The Wilkinson Prize was established to honor the outstanding contributions of Dr. It is awarded every four years to the entry that best addresses all phases of the preparation of numerical software at the International Congress on Industrial and Applied Mathematics by Argonne National Laboratory , the National Physical Laboratory and the Numerical Algorithms Group. Previous Wilkinson Prize winners include. As the dolfin-adjoint project details on its Web site, the need for adjoints of partial differential equations PDEs pervades science and engineering.
Adjoints enable the study of the sensitivity and stability of physical systems, and the optimization of designs subject to constraints.
While deriving the adjoint model associated with a linear stationary forward model is straightforward, the derivation and implementation of adjoint models for nonlinear or time-dependent models is notoriously difficult. Dolfin-adjoint solves this problem by automatically analyzing and exploiting the high-level mathematical structure inherent in finite element methods.
Speaking of the winning software, Mike Dewar, Chair of the Wilkinson Prize Board of Trustees and Chief Technical Officer at NAG said "dolfin-adjoint is an excellent piece of software that can solve problems in a range of application areas. Through its elegant use of high-level abstractions it makes performing what is usually a very challenging piece of computation seem extremely natural.
The dolfin-adjoint project automatically derives the discrete adjoint and tangent linear models from a forward model written in the Python interface to DOLFIN. These adjoint and tangent linear models are key ingredients in many important algorithms, such as data assimilation, optimal control, sensitivity analysis, design optimization and error estimation. Such models have made an enormous impact in fields such as meteorology and oceanography, but their use in other scientific fields has been hampered by the great practical difficulty of their derivation and implementation.
The project site explains that the traditional approach to deriving adjoint and tangent linear models is called algorithmic differentiation also called automatic differentiation.gaivabpostti.tk
The Art of Differentiating Computer Programs | Society for Industrial and Applied Mathematics
The fundamental idea of algorithmic differentiation is to treat the model as a sequence of elementary instructions. An elementary instruction is a simple operation such as addition, multiplication or exponentiation. The specification typically involves several layers of abstractions, but we may think of it as a single computer program in an imperative language like Fortran and C, or systems like Mathematica and Maple. By applying the above simple rules recursively we must arrive at the derivatives of elementary operations and functions that are part of the programming environment.
The total set of variables is therefore given by. Then we may decompose the formula into the following sequence of elemental operations:. More generally, we will consider three-part function evaluation procures of the form given in Table 2. In actual computer programs some of the intermediate quantities v i will share the same memory location because they are not needed at the same time. For example, in the simple example above v 4 and v 5 may overwrite v 1 or v 3. That makes no difference for the forward mode of differentiation, but poses a challenge to the reverse mode as we will see.
For the time being we will stay with our single assignment assumption that each variable v i has its own memory location and occurs exactly once on the left-hand side of an instruction. For details see the standard reference [ 15 ]. Here n i , the number of arguments, is equal to 1 for unary nonlinear functions and to 2 for binary arithmetic operations.
In other words, we need to compute. For example, in case of a multiplication. Thus we have in both cases two extra multiplications and either one or two extra additions. After the preparation we can now develop the basic modes of algorithmic differentiation. Then each corresponding intermediate value v i has a linearization. In the little example above we obtain the extended procedure depicted in Table 4. It is quite clear that the computational effort for propagating the directional derivatives i on top of the v i is just a small multiple of propagating the v i by themselves.
More precisely, in terms of the complexity measure OPS we obtain by 1. In this estimate we have assumed that computational cost is essentially additive, which ignores for example delays due to the scheduling between the various subtasks and gains that might be made on a multicore machine by parallel executions of several threads.
The resulting complexity estimate is.
This bound can be reduced in theory and practice if one executes Table 3 in vector mode, i. Then common intermediate quantities can be reused and the vector operations are likely to run quite fast for a moderate number n of independent variables.
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This technique of row compression is described in some detail in [ 15 ] and reduces the complexity growth to a multiple of rather than n. Recently, there has been a steadily growing interest in a process called adjoining scientific and industrial codes.
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The concept of adjoints applies originally to algebraic and differential equations on function spaces. Its discrete analog is what is called the reverse mode of differentiation. Instead of propagating the dot quantities i that represent sensitivities of intermediates with respect to independent variables forwards, the reverse mode propagates backwards the bar quantities.
What at first may just seem some notational manipulation turns out to be an exciting and fundamental result. The same is true for the initialization of the y l - i , which are input variables, and the deinitialization of the x j , which are output variables. Apparently the first author to write down this reverse procedure was Seppo Linnainmaa, who listed it in Fortran at the end of his Master Thesis [ 23 ] , which is otherwise written in Finnish. He interpreted and used the quantitie v i to estimate the propagation of errors in complicated programs.
For more information on the history of the reverse mode see [ 16 ]. The validity of Table 5 can be derived from the classical rules of differentiation using either directed acyclic graphs or matrix products [ 15 ]. It should be noted that Table 5 is, in contrast to Table 3 , not a single assignment code. This can be seen in the adjoint procedure Table 6 , which needs to follow our little example program Table 1. With respect to the computational cost we find by 1. Here we account for executing the forward sweep Table 2 and the reverse sweep Table 5 after one another.
In other words, as Wolfe [ 36 ] observed in , gradients can 'always' be computed at a small multiple of the cost of computing the underlying function, irrespective of n , the number of independent variables, which may be huge. This interpretation was used amongst others by the oceanographer Thacker in [ 32 ].
It might be used to identify critical and calm parts of an evaluation process, possibly suggesting certain simplifications, e. To highlight the properties of the reverse mode, let us consider a very simple example of variable dimension that was originally suggested by the late Arthur Sedgewick, the PhD supervisor of Bert Speelpenning at the University of Urbana Champaign.
They considered a simple product and its gradient. Computing each gradient component independently would require n 2 - O n multiplications and there are cheaper alternatives. However, this approach is not entirely convincing since divisions are much more expensive than multiplications and may lead to a NaN if some component x j is zero. Now let us apply the reverse mode. Starting from the natural, forward evaluation loop. Here we have eliminated the assignments from the x j to the v j - n and vice versa from v j- n to x j for the sake of readability.
We could also have replaced the incremental assignments by direct assignments since no intermediate occurs more than once as an argument.
Algorithmic Differentiation (2016-2017)
That would save n zero initializations and the same number of additions. Of course, the key observation is that this gradient procedure requires at most 4 times as many arithmetic operations as the evaluation of the function itself and involves no tests and branching. This effect was observed by Arthur S. That suggestion led Bert S. When m and n are similar, the forward mode is still preferable because of the following memory issue.
The dot quantities i and j can and should share the same memory location exactly when that is true for v i and j. The results will still be consistent in that i is the directional derivative of v i ,even if there were some unintended overwriting, for example through the aliasing of calling parameters. The situation is radically different in the reverse mode. Hence, the old value of v j must be written on a stack just before it is overwritten and then recuperated on the reverse sweep. Alternatively, some AD implementations prefer to store the partials c ij.
Related The Art of Differentiating Computer Programs: An Introduction to Algorithmic Differentiation
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