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**Fast reverse-mode automatic differentiation using expression templates in C++**

Robin Hogan University of Reading

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**Overview Spaceborne radar and lidar Adjoint coding**

Automatic differentiation New approach Testing with lidar multiple-scattering forward models

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**Spaceborne radar, lidar and radiometers**

EarthCare The A-Train NASA 700-km orbit CloudSat 94-GHz radar (launch 2006) Calipso 532/1064-nm depol. lidar MODIS multi-wavelength radiometer CERES broad-band radiometer AMSR-E microwave radiometer EarthCARE: launch 2015(?) ESA+JAXA 400-km orbit: more sensitive 94-GHz Doppler radar 355-nm HSRL/depol. lidar Multispectral imager Broad-band radiometer Heart-warming name

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**What do CloudSat and Calipso see?**

Cloudsat radar Radar: ~D6, detects whole profile, surface echo provides integral constraint Lidar: ~D2, more sensitive to thin cirrus and liquid but attenuated Radar-lidar ratio provides size D CALIPSO lidar Target classification Insects Aerosol Rain Supercooled liquid cloud Warm liquid cloud Ice and supercooled liquid Ice Clear No ice/rain but possibly liquid Ground Delanoe and Hogan (2008, 2010)

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**Ingredients developed Implement previous work Not yet developed**

Unified retrieval 1. New ray of data: define state vector Use classification to specify variables describing each species at each gate Ice: extinction coefficient, N0’, lidar extinction-to-backscatter ratio Liquid: extinction coefficient and number concentration Rain: rain rate, drop diameter and melting ice Aerosol: extinction coefficient, particle size and lidar ratio 3a. Radar model Including surface return and multiple scattering 3b. Lidar model Including HSRL channels and multiple scattering 3c. Radiance model Solar and IR channels 4. Compare to observations Check for convergence 6. Iteration method Derive a new state vector Adjoint of full forward model Quasi-Newton scheme 3. Forward model Not converged Converged Proceed to next ray of data 2. Convert state vector to radar-lidar resolution Often the state vector will contain a low resolution description of the profile 7. Calculate retrieval error Error covariances and averaging kernel Ingredients developed Implement previous work Not yet developed

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**Unified retrieval: Forward model**

From state vector x to forward modelled observations H(x)... Ice & snow Liquid cloud Rain Aerosol x Adjoint of radar model (vector) Adjoint of lidar model (vector) Adjoint of radiometer model Gradient of cost function (vector) xJ=HTR-1[y–H(x)] Vector-matrix multiplications: around the same cost as the original forward operations Adjoint of radiative transfer models yJ=R-1[y–H(x)] Ice/radar Liquid/radar Rain/radar Ice/lidar Liquid/lidar Rain/lidar Aerosol/lidar Ice/radiometer Liquid/radiometer Rain/radiometer Aerosol/radiometer Lookup tables to obtain profiles of extinction, scattering & backscatter coefficients, asymmetry factor Radar scattering profile Lidar scattering profile Radiometer scattering profile Sum the contributions from each constituent Radar forward modelled obs Lidar forward modelled obs Radiometer fwd modelled obs H(x) Radiative transfer models

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**Radiative transfer models**

Observation Model Speed Status Radar reflectivity factor Multiscatter: single scattering option N OK Radar reflectivity factor in deep convection Multiscatter: single scattering plus TDTS MS model (Hogan and Battaglia 2008) N2 Radar Doppler velocity Single scattering OK if no NUBF; fast MS model with Doppler does not exist Not available for MS HSRL lidar in ice and aerosol Multiscatter: PVC model (Hogan 2008) HSRL lidar in liquid cloud Multiscatter: PVC plus TDTS models Lidar depolarization Multiscatter: under development In progress Infrared radiances Delanoe and Hogan (2008) two-stream source function method No adjoint RTTOV (EUMETSAT license) Disappointing accuracy for clouds Solar radiances LIDORT (permissive license) Testing After much pain have hand-coded adjoint for multiscatter model (in C) but still need adjoint for all the rest of the algorithm (in C++)

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**Adjoint and Jacobian coding**

Variational retrieval methods are posed as: “find the vector x that minimises the cost function J(x)” Two common minimization methods: The quasi-Newton method requires the “adjoint code” to compute the gradient ∂J/∂x for any x The Gauss-Newton method writes the observational part of the cost function as the sum of the squared deviation of the observations from their forward modelled counterparts y, and requires a code to compute the Jacobian matrix H = ∂y/∂x Since J(x) is complicated (containing all of our radiative transfer models), the code to generate ∂J/∂x or ∂y/∂x is even more complicated Can it be generated automatically?

