src/gromacs/math/gausstransform.cpp

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  35 /*! \internal \file
  36  * \brief
  37  * Implements Gaussian function evaluations on lattices and related functionality
  38  *
  39  * \author Christian Blau <blau@kth.se>
  40  *
  41  * \ingroup module_math
  42  */
  43 #include "gmxpre.h"
  44
  45 #include "gausstransform.h"
  46
  47 #include <cmath>
  48
  49 #include <algorithm>
  50 #include <array>
  51
  52 #include "gromacs/math/functions.h"
  53 #include "gromacs/math/multidimarray.h"
  54 #include "gromacs/math/utilities.h"
  55
  56 namespace gmx
  57 {
  58
  59 /********************************************************************
  60  * GaussianOn1DLattice::Impl
  61  */
  62
  63 class GaussianOn1DLattice::Impl
  64 {
  65 public:
  66     Impl(int numGridPointsForSpreadingHalfWidth, real sigma);
  67     ~Impl()                 = default;
  68     Impl(const Impl& other) = default;
  69     Impl& operator=(const Impl& other) = default;
  70
  71     /*! \brief evaluate Gaussian function at all lattice points
  72      * \param[in] amplitude the amplitude of the Gaussian
  73      * \param[in] dx distance from the center
  74      */
  75     void spread(double amplitude, real dx);
  76     //! Largest distance in number of gridpoints from 0
  77     int numGridPointsForSpreadingHalfWidth_;
  78     /*! \brief Avoid overflow for E2^offset and underflow for E3(i).
  79      *
  80      * Occurs when sigma is much smaller than numGridPointsForSpreadingHalfWidth_.
  81      *
  82      * E2^offset smaller than maximum float requires
  83      * \f$exp(dx / (2*square(sigma))^numGridPointsForSpreadingHalfWidth_ \leq max_float \f$
  84      * The maximum expected distance of the Gaussian center to the next lattice point is dx = 0.5,
  85      * thus the maximum spread distance here is \f$4 * sigma^2 * \log(\mathrm{maxfloat})\f$ .
  86      *
  87      * E3(i) larger than minmium float requires
  88      * exp(i^2 / 2*(sigma)^2) > min_float
  89      * Thus the maximum spread distance here is \f$\sigma \sqrt(-2\log(\mathrm{minfloat}))\f$
  90      */
  91     int maxEvaluatedSpreadDistance_;
  92     //! Width of the Gaussian function
  93     double sigma_;
  94     //! The result of the spreading calculation
  95     std::vector<float> spreadingResult_;
  96     //! Pre-calculated exp(-gridIndex^2/2 * (sigma^2)) named as in Greengard2004
  97     std::vector<float> e3_;
  98     /*! \brief Equal to std::floor(std::log(std::numeric_limits<float>::max())).
  99      * Above expression is not constexpr and a const variable would implicitly delete default copy
 100      * assignment. Therefore resorting to setting number manually.
 101      */
 102     static constexpr double c_logMaxFloat = 88.72284;
 103     static constexpr double c_logMinFloat = -87.33654;
 104 };
 105
 106 GaussianOn1DLattice::Impl::Impl(int numGridPointsForSpreadingHalfWidth, real sigma) :
 107     numGridPointsForSpreadingHalfWidth_(numGridPointsForSpreadingHalfWidth),
 108     sigma_(sigma),
 109     spreadingResult_(2 * numGridPointsForSpreadingHalfWidth + 1)
 110 {
 111     maxEvaluatedSpreadDistance_ =
 112             std::min(numGridPointsForSpreadingHalfWidth_,
 113                      static_cast<int>(std::floor(4 * square(sigma) * c_logMaxFloat)) - 1);
 114     maxEvaluatedSpreadDistance_ =
 115             std::min(maxEvaluatedSpreadDistance_,
 116                      static_cast<int>(std::floor(sigma * sqrt(-2.0 * c_logMinFloat))) - 1);
 117
 118     std::generate_n(std::back_inserter(e3_), maxEvaluatedSpreadDistance_ + 1,
 119                     [sigma, latticeIndex = 0]() mutable {
 120                         return std::exp(-0.5 * square(latticeIndex++ / sigma));
 121                     });
 122
 123     std::fill(std::begin(spreadingResult_), std::end(spreadingResult_), 0.);
 124 };
 125
 126 void GaussianOn1DLattice::Impl::spread(double amplitude, real dx)
 127 {
 128     /* The spreading routine implements the fast gaussian gridding as in
 129      *
 130      * Leslie Greengard and June-Yub Lee,
 131      * "Accelerating the Nonuniform Fast Fourier Transform"
 132      * SIAM REV 2004 Vol. 46, No. 3, pp. 443-454 DOI. 10.1137/S003614450343200X
 133      *
 134      * Following the naming conventions for e1, e2 and e3, nu = 1, m = numGridPointsForSpreadingHalfWidth_.
