoint equations to solve a variational adjustment problem with advective constraints

The use of adjoint equations to solve a variational adjustment problem with advective constraints

The use of adjoint equations to solve a variational adjustment problem with advective constraints

A methodology is developed to guarantee time continuity in a sequence of analyses. Coupling is accomplished by requiring the least squares minimization of adjustment to the analyses subject to dynamic constraints. In this paper, the analyses are assumed to be governed by advective constraints such as those used in vorticity conservation models; however, the method can easily be applied to other constraints. The results correspond to the application of a strong constraint as introduced by Sasaki, but the procedure used to accomplish the minimization is an alternative to the traditional methods for solution of the Euler-Lagrange equation. The method allows easier inclusion of more time levels in the analysis sequence as well as accommodation of more complicated constraints. The method is tested using both simulated and real data. The simulated data studies use a one-dimensional advection equation with progressively more complicated dynamics: constant advection velocity, spatially varying advection velocity, and nonlinear advection. The real data case study uses an advection of quasi-geostrophic potential vorticity constraint to examine the height adjustment process for three time periods on 6 March 1982. Separate studies are made for analyses derived from VAS and RAOB data. The results of this study indicate the method has excellent potential to reduce the random component of the analysis errors. DOI: 10.1111/j.1600-0870.1985.tb00430.x