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 assignment.
 100          * 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_ = std::min(numGridPointsForSpreadingHalfWidth_, static_cast<int>(std::floor(4 * square(sigma) * c_logMaxFloat)) - 1);
 112     maxEvaluatedSpreadDistance_ = std::min(maxEvaluatedSpreadDistance_, static_cast<int>(std::floor(sigma * sqrt(-2.0 * c_logMinFloat))) - 1);
 113
 114     std::generate_n(std::back_inserter(e3_), maxEvaluatedSpreadDistance_ + 1,
 115                     [sigma, latticeIndex = 0]() mutable {
 116                         return std::exp(-0.5 * square(latticeIndex++ / sigma));
 117                     });
 118
 119     std::fill(std::begin(spreadingResult_), std::end(spreadingResult_), 0.);
 120 };
 121
 122 void GaussianOn1DLattice::Impl::spread(double amplitude, real dx)
 123 {
 124     /* The spreading routine implements the fast gaussian gridding as in
 125      *
 126      * Leslie Greengard and June-Yub Lee,
 127      * "Accelerating the Nonuniform Fast Fourier Transform"
 128      * SIAM REV 2004 Vol. 46, No. 3, pp. 443-454 DOI. 10.1137/S003614450343200X
 129      *
 130      * Following the naming conventions for e1, e2 and e3, nu = 1, m = numGridPointsForSpreadingHalfWidth_.
 131      *
 132      * Speed up is achieved by factorization of the exponential that is evaluted
 133      * at regular lattice points i, where the distance from the
 134      * Gaussian center is \f$x-i\f$:
 135      *
 136      * \f[
 137      *      a * \exp(-(x^2-2*i*x+ i^2)/(2*\sigma^2)) =
 138      *      a * \exp(-x^2/2*\sigma^2) * \exp(x/\sigma^2)^i * \exp(i/2*sigma^2) =
 139      *      e_1(x) * e_2(x)^i * e_3(i)
 140      * \f]
 141      *
 142      * Requiring only two exp evaluations per spreading operation.
 143      *
 144      */
 145     const double e1 = amplitude * exp(-0.5 * dx * dx / square(sigma_)) / (sqrt(2 * M_PI) * sigma_);
 146     spreadingResult_[numGridPointsForSpreadingHalfWidth_] = e1;
 147
 148     const double e2 = exp(dx / square(sigma_));
 149
 150     double       e2pow = e2; //< powers of e2, e2^offset
 151
 152     // Move outwards from mid-point, using e2pow value for both points simultaneously
 153     //      o    o    o<----O---->o    o    o
 154     for (int offset = 1; offset < maxEvaluatedSpreadDistance_; offset++)
 155     {
 156         const double e1_3 = e1 * e3_[offset];
 157         spreadingResult_[numGridPointsForSpreadingHalfWidth_ + offset] = e1_3 * e2pow;
 158         spreadingResult_[numGridPointsForSpreadingHalfWidth_ - offset] = e1_3 / e2pow;
 159         e2pow *= e2;
 160     }
 161     // separate statement for gridpoints at the end of the range avoids
 162     // overflow for large sigma and saves one e2 multiplication operation
 163     spreadingResult_[numGridPointsForSpreadingHalfWidth_ - maxEvaluatedSpreadDistance_] = (e1 / e2pow) * e3_[maxEvaluatedSpreadDistance_];
 164     spreadingResult_[numGridPointsForSpreadingHalfWidth_ + maxEvaluatedSpreadDistance_] = (e1 * e2pow) * e3_[maxEvaluatedSpreadDistance_];
 165 }
 166
 167 /********************************************************************
 168  * GaussianOn1DLattice
 169  */
 170
 171 GaussianOn1DLattice::GaussianOn1DLattice(int numGridPointsForSpreadingHalfWidth_, real sigma) : impl_(new Impl(numGridPointsForSpreadingHalfWidth_, sigma))
 172 {
 173 }
 174
 175 GaussianOn1DLattice::~GaussianOn1DLattice () {}
 176
 177 void GaussianOn1DLattice::spread(double amplitude, real dx)
 178 {
 179     impl_->spread(amplitude, dx);
 180 }
 181
 182 ArrayRef<const float> GaussianOn1DLattice::view()
 183 {
 184     return impl_->spreadingResult_;
 185 }
 186
 187 GaussianOn1DLattice::GaussianOn1DLattice(const GaussianOn1DLattice &other)
 188     : impl_(new Impl(*other.impl_))
 189 {
 190 }
 191
 192 GaussianOn1DLattice &GaussianOn1DLattice::operator=(const GaussianOn1DLattice &other)
 193 {
 194     *impl_ = *other.impl_;
 195     return *this;
 196 }
 197
 198 GaussianOn1DLattice::GaussianOn1DLattice(GaussianOn1DLattice &&) noexcept = default;
 199
 200 GaussianOn1DLattice &GaussianOn1DLattice::operator=(GaussianOn1DLattice &&) noexcept = default;
 201
 202 namespace
 203 {
 204
 205 //! rounds real-valued coordinate to the closest integer values
 206 IVec closestIntegerPoint(const RVec &coordinate)
 207 {
 208     return {
 209                roundToInt(coordinate[XX]),
 210                roundToInt(coordinate[YY]),
 211                roundToInt(coordinate[ZZ])
 212     };
 213 }
 214
 215 /*! \brief Substracts a range from a three-dimensional integer coordinate and ensures
 216  * the resulting coordinate is within a lattice.