 135      *
 136      * Speed up is achieved by factorization of the exponential that is evaluted
 137      * at regular lattice points i, where the distance from the
 138      * Gaussian center is \f$x-i\f$:
 139      *
 140      * \f[
 141      *      a * \exp(-(x^2-2*i*x+ i^2)/(2*\sigma^2)) =
 142      *      a * \exp(-x^2/2*\sigma^2) * \exp(x/\sigma^2)^i * \exp(i/2*sigma^2) =
 143      *      e_1(x) * e_2(x)^i * e_3(i)
 144      * \f]
 145      *
 146      * Requiring only two exp evaluations per spreading operation.
 147      *
 148      */
 149     const double e1 = amplitude * exp(-0.5 * dx * dx / square(sigma_)) / (sqrt(2 * M_PI) * sigma_);
 150     spreadingResult_[numGridPointsForSpreadingHalfWidth_] = e1;
 151
 152     const double e2 = exp(dx / square(sigma_));
 153
 154     double e2pow = e2; //< powers of e2, e2^offset
 155
 156     // Move outwards from mid-point, using e2pow value for both points simultaneously
 157     //      o    o    o<----O---->o    o    o
 158     for (int offset = 1; offset < maxEvaluatedSpreadDistance_; offset++)
 159     {
 160         const double e1_3                                              = e1 * e3_[offset];
 161         spreadingResult_[numGridPointsForSpreadingHalfWidth_ + offset] = e1_3 * e2pow;
 162         spreadingResult_[numGridPointsForSpreadingHalfWidth_ - offset] = e1_3 / e2pow;
 163         e2pow *= e2;
 164     }
 165     // separate statement for gridpoints at the end of the range avoids
 166     // overflow for large sigma and saves one e2 multiplication operation
 167     spreadingResult_[numGridPointsForSpreadingHalfWidth_ - maxEvaluatedSpreadDistance_] =
 168             (e1 / e2pow) * e3_[maxEvaluatedSpreadDistance_];
 169     spreadingResult_[numGridPointsForSpreadingHalfWidth_ + maxEvaluatedSpreadDistance_] =
 170             (e1 * e2pow) * e3_[maxEvaluatedSpreadDistance_];
 171 }
 172
 173 /********************************************************************
 174  * GaussianOn1DLattice
 175  */
 176
 177 GaussianOn1DLattice::GaussianOn1DLattice(int numGridPointsForSpreadingHalfWidth_, real sigma) :
 178     impl_(new Impl(numGridPointsForSpreadingHalfWidth_, sigma))
 179 {
 180 }
 181
 182 GaussianOn1DLattice::~GaussianOn1DLattice() {}
 183
 184 void GaussianOn1DLattice::spread(double amplitude, real dx)
 185 {
 186     impl_->spread(amplitude, dx);
 187 }
 188
 189 ArrayRef<const float> GaussianOn1DLattice::view()
 190 {
 191     return impl_->spreadingResult_;
 192 }
 193
 194 GaussianOn1DLattice::GaussianOn1DLattice(const GaussianOn1DLattice& other) :
 195     impl_(new Impl(*other.impl_))
 196 {
 197 }
 198
 199 GaussianOn1DLattice& GaussianOn1DLattice::operator=(const GaussianOn1DLattice& other)
 200 {
 201     *impl_ = *other.impl_;
 202     return *this;
 203 }
 204
 205 GaussianOn1DLattice::GaussianOn1DLattice(GaussianOn1DLattice&&) noexcept = default;
 206
 207 GaussianOn1DLattice& GaussianOn1DLattice::operator=(GaussianOn1DLattice&&) noexcept = default;
 208
 209 namespace
 210 {
 211
 212 //! rounds real-valued coordinate to the closest integer values
 213 IVec closestIntegerPoint(const RVec& coordinate)
 214 {
 215     return { roundToInt(coordinate[XX]), roundToInt(coordinate[YY]), roundToInt(coordinate[ZZ]) };
 216 }
 217
 218 /*! \brief Substracts a range from a three-dimensional integer coordinate and ensures
 219  * the resulting coordinate is within a lattice.
 220  * \param[in] index point in lattice
 221  * \param[in] range to be shifted
 222  * \returns Shifted index or zero if shifted index is smaller than zero.
 223  */
 224 IVec rangeBeginWithinLattice(const IVec& index, const IVec& range)
 225 {
 226     return elementWiseMax({ 0, 0, 0 }, index - range);
 227 }
 228
 229 /*! \brief Adds a range from a three-dimensional integer coordinate and ensures
 230  * the resulting coordinate is within a lattice.