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**Approaches to adjoint coding**

Do it by hand (e.g. ECMWF) Painful and time consuming to debug Generates the most efficient code Do it numerically: perturb each element of x one by one Inefficient and infeasible for large x Subject to round-off error What I’m using at the moment with Unified Algorithm Automatic differentiation 1: Use a source-to-source compiler E.g. TAPENADE/TAF/TAC++ generate adjoint source file from algorithm file: generates quite efficient code Comercial: 5k/year for TAF/TAC++ academic license and need permission to distribute generated source code TAPENADE requires to upload file to server Limited support for C++ classes and no support for C++ templates Automatic differentiation 2: Use an operator overloading technique E.g. CppAD, ADOL-C, in principle can work with any language features Typically 25 times slower than hand-coded adjoint! Can we do better?

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Simple example Consider simple algorithm y(x0, x1) contrived for didactic purposes: Implemented in C or Fortran90 as: Task: given ∂J/∂y, we want to compute ∂J/∂x0 and ∂J/∂x1 function algorithm(x) result(y) implicit none real, intent(in) :: x(2) real :: y real :: s y = 4.0 s = 2.0*x(1) + 3.0*x(2)*x(2) y = y * sin(s) return endfunction double algorithm(const double x[2]) { double y = 4.0; double s = 2.0*x[0] + 3.0*x[1]*x[1]; y *= sin(s); return y; }

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**Creating the adjoint code 1**

Differentiate the algorithm: Write each statement in matrix form: Transpose the matrix to get equivalent adjoint statement: Consider dy as the derivative of y with respect to something Consider d*y as dJ/dy

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**Creating the adjoint code 2**

Apply adjoint statements in reverse order: Reverse mode: Forward mode: double algorithm_AD(const double x[2], double y_AD[1], double x_AD[2]) { double y = 4.0; double s = 2.0*x[0] + 3.0*x[1]*x[1]; y *= sin(s); /* Adjoint part: */ double s_AD = 0.0; y_AD[0] += sin(s) * y_AD[0]; s_AD += y * cos(s) * y_AD[0]; x_AD[0] += 3.0 * s_AD; x_AD[1] += 6.0 * x[0] * s_AD; s_AD = 0.0; y_AD[0] = 0.0; return y; } Note: need to store intermediate values for the reverse pass Hand-coding is time-consuming and error prone for large codes

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**Automatic differentiation**

We want something like this (now in C++): Operators (e.g. +–*/) and functions (e.g. sin, exp, log) applying to adouble objects are overloaded not only to return the result of the operation, but also to store the gradient information in stack Libraries CppAD, SACADO and ADOL-C do this but the result is around 25 times slower than hand-coded adjoints… why? adouble algorithm(const adouble x[2]) { adouble y = 4.0; adouble s = 2.0*x[0] + 3.0*x[1]*x[1]; y *= sin(s); return y; } // Main code Stack stack; // Object where info will be stored adouble x[2] = {…, …} // Set algorithm inputs adouble y = algorithm(x); // Run algorithm and store info in stack y.set_gradient(y_AD); // Set dJ/dy stack.reverse(); // Run adjoint code from stored info x_AD[0] = x[0].get_gradient(); // Save resulting values of dJ/dx0 x_AD[1] = x[1].get_gradient(); // ... and dJ/dx1 Simple change: label “active” variables as a new type

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**Minimum necessary storage**

What is the minimum necessary storage to store these statements? If we label each gradient by an integer (since they’re unknown in forward pass) then we need two stacks that can be added to as the algorithm progresses: Can then run backwards through stack to compute adjoints Statement stack Operation stack Index to LHS gradient Index to first operation 2 (dy) 3 (ds) 2 … # Multiplier Index to RHS gradient 2.0 0 (dx0) 1 6.0x1 1 (dx1) 2 sin(s) 2 (dy) 3 y cos(s) 3 (ds) 4 …

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**Adjoint algorithm is simple**

Need to cope with three different types of differential statement: Reverse mode: Forward mode: Equivalent adjoint statements: General differential statement: for i = 0 to n:

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**…which can be coded as follows**

