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## 2. Calculator Interface

This document describes the layer between the form factor kernels and the model calculator which implements the dispersity and magnetic SLD calculations. There are three separate implementations of this layer, `kernelcl` for OpenCL, which operates on a single Q value at a time, `kerneldll` for the DLL, which loops over a vector of Q values, and `kernelpy` for python models which operates on vector Q values.

Each implementation provides three different calls Iq, Iqxy and Imagnetic for 1-D, 2-D and 2-D magnetic kernels respectively. The C code is defined in kernel_iq.c, with the minor differences between OpenCL and DLL handled by #ifdef statements.

The kernel call looks as follows:

    kernel void KERNEL_NAME(
        int nq,                  // Number of q values in the q vector
        int pd_start,            // Starting position in the dispersity loop
        int pd_stop,             // Ending position in the dispersity loop
        ProblemDetails *details, // dispersity info
        double *values,          // Value and weights vector
        double *q,               // q or (qx,qy) vector
        double *result,          // returned I(q), with result[nq] = pd_weight
        double cutoff)           // dispersity weight cutoff

The details for OpenCL and the python loop are slightly different, but these data structures are common.

nq indicates the number of q values that will be calculated.

The pd_start and pd_stop parameters set the range of the dispersity loop to compute for the current kernel call. Give a dispersity calculation with 30 weights for length and 30 weights for radius for example, there are a total of 900 calls to the form factor required to compute the kernel. These might be done in chunks of 100, so the first call would start at zero and stop after 100 iterations. The second call would then have to set the length index to 3 and the radius index to 10 for a position of 3*30+10=100, and could then proceed to position 200. This allows us to interrupt the calculation in the middle of a long dispersity loop without having to do special tricks with the C code. More importantly, it stops the OpenCL kernel in a reasonable time; because the GPU is used by the operating system to show its windows, if a GPU kernel runs too long then it will be automatically killed and no results will be returned to the caller.

The ProblemDetails structure is a direct map of the `details.CallDetails` buffer. This indicates which parameters have dispersity, and where in the values vector the values and weights can be found. For each parameter with dispersity there is a parameter id, the length of the dispersity loop for that parameter, the offset of the parameter values in the pd value and pd weight vectors and the ‘stride’ from one index to the next, which is used to translate between the position in the dispersity loop and the particular parameter indices. The num_eval field is the total size of the dispersity loop. num_weights is the number of elements in the pd value and pd weight vectors. num_active is the number of non-trivial pd loops (parameters with dispersity should be ordered by decreasing pd vector length, with a length of 1 meaning no dispersity). Oriented objects in 2-D need a cos(theta) spherical correction on the angular variation in order to preserve the ‘surface area’ of the weight distribution. theta_par is the id of the polar coordinate parameter if there is one.

The values vector consists of the fixed values for the model plus pd value and pd weight vectors. There are NUM_VALUES fixed values for the model, which includes the initial two slots for scale and background, the NUM_PARS values for the kernel parameters, three slots for the applied magnetism, and three slots for each of the SLD parameters for the sample magnetiziation (Mx, My, Mz). Sample magnetization is translated from (M, theta, phi) to (Mx, My, Mz) before the kernel is called. After the fixed values comes the pd value vector, with the dispersity values for each parameter stacked one after the other. The order isn’t important since the location for each parameter is stored in the pd_offset field of the ProblemDetails structure, but the values do need to be contiguous. After num_weights values, the pd weight vector is stored, with the same configuration as the pd value vector. Note that the pd vectors can contain values that are not in the dispersity loop; this is used by `mixture.MixtureKernel` to make it easier to call the various component kernels.

The q vector contains one value for each q for Iq kernels, or a pair of (qx,qy) values for each q for Iqxy and Imagnetic kernels. OpenCL pads all vectors to 32 value boundaries just to be safe, including the values vector and the the results vector.

The results vector contains one slot for each of the nq values, plus one extra slot at the end for the weight normalization accumulated across all points in the dispersity mesh. This is required when the dispersity loop is broken across several kernel calls.

cutoff is a importance cutoff so that points which contribute negligibly to the total scattering can be skipped without calculating them.

`generate.make_source()` defines the following C macros:

- USE_OPENCL is defined if running in opencl

- MAX_PD is the maximum depth of the dispersity loop [model specific]

- NUM_PARS is the number of parameter values in the kernel. This may be more than the number of parameters if some of the parameters are vector values.

- NUM_VALUES is the number of fixed values, which defines the offset in the value list to the dispersity value and weight vectors.

- NUM_MAGNETIC is the number of magnetic SLDs

- MAGNETIC_PARS is a comma separated list of the magnetic SLDs, indicating their locations in the values vector.

- KERNEL_NAME is the name of the function being declared

- PARAMETER_TABLE is the declaration of the parameters to the kernel:

  > Cylinder:

  >     #define PARAMETER_TABLE \
  >     double length; \
  >     double radius; \
  >     double sld; \
  >     double sld_solvent;

  > Note: scale and background are never included

  > Multi-shell cylinder (10 shell max):

  >     #define PARAMETER_TABLE \
  >     double num_shells; \
  >     double length; \
  >     double radius[10]; \
  >     double sld[10]; \
  >     double sld_solvent;

- CALL_IQ(q, i, var) is the declaration of a call to the kernel:

  > Cylinder:

  >     #define CALL_IQ(q, i, var) Iq(q[i], \
  >     var.length, \
  >     var.radius, \
  >     var.sld, \
  >     var.sld_solvent)

  > Multi-shell cylinder:

  >     #define CALL_IQ(q, i, var) Iq(q[i], \
  >     var.num_shells, \
  >     var.length, \
  >     var.radius, \
  >     var.sld, \
  >     var.sld_solvent)