 231  * \param[in] index point in lattice
 232  * \param[in] extents extent of the lattice
 233  * \param[in] range to be shifted
 234  * \returns Shifted index or the lattice extent if shifted index is larger than the extent
 235  */
 236 IVec rangeEndWithinLattice(const IVec& index, const dynamicExtents3D& extents, const IVec& range)
 237 {
 238     IVec extentAsIvec(static_cast<int>(extents.extent(ZZ)), static_cast<int>(extents.extent(YY)),
 239                       static_cast<int>(extents.extent(XX)));
 240     return elementWiseMin(extentAsIvec, index + range);
 241 }
 242
 243
 244 } // namespace
 245
 246 /********************************************************************
 247  * OuterProductEvaluator
 248  */
 249
 250 mdspan<const float, dynamic_extent, dynamic_extent> OuterProductEvaluator::
 251                                                     operator()(ArrayRef<const float> x, ArrayRef<const float> y)
 252 {
 253     data_.resize(ssize(x), ssize(y));
 254     for (gmx::index xIndex = 0; xIndex < ssize(x); ++xIndex)
 255     {
 256         const auto xValue = x[xIndex];
 257         std::transform(std::begin(y), std::end(y), begin(data_.asView()[xIndex]),
 258                        [xValue](float yValue) { return xValue * yValue; });
 259     }
 260     return data_.asConstView();
 261 }
 262
 263 /********************************************************************
 264  * IntegerBox
 265  */
 266
 267 IntegerBox::IntegerBox(const IVec& begin, const IVec& end) : begin_{ begin }, end_{ end } {}
 268
 269 const IVec& IntegerBox::begin() const
 270 {
 271     return begin_;
 272 }
 273 const IVec& IntegerBox::end() const
 274 {
 275     return end_;
 276 }
 277
 278 bool IntegerBox::empty() const
 279 {
 280     return !((begin_[XX] < end_[XX]) && (begin_[YY] < end_[YY]) && (begin_[ZZ] < end_[ZZ]));
 281 }
 282
 283 IntegerBox spreadRangeWithinLattice(const IVec& center, dynamicExtents3D extent, IVec range)
 284 {
 285     const IVec begin = rangeBeginWithinLattice(center, range);
 286     const IVec end   = rangeEndWithinLattice(center, extent, range);
 287     return { begin, end };
 288 }
 289 /********************************************************************
 290  * GaussianSpreadKernel
 291  */
 292
 293 IVec GaussianSpreadKernelParameters::Shape::latticeSpreadRange() const
 294 {
 295     DVec range(std::ceil(sigma_[XX] * spreadWidthMultiplesOfSigma_),
 296                std::ceil(sigma_[YY] * spreadWidthMultiplesOfSigma_),
 297                std::ceil(sigma_[ZZ] * spreadWidthMultiplesOfSigma_));
 298     return range.toIVec();
 299 }
 300
 301 /********************************************************************
 302  * GaussTransform3D::Impl
 303  */
 304
 305 /*! \internal \brief
 306  * Private implementation class for GaussTransform3D.
 307  */
 308 class GaussTransform3D::Impl
 309 {
 310 public:
 311     //! Construct from extent and spreading width and range
 312     Impl(const dynamicExtents3D& extent, const GaussianSpreadKernelParameters::Shape& kernelShapeParameters);
 313     ~Impl() = default;
 314     //! Copy constructor
 315     Impl(const Impl& other) = default;
 316     //! Copy assignment
 317     Impl& operator=(const Impl& other) = default;
 318     //! Add another gaussian
 319     void add(const GaussianSpreadKernelParameters::PositionAndAmplitude& localParamters);
 320     //! The width of the Gaussian in lattice spacing units
 321     BasicVector<double> sigma_;
 322     //! The spread range in lattice points
 323     IVec spreadRange_;
 324     //! The result of the Gauss transform
 325     MultiDimArray<std::vector<float>, dynamicExtents3D> data_;
 326     //! The outer product of a Gaussian along the z and y dimension
 327     OuterProductEvaluator outerProductZY_;
 328     //! The three one-dimensional Gaussians, whose outer product is added to the Gauss transform
 329     std::array<GaussianOn1DLattice, DIM> gauss1d_;
 330 };
 331
 332 GaussTransform3D::Impl::Impl(const dynamicExtents3D&                      extent,
 333                              const GaussianSpreadKernelParameters::Shape& kernelShapeParameters) :
 334     sigma_{ kernelShapeParameters.sigma_ },