1. Loop over derivative statements in reverse order 2. Save gradient 3. Skip if gradient equals 0 (big optimization) 4. Loop over operations 5. Update an adjoint This does the right thing in our three cases: Zero on RHS One or more gradients on RHS Same gradient on LHS and RHS

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**“Dual numbers” approach**

How can these stacks be created? Consider what happens when compiler sees this line: Compiler splits this up into two parts with temporary t: We could define adouble as “dual number” [x, dx] (invented by Clifford 1873) and then overload sin and operator*: [sin(s), cos(s)*ds] = sin([s, ds]) [y*t, t*dy+y*dt] = [y, dy] * [t, dt] This would correctly apply but only if the gradient terms on the right-hand-side are known! This is not useful for the reverse-mode (adjoint) when we want to store a symbolic representation of the gradient on the forward sweep which is then filled on the reverse sweep Dual numbers are used in some forward-mode-only (tangent linear) automatic differentiation tools. y = y * sin(s) adouble t = sin(s) y = operator*(y, t)

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**So how do CppAD & ADOL-C work?**

In the forward pass they store the whole algorithm symbolically, not just the derivative form! This means every operator and function needs to be stored symbolically (e.g. 0 for plus, 1 for minus, 42 for atan etc) The stored algorithm can then be analysed to generate an adjoint function This all happens behind the scenes so easy to use, but not surprising that it is 25 times slower than a hand-coded adjoint

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**Computational graphs operator* y sin s**

The basic problem is that standard operator overloading can only pass information from the most nested operation outwards Pass y sin(s) to be new y operator* Pass value of sin(s) y sin s

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**Implementing the chain rule**

Differentiate multiply operator Differentiate sine function

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**Computational graph 2 operator* sin y s**

Clearly differentiation most naturally involves passing information in the opposite sense Each node representing arbitrary function or operator y(a) needs to be able to take a real number w and pass wdy/da down the chain Binary function or operator y(a,b) would pass wdy/da to one argument and wdy/db to other At the end of the chain, store the result on the stack But how do we implement this? operator* sin y s Pass y Pass y cos(s) Pass sin(s) Add sin(s)dy to stack Add y cos(s)ds to stack

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What is a template? Templates are a key ingredient to generic programming in C++ Imagine we have a function like this: We want it to work with any numerical type (single precision, complex numbers etc) but don’t want to laboriously define a new overloaded function for each possible type Can use a function template: double cube(const double x) { double y = x*x*x; return y; } template <typename Type> Type cube(Type x) { Type y = x*x*x; return y; } double a = 1.0; b = cube(a); // compiler creates function cube<double> complex<double> c(1.0, 2.0); // c = 1 + 2i d = cube(c); // compiler creates function cube<complex<double> >

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**What is an expression template?**

C++ also supports class templates Veldhuizen (1995) used this feature to introduce the idea of Expression Templates to optimize array operations and make C++ as fast as Fortran-90 for array-wise operations We use it as a way to pass information in both directions through the expression tree: sin(A) for an argument of arbitrary type A is overloaded to return an object of type Sin<A> operator*(A,B) for arguments of arbitrary type A and B is overloaded to return an object of type Multiply<A,B> Now when we compile the statement “y=y*sin(x)”: The right-hand-side resolves to an object “RHS” of type Multiply<adouble,Sin<adouble> > The overloaded assignment operator first calls RHS.value() to get y It then calls RHS.calc_gradient(), to add entries to operation stack Multiply and Sin are defined with member functions so that they can correctly pass information up and down the expression tree

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**Multiply<adouble,Sin<adouble> >**

New approach operator* sin y s Pass y Pass y cos(s) Pass sin(s) Add sin(s)dy to stack Add y cos(s)ds to stack Each function and operator y(a) implements a function calc_gradient that takes a real number w and passes wdy/da down the chain: The following types are passed up the chain at compile time: Multiply<adouble,Sin<adouble> > operator* adouble Sin<adouble> y sin adouble s

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**Implementation of Sin<A>**

// Definition of Sin class template <class A> class Sin : public Expression<Sin<A> > { public: // Member functions // Constructor: store reference to a and its numerical value Sin(const Expression<A>& a) : a_(a), a_value_(a.value()) { } // Return the value double value() const { return sin(a_value_); } // Compute derivative and pass to a void calc_gradient(Stack& stack, double multiplier) const { a_.calc_gradient(stack, cos(a_value_)*multiplier); } private: // Data members const A& a_; // A reference to the object double a_value_; // The numerical value of object }; // Overload the sin function: it returns a Sin<A> object inline Sin<A> sin(const Expression<A>& a) { return Sin<A>(a); } …Adept library has done this for all operators and functions