 217  * \param[in] index point in lattice
 218  * \param[in] range to be shifted
 219  * \returns Shifted index or zero if shifted index is smaller than zero.
 220  */
 221 IVec rangeBeginWithinLattice(const IVec &index, const IVec &range)
 222 {
 223     return elementWiseMax({0, 0, 0}, index - range);
 224 }
 225
 226 /*! \brief Adds a range from a three-dimensional integer coordinate and ensures
 227  * the resulting coordinate is within a lattice.
 228  * \param[in] index point in lattice
 229  * \param[in] extents extent of the lattice
 230  * \param[in] range to be shifted
 231  * \returns Shifted index or the lattice extent if shifted index is larger than the extent
 232  */
 233 IVec rangeEndWithinLattice(const IVec &index, const dynamicExtents3D &extents, const IVec &range)
 234 {
 235     IVec extentAsIvec(static_cast<int>(extents.extent(ZZ)), static_cast<int>(extents.extent(YY)), static_cast<int>(extents.extent(XX)));
 236     return elementWiseMin(extentAsIvec, index + range);
 237 }
 238
 239
 240 }   // namespace
 241
 242 /********************************************************************
 243  * OuterProductEvaluator
 244  */
 245
 246 mdspan<const float, dynamic_extent, dynamic_extent>
 247 OuterProductEvaluator::operator()(ArrayRef<const float> x, ArrayRef<const float> y)
 248 {
 249     data_.resize(ssize(x), ssize(y));
 250     for (gmx::index xIndex = 0; xIndex < ssize(x); ++xIndex)
 251     {
 252         const auto xValue = x[xIndex];
 253         std::transform(std::begin(y), std::end(y), begin(data_.asView()[xIndex]),
 254                        [xValue](float yValue) { return xValue * yValue; });
 255     }
 256     return data_.asConstView();
 257 }
 258
 259 /********************************************************************
 260  * IntegerBox
 261  */
 262
 263 IntegerBox::IntegerBox(const IVec &begin, const IVec &end) : begin_ {begin}, end_ {
 264     end
 265 }
 266 {}
 267
 268 const IVec &IntegerBox::begin() const{return begin_; }
 269 const IVec &IntegerBox::end() const { return end_; }
 270
 271 bool IntegerBox::empty() const { return !((begin_[XX] < end_[XX] ) && (begin_[YY] < end_[YY]) && (begin_[ZZ] < end_[ZZ])); }
 272
 273 IntegerBox spreadRangeWithinLattice(const IVec &center, dynamicExtents3D extent, IVec range)
 274 {
 275     const IVec begin = rangeBeginWithinLattice(center, range);
 276     const IVec end   = rangeEndWithinLattice(center, extent, range);
 277     return {
 278                begin, end
 279     };
 280 }
 281 /********************************************************************
 282  * GaussianSpreadKernel
 283  */
 284
 285 IVec GaussianSpreadKernelParameters::Shape::latticeSpreadRange() const
 286 {
 287     DVec range(std::ceil(sigma_[XX] * spreadWidthMultiplesOfSigma_), std::ceil(sigma_[YY] * spreadWidthMultiplesOfSigma_), std::ceil(sigma_[ZZ] * spreadWidthMultiplesOfSigma_));
 288     return range.toIVec();
 289 }
 290
 291 /********************************************************************
 292  * GaussTransform3D::Impl
 293  */
 294
 295 /*! \internal \brief
 296  * Private implementation class for GaussTransform3D.
