Adjoint method

Results from this model were verified by Neupauer and Wilson [51] using the adjoint method. In this method, the forward governing equation, with concentration as the dependent variable, is replaced by the adjoint equation, with the adjoint state as the dependent variable. They showed that backward-in-time location and travel time probabilities are adjoint states of the forward-in-time resident concentration. In this and the follow-up paper, Neupauer and Wilson [51,52] presented the adjoint method as a formal framework for obtaining the backward-in-time probabilities for multidimensional problems and more complex domain geometries. 

Neupauer RM, Wilson JL (1999) Adjoint method for obtaining backward-in-time location and travel time probabilities of a conservative groundwater contaminant. Water Resour Res 35 3389-3398... 

Heberton, C.I., Russel, T.F., Konikow, L.F., and Hornberger, G.Z., A Three-Dimensional Finite-Volume Eulerian-Lagrangian Localized Adjoint Method (ELLAM) for Solute-Transport Modeling, Water Resources Investigations Report 00-4087, U.S. Geological Survey, Reston, VA, 2000. 

We have used sensitivity equation methods (Leis and Kramer, 1985) for gradient evaluation as these are simple and efficient for problems with few parameters and constraints. In general, the balance in efficiency between sensitivity and adjoint methods depends on the type of problem being addressed. Adjoint methods are particularly advantageous for optimal control problems in which the inputs are represented as a large number of piecewise constant input values and few interior point constraints exist. Sensitivity methods are preferable for problems with few parameters and many constraints. 

Adjoint method Solution of n ODEs of (4.A.3) once, followed by solution of n ODEs of (4.A.11) once, followed by evaluation of an integral (4. A. 13) for each concentration... 

Appendix 4 Sensitivity/Uncertainty Analysis of Atmospheric Chemical Mechanisms 219 4.A. 1 Sensitivity Coefficients 222 4.A.2 The Direct Decoupled Method 223 4. A.3 Adjoint Methods 224 4.A.4 Green s Function Methods 224 References 226 Problems 230... 

Two widely used practical inverse approaches are presented in the next two sections ( Estimating absolute velocities and nutrient fluxes across sections and Estimating carbon export fluxes with the adjoint method ). These serve as examples to describe details of the mathematical methods and to list achievable results. The first method, the section inverse approach, infers nutrient, carbon, and tracer fluxes across sets of sections, based on hydro-graphic, tracer, and nutrient data along these sections. The second example describes an application of the adjoint method for the determination of ocean currents, biological productivity, and downward particle fluxes. This method is specifically adapted for the utilization of many different tracers and can handle problems with heterogeneous and sparse data coverage. 

Estimating Carbon Export Fiuxes with the Adjoint Method... 

The method (2.18)-(2.19) is called the Symplectic Euler method. Its adjoint method has a similar structure ... 

Given a method with adjoint method it is possible to obtain the adjoint of the adjoint method but, as we might expect, the adjoint of the adjoint is the... 

Example 2.5 (Verlet is Symplectic) For the Symplectic Euler method /, and its adjoint method consider the composition... 

When we construct a composition of a method with its adjoint method, we see readily that it is symmetric, i.e. self-adjoint, since... 

Is it true that any symplectic method will have a symplectic adjoint method Either give a proof or find a counterexample. 

Recall that the Stormer-Verlet method could be constructed by composing steps using Symplectic Euler and its adjoint method. Using more complicated methods it is possible to build higher order schemes. It seems natural that a similar procedure should be possible in the constrained setting. But what, precisely, is the adjoint method in the case of (4.20)-(4.24) ... 

Recall that, for a given method Qh the definition of the adjoint method Ql is given... 

Loulou, T., and Scott, E. P., 2000 2-D Thermal Dose Optimization in High Intensity Focused Ultrasound Treatments Using the Adjoint Method, ASME HTD—Vol. 368/BED—Vol. 47, Advances in Heat and Mass Transfer in Biotechnology—2(X)0, pp. 67—71. 

Response sensitivity computation can be performed using different methods, such as the forward/backward/central Finite Difference Method (FDM) (Kleiber et al. 1997, Conte et al. 2003, 2004), the Adjoint Method (AM) (Kleiber et al. 1997), the Perturbation Method (PM) (Kleiber Hien 1992), and the Direct Differentiation Method... 

I think the first step in this is sensitivity analysis, for which there has been a lot of work in past decade. The forward method of sensitivity analysis for differential equations has been thoroughly investigated, and software is available. The adjoint method is a much more powerful method of sensitivity analysis for some problems, but it is more difficult to implement, although good progress has been made. Next it moved to PDEs, where progress is being made for boundary conditions and adaptive meshes. 

Forward and adjoint methods—PDE methods with adaptive mesh refinement... 

When the number of unknowns is large, as for example in the case that all grid block property values are regarded as unknown parameters, then it is impractical to find the derivatives of the objective function. By treating the fluid flow model as a constraint, the adjoint method can be used. This is, conceptually, the same adjoint method as described in Section 4.5. The method exploits the fact that the dot product of the gradient with a small number of fixed vectors can be obtained at relatively low cost. However, this means that a lower order optimisation method, such as the conjugate gradient technique must be used. This is why for small numbers of unknowns the direct technique is best. 

For each control variable, the sensitivity method involves an additional Hnear IVP to be solved. Thus for a large number of parameters, this method is not reasonable. Another method is more feasible which avoids calculating the sensitivity of the state directly, the so-called adjoint method which enables to compute the sensitivities in one shot. 

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