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**Optimizations Why are expression templates fast?**

Compound types representing complex expressions are known at compile time C++ automatically inlines function calls between objects in an expression, leaving little more than the operations you would put in a hand-coded application of the chain rule Further optimizations: Stack object keeps memory allocated between calls to avoid time spent allocating incrementally more memory If the Jacobian is computed it is done in strips to exploit vectorization (SSE/SSE2 on Intel) and loop unrolling The current stack is accessed by a global but thread-local variable, rather than storing a link to the stack in every adouble object (as in CppAD and ADOL-C)

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**Testing using lidar multiple scattering models**

Photon Variance-Covariance method for small-angle multiple scattering Hogan (JAS 2008) Somewhat similar to a monochromatic radiance model Four coupled ODEs are integrated forward in space Several variables at N gates give N output signals Computational cost proportional to N Time-dependent two-stream method for wide-angle multiple scattering Hogan and Battaglia (JAS 2008) Similar to a time-dependent 1D advection model Four coupled PDEs are integrated forward in time Several variables at N gates gives N output signals Computational cost proportional to N 2

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**Simulation of 3D photon transport**

Animation of scalar flux (I++I–) Colour scale is logarithmic Represents 5 orders of magnitude Domain properties: 500-m thick 2-km wide Optical depth of 20 No absorption In this simulation the lateral distribution is Gaussian at each height and each time

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**Benchmark results Only 5-20% slower than hand-coded adjoint**

Time relative to original code, gcc-4.4, Pentium 2.5 GHz, 2 MB cache Only 5-20% slower than hand-coded adjoint Adjoint PVC N=50 TDTS N=50 Hand-coded adjoint 3.0 ( ) 3.6 ( ) New C++ library: Adept 3.5 ( ) 3.8 ( ) ADOL-C 25 (18+7) 20 (15+5) CppAD 29 (15+7+7) 34 (17+8+9) 5-9 times faster than leading libraries providing same functionality Full Jacobian (50x350) PVC N=50 TDTS N=50 New C++ library: Adept 20 ADOL-C 83 69 CppAD 352 470 4-20 times faster for 50x350 Jacobian

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Outlook New library Adept (Automatic Differentiation using Expression Templates) produces adjoint with minimum difficulty for user No knowledge of templates required by user at all Simple and efficient to compute Jacobian matrix as well Freely available at Typically 5-20% slower than hand-coded adjoints But immeasurably faster in terms of programmer time Code is complete for applying to any C code with real numbers Further development desirable: Complex numbers Use within C++ matrix/vector libraries, particularly those that already use Expression Templates (like the one I use for the Unified Algorithm) Easily facilitate checkpointing so large codes don’t exhaust memory Automatically compute higher-order derivatives (e.g. Hessian matrix) Potential for student projects to get small data assimilation systems up and running and efficient quickly Impossible to apply in Fortran: no template capability!

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**Minimizing the cost function**

Gradient of cost function (a vector) Gauss-Newton method Rapid convergence (instant for linear problems) Get solution error covariance “for free” at the end Levenberg-Marquardt is a small modification to ensure convergence Need the Jacobian matrix H of every forward model: can be expensive for larger problems as forward model may need to be rerun with each element of the state vector perturbed and 2nd derivative (the Hessian matrix): Gradient Descent methods Fast adjoint method to calculate xJ means don’t need to calculate Jacobian Disadvantage: more iterations needed since we don’t know curvature of J(x) Quasi-Newton method to get the search direction (e.g. L-BFGS used by ECMWF): builds up an approximate inverse Hessian A for improved convergence Scales well for large x Poorer estimate of the error at the end

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**Time-dependent 2-stream approx.**

Describe diffuse flux in terms of outgoing stream I+ and incoming stream I–, and numerically integrate the following coupled PDEs: These can be discretized quite simply in time and space (no implicit methods or matrix inversion required) Time derivative Remove this and we have the time-independent two-stream approximation Source Scattering from the quasi-direct beam into each of the streams Gain by scattering Radiation scattered from the other stream Loss by absorption or scattering Some of lost radiation will enter the other stream Spatial derivative Transport of radiation from upstream Hogan and Battaglia (2008, J. Atmos. Sci.)

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