 335     spreadRange_{ kernelShapeParameters.latticeSpreadRange() },
 336     data_{ extent },
 337     gauss1d_({ GaussianOn1DLattice(spreadRange_[XX], sigma_[XX]),
 338                GaussianOn1DLattice(spreadRange_[YY], sigma_[YY]),
 339                GaussianOn1DLattice(spreadRange_[ZZ], sigma_[ZZ]) })
 340 {
 341 }
 342
 343 void GaussTransform3D::Impl::add(const GaussianSpreadKernelParameters::PositionAndAmplitude& localParameters)
 344 {
 345     const IVec closestLatticePoint = closestIntegerPoint(localParameters.coordinate_);
 346     const auto spreadRange =
 347             spreadRangeWithinLattice(closestLatticePoint, data_.asView().extents(), spreadRange_);
 348
 349     // do nothing if the added Gaussian will never reach the lattice
 350     if (spreadRange.empty())
 351     {
 352         return;
 353     }
 354
 355     for (int dimension = XX; dimension <= ZZ; ++dimension)
 356     {
 357         // multiply with amplitude so that Gauss3D = (amplitude * Gauss_x) * Gauss_y * Gauss_z
 358         const float gauss1DAmplitude = dimension > XX ? 1.0 : localParameters.amplitude_;
 359         gauss1d_[dimension].spread(gauss1DAmplitude, localParameters.coordinate_[dimension]
 360                                                              - closestLatticePoint[dimension]);
 361     }
 362
 363     const auto spreadZY         = outerProductZY_(gauss1d_[ZZ].view(), gauss1d_[YY].view());
 364     const auto spreadX          = gauss1d_[XX].view();
 365     const IVec spreadGridOffset = spreadRange_ - closestLatticePoint;
 366
 367     // \todo optimize these loops if performance critical
 368     // The looping strategy uses that the last, x-dimension is contiguous in the memory layout
 369     for (int zLatticeIndex = spreadRange.begin()[ZZ]; zLatticeIndex < spreadRange.end()[ZZ]; ++zLatticeIndex)
 370     {
 371         const auto zSlice = data_.asView()[zLatticeIndex];
 372
 373         for (int yLatticeIndex = spreadRange.begin()[YY]; yLatticeIndex < spreadRange.end()[YY]; ++yLatticeIndex)
 374         {
 375             const auto  ySlice      = zSlice[yLatticeIndex];
 376             const float zyPrefactor = spreadZY(zLatticeIndex + spreadGridOffset[ZZ],
 377                                                yLatticeIndex + spreadGridOffset[YY]);
 378
 379             for (int xLatticeIndex = spreadRange.begin()[XX]; xLatticeIndex < spreadRange.end()[XX];
 380                  ++xLatticeIndex)
 381             {
 382                 const float xPrefactor = spreadX[xLatticeIndex + spreadGridOffset[XX]];
 383                 ySlice[xLatticeIndex] += zyPrefactor * xPrefactor;
 384             }
 385         }
 386     }
 387 }
 388
 389 /********************************************************************
 390  * GaussTransform3D
 391  */
 392
 393 GaussTransform3D::GaussTransform3D(const dynamicExtents3D&                      extent,
 394                                    const GaussianSpreadKernelParameters::Shape& kernelShapeParameters) :
 395     impl_(new Impl(extent, kernelShapeParameters))
 396 {
 397 }
 398
 399 void GaussTransform3D::add(const GaussianSpreadKernelParameters::PositionAndAmplitude& localParameters)
 400 {
 401     impl_->add(localParameters);
 402 }
 403
 404 void GaussTransform3D::setZero()
 405 {
 406     std::fill(begin(impl_->data_), end(impl_->data_), 0.);
 407 }
 408
 409 basic_mdspan<float, dynamicExtents3D> GaussTransform3D::view()
 410 {
 411     return impl_->data_.asView();
 412 }
 413
 414 basic_mdspan<const float, dynamicExtents3D> GaussTransform3D::constView() const
 415 {
 416     return impl_->data_.asConstView();
 417 }
 418
 419 GaussTransform3D::~GaussTransform3D() {}
 420
 421 GaussTransform3D::GaussTransform3D(const GaussTransform3D& other) : impl_(new Impl(*other.impl_)) {}
 422
 423 GaussTransform3D& GaussTransform3D::operator=(const GaussTransform3D& other)
 424 {
 425     *impl_ = *other.impl_;
 426     return *this;
 427 }
 428
 429 GaussTransform3D::GaussTransform3D(GaussTransform3D&&) noexcept = default;
 430
 431 GaussTransform3D& GaussTransform3D::operator=(GaussTransform3D&&) noexcept = default;
 432
 433 } // namespace gmx