 297  */
 298 class GaussTransform3D::Impl
 299 {
 300     public:
 301         //! Construct from extent and spreading width and range
 302         Impl(const dynamicExtents3D                       &extent,
 303              const GaussianSpreadKernelParameters::Shape  &kernelShapeParameters);
 304         ~Impl()                            = default;
 305         //! Copy constructor
 306         Impl(const Impl &other)            = default;
 307         //! Copy assignment
 308         Impl &operator=(const Impl &other) = default;
 309         //! Add another gaussian
 310         void add(const GaussianSpreadKernelParameters::PositionAndAmplitude &localParamters );
 311         //! The width of the Gaussian in lattice spacing units
 312         BasicVector<double>   sigma_;
 313         //! The spread range in lattice points
 314         IVec                  spreadRange_;
 315         //! The result of the Gauss transform
 316         MultiDimArray<std::vector<float>, dynamicExtents3D> data_;
 317         //! The outer product of a Gaussian along the z and y dimension
 318         OuterProductEvaluator                               outerProductZY_;
 319         //! The three one-dimensional Gaussians, whose outer product is added to the Gauss transform
 320         std::array<GaussianOn1DLattice, DIM>                gauss1d_;
 321 };
 322
 323 GaussTransform3D::Impl::Impl(const dynamicExtents3D                       &extent,
 324                              const GaussianSpreadKernelParameters::Shape  &kernelShapeParameters)
 325     : sigma_ {kernelShapeParameters.sigma_ },
 326 spreadRange_ {
 327     kernelShapeParameters.latticeSpreadRange()
 328 },
 329 data_ {
 330     extent
 331 },
 332 gauss1d_( {GaussianOn1DLattice(spreadRange_[XX], sigma_[XX]),
 333            GaussianOn1DLattice(spreadRange_[YY], sigma_[YY]),
 334            GaussianOn1DLattice(spreadRange_[ZZ], sigma_[ZZ]) } )
 335 {
 336 }
 337
 338 void GaussTransform3D::Impl::add(const GaussianSpreadKernelParameters::PositionAndAmplitude &localParameters)
 339 {
 340     const IVec closestLatticePoint = closestIntegerPoint(localParameters.coordinate_);
 341     const auto spreadRange         = spreadRangeWithinLattice(closestLatticePoint, data_.asView().extents(), spreadRange_);
 342
 343     // do nothing if the added Gaussian will never reach the lattice
 344     if (spreadRange.empty())
 345     {
 346         return;
 347     }
 348
 349     for (int dimension = XX; dimension <= ZZ; ++dimension)
 350     {
 351         // multiply with amplitude so that Gauss3D = (amplitude * Gauss_x) * Gauss_y * Gauss_z
 352         const float gauss1DAmplitude = dimension > XX ? 1.0 : localParameters.amplitude_;
 353         gauss1d_[dimension].spread(gauss1DAmplitude, localParameters.coordinate_[dimension] - closestLatticePoint[dimension]);
 354     }
 355
 356     const auto spreadZY         = outerProductZY_(gauss1d_[ZZ].view(), gauss1d_[YY].view());
 357     const auto spreadX          = gauss1d_[XX].view();
 358     const IVec spreadGridOffset = spreadRange_ - closestLatticePoint;
 359
 360     // \todo optimize these loops if performance critical
 361     // The looping strategy uses that the last, x-dimension is contiguous in the memory layout
 362     for (int zLatticeIndex = spreadRange.begin()[ZZ]; zLatticeIndex < spreadRange.end()[ZZ]; ++zLatticeIndex)
 363     {
 364         const auto zSlice = data_.asView()[zLatticeIndex];
 365
 366         for (int yLatticeIndex = spreadRange.begin()[YY]; yLatticeIndex < spreadRange.end()[YY]; ++yLatticeIndex)
 367         {
 368             const auto  ySlice      = zSlice[yLatticeIndex];
 369             const float zyPrefactor = spreadZY(zLatticeIndex + spreadGridOffset[ZZ], yLatticeIndex + spreadGridOffset[YY]);
 370
 371             for (int xLatticeIndex = spreadRange.begin()[XX]; xLatticeIndex < spreadRange.end()[XX]; ++xLatticeIndex)
 372             {
 373                 const float xPrefactor = spreadX[xLatticeIndex + spreadGridOffset[XX]];
 374                 ySlice[xLatticeIndex] += zyPrefactor * xPrefactor;
 375             }
 376         }
 377     }
 378 }
 379
 380 /********************************************************************
 381  * GaussTransform3D
 382  */
 383
 384 GaussTransform3D::GaussTransform3D(const dynamicExtents3D                       &extent,
 385                                    const GaussianSpreadKernelParameters::Shape  &kernelShapeParameters) : impl_(new Impl(extent, kernelShapeParameters))
 386 {
 387 }
 388
 389 void GaussTransform3D::add(const GaussianSpreadKernelParameters::PositionAndAmplitude &localParameters)
 390 {
 391     impl_->add(localParameters);
 392 }
 393
 394 void GaussTransform3D::setZero()
 395 {
 396     std::fill(begin(impl_->data_), end(impl_->data_), 0.);
 397 }
 398
 399 basic_mdspan<const float, dynamicExtents3D> GaussTransform3D::view()
 400 {
 401     return impl_->data_.asConstView();
 402 }
 403
 404 GaussTransform3D::~GaussTransform3D()
 405 { }
 406
 407 GaussTransform3D::GaussTransform3D(const GaussTransform3D &other)
 408     : impl_(new Impl(*other.impl_))
 409 {
 410 }
 411
 412 GaussTransform3D &GaussTransform3D::operator=(const GaussTransform3D &other)
 413 {
 414     *impl_ = *other.impl_;
 415     return *this;
 416 }
 417
 418 GaussTransform3D::GaussTransform3D(GaussTransform3D &&) noexcept = default;
 419
 420 GaussTransform3D &GaussTransform3D::operator=(GaussTransform3D &&) noexcept = default;
 421
 422 } // namespace gmx