  > Cylinder2D:

  >     #define CALL_IQ(q, i, var) Iqxy(qa, qc, \
  >     var.length, \
  >     var.radius, \
  >     var.sld, \
  >     var.sld_solvent)

- CALL_VOLUME(var) is similar, but for calling the form volume:

      #define CALL_VOLUME(var) \
      form_volume(var.length, var.radius)

There is an additional macro that can be defined within the model.c file:

- INVALID(var) is a test for model parameters in the correct range:

  > Cylinder:

  >     #define INVALID(var) 0

  > BarBell:

  >     #define INVALID(var) (var.bell_radius < var.radius)

  > Model with complicated constraints:

  >     inline bool constrained(p1, p2, p3) { return expression; }
  >     #define INVALID(var) constrained(var.p1, var.p2, var.p3)

Our design supports a limited number of dispersity loops, wherein we need to cycle through the values of the dispersity, calculate the I(q, p) for each combination of parameters, and perform a normalized weighted sum across all the weights. Parameters may be passed to the underlying calculation engine as scalars or vectors, but the dispersity calculator treats the parameter set as one long vector.

Let’s assume we have 8 parameters in the model, two of which allow dispersity. Since this is a 1-D model the orientation parameters won’t be used:

    0: scale        {scl = constant}
    1: background   {bkg = constant}
    2: radius       {r = vector of 10pts}
    3: length       {l = vector of 30pts}
    4: sld          {s1 = constant/(radius**2*length)}
    5: sld_solvent  {s2 = constant}
    6: theta        {not used}
    7: phi          {not used}

This generates the following call to the kernel. Note that parameters 4 and 5 are treated as having dispersity even though they don’t — this is because it is harmless to do so and simplifies the looping code:

    MAX_PD = 4
    NUM_PARS = 6          // kernel parameters only
    NUM_VALUES = 17       // all values, including scale and background
    NUM_MAGNETIC = 2      // two parameters might be magnetic
    MAGNETIC_PARS = 4, 5  // they are sld and sld_solvent

    details {
        pd_par = {3, 2, 4, 5}         // parameters *radius* and *length* vary
        pd_length = {30, 10, 1, 1}    // *length* has more, so it is first
        pd_offset = {10, 0, 31, 32}   // *length* starts at index 10 in weights
        pd_stride = {1, 30, 300, 300} // cumulative product of pd length
        num_eval = 300   // 300 values in the dispersity loop
        num_weights = 42 // 42 values in the pd vector
        num_active = 2   // only the first two pd are active
        theta_var =  6   // spherical correction
    }

    values = { scl, bkg,                                  // universal
               r, l, s1, s2, theta, phi,                  // kernel pars
               in spin, out spin, spin angle,             // applied magnetism
               mx s1, my s1, mz s1, mx s2, my s2, mz s2,  // magnetic slds
               r0, .., r9, l0, .., l29, s, s2,            // pd values
               r0, .., r9, l0, .., l29, s, s2}            // pd weights

    nq = 130
    q = { q0, q1, ..., q130, x, x }  # pad to 8 element boundary
    result = {r1, ..., r130, pd_norm, x }

The dispersity parameters are stored in as an array of parameter indices, one for each parameter, stored in pd_par[n]. Parameters which do not support dispersity do not appear in this array. Each dispersity parameter has a weight vector whose length is stored in pd_length[n]. The weights are stored in a contiguous vector of weights for all parameters, with the starting position for the each parameter stored in pd_offset[n]. The values corresponding to the weights are stored together in a separate weights[] vector, with offset stored in par_offset[pd_par[n]]. Dispersity parameters should be stored in decreasing order of length for highest efficiency.

We limit the number of dispersity dimensions to MAX_PD (currently 4), though some models may have fewer if they have fewer dispersity parameters. The main reason for the limit is to reduce code size. Each additional dispersity parameter requires a separate dispersity loop. If more than 4 levels of dispersity are needed, then we need to switch to a monte carlo importance sampling algorithm with better performance for high-dimensional integrals.

Constraints between parameters are not supported. Instead users will have to define a new model with the constraints built in by making a copy of the existing model. Mac provides OpenCL and we are supplying the tinycc compiler for Windows so this won’t be a complete limitation, but it is rather inconvenient. The process could perhaps be automated so that there is no code copying required, just an alternate CALL_IQ macro that implements the constraints. Think carefully about constraining theta since the polar coordinates normalization is tied to this parameter.

If there is no dispersity we pretend that we have a disperisty mesh over a single parameter with a single point in the distribution, giving pd_start=0 and pd_stop=1.

The problem details structure could be allocated and sent in as an integer array using the read-only flag. This would allow us to copy it once per fit along with the weights vector, since features such as the number of disperity points per pd parameter won’t change between function evaluations. A new parameter vector must be sent for each I(q) evaluation. This is not currently implemented, and would require some resturcturing of the `sasview_model.SasviewModel` interface.

The results array will be initialized to zero for dispersity loop entry zero, and preserved between calls to [start, stop] so that the results accumulate by the time the loop has completed. Background and scale will be applied when the loop reaches the end. This does require that the results array be allocated read-write, which is less efficient for the GPU, but it makes the calling sequence much more manageable.

For accuracy we may want to introduce Kahan summation into the integration:

    double accumulated_error = 0.0;
    ...
    #if USE_KAHAN_SUMMATION
        const double y = next - accumulated_error;
        const double t = ret + y;
        accumulated_error = (t - ret) - y;
        ret = t;
    #else
        ret += next;
    #endif

This will require accumulated error for each I(q) value to be preserved between kernel calls to implement this fully. The kernel_iq.c code, which loops over q for each parameter set in the dispersity loop, will also need the accumulation vector.