Line data Source code
1 : /* +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
2 : Copyright (c) 2016-2021 The VES code team
3 : (see the PEOPLE-VES file at the root of this folder for a list of names)
4 :
5 : See http://www.ves-code.org for more information.
6 :
7 : This file is part of VES code module.
8 :
9 : The VES code module is free software: you can redistribute it and/or modify
10 : it under the terms of the GNU Lesser General Public License as published by
11 : the Free Software Foundation, either version 3 of the License, or
12 : (at your option) any later version.
13 :
14 : The VES code module is distributed in the hope that it will be useful,
15 : but WITHOUT ANY WARRANTY; without even the implied warranty of
16 : MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
17 : GNU Lesser General Public License for more details.
18 :
19 : You should have received a copy of the GNU Lesser General Public License
20 : along with the VES code module. If not, see <http://www.gnu.org/licenses/>.
21 : +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ */
22 :
23 :
24 : #include "BasisFunctions.h"
25 : #include "GridLinearInterpolation.h"
26 : #include "tools/Grid.h"
27 : #include "VesTools.h"
28 : #include "WaveletGrid.h"
29 : #include "core/ActionRegister.h"
30 : #include "tools/Exception.h"
31 : #include "core/PlumedMain.h"
32 :
33 :
34 : namespace PLMD {
35 : namespace ves {
36 :
37 :
38 : //+PLUMEDOC VES_BASISF BF_WAVELETS
39 : /*
40 : Daubechies Wavelets basis functions.
41 :
42 : Note: at the moment only bases with a single level of scaling functions are usable, as multiscale optimization is not yet implemented.
43 :
44 : This basis set uses Daubechies Wavelets \cite daubechies_ten_1992 to construct a complete and orthogonal basis. See \cite ValssonPampel_Wavelets_2022 for full details.
45 :
46 : The basis set is based on using a pair of functions, the scaling function (or father wavelet) \f$\phi\f$ and the wavelet function (or mother wavelet) \f$\psi\f$.
47 : They are defined via the two-scale relations for scale \f$j\f$ and shift \f$k\f$:
48 :
49 : \f{align*}{
50 : \phi_k^j \left(x\right) = 2^{-j/2} \phi \left( 2^{-j} x - k\right)\\
51 : \psi_k^j \left(x\right) = 2^{-j/2} \psi \left( 2^{-j} x - k\right)
52 : \f}
53 :
54 : The exact properties are set by choosing filter coefficients, e.g. choosing \f$h_k\f$ for the father wavelet:
55 :
56 : \f[
57 : \phi\left(x\right) = \sqrt{2} \sum_k h_k\, \phi \left( 2 x - k\right)
58 : \f]
59 :
60 : The filter coefficients by Daubechies result in an orthonormal basis of all integer shifted functions:
61 : \f[
62 : \int \phi(x+i) \phi(x+j) \mathop{}\!\mathrm{d}x = \delta_{ij} \quad \text{for} \quad i,j \in \mathbb{Z}
63 : \f]
64 :
65 : Because no analytic formula for these wavelets exist, they are instead constructed iteratively on a grid.
66 : The method of construction is close to the "Vector cascade algorithm" described in \cite strang_wavelets_1997 .
67 : The needed filter coefficients of the scaling function are hardcoded, and were previously generated via a python script.
68 : Currently the "maximum phase" type (Db) and the "least asymmetric" (Sym) type are implemented.
69 : We recommend to use Symlets.
70 :
71 : As an example two adjacent basis functions of both Sym8 (ORDER=8, TYPE=SYMLET) and Db8 (ORDER=8, TYPE=DAUBECHIES) is shown in the figure.
72 : The full basis consists of shifted wavelets in the full specified interval.
73 :
74 : \image html ves_basisf-wavelets.png
75 :
76 :
77 : \par Specify the wavelet type
78 :
79 : The TYPE keyword sets the type of Wavelet, at the moment "DAUBECHIES" and "SYMLETS" are available.
80 : The specified ORDER of the basis corresponds to the number of vanishing moments of the wavelet, i.e. if TYPE was specified as "DAUBECHIES" an order of 8 results in Db8 wavelets.
81 :
82 :
83 : \par Specify the number of functions
84 :
85 : The resulting basis set consists of integer shifts of the wavelet with some scaling \f$j\f$,
86 : \f[
87 : V(x) = \sum_i \alpha_i * \phi_i (x) = \sum_i \alpha_i * \phi(\frac{x+i}{j})
88 : \f]
89 : with the variational parameters \f$ \alpha \f$.
90 : Additionally a constant basis function is included.
91 :
92 : There are two different ways to specify the number of used basis functions implemented.
93 : You can either specify the scale or alternatively a fixed number of basis function.
94 :
95 : Coming from the multiresolution aspect of wavelets, you can set the scale of the father wavelets, i.e. the largest scale used for approximation.
96 : This can be done with the FUNCTION_LENGTH keyword.
97 : It should be given in the same units as the used CV and specifies the length (of the domain interval) of the individual father wavelet functions.
98 :
99 : Alternatively a fixed number of basis functions for the bias expansion can be specified with the NUM_BF keyword, which will set the scale automatically to match the desired number of functions.
100 : Note that this also includes the constant function.
101 :
102 : If you do not specify anything, it is assumed that the range of the bias should match the scale of the wavelet functions.
103 : More precise, the basis functions are scaled to match the specified size of the CV space (MINIMUM and MAXIMUM keywords).
104 : This has so far been a good initial choice.
105 :
106 : If the wavelets are scaled to match the CV range exactly there would be \f$4*\text{ORDER} -3\f$ basis functions whose domain is at least partially in this region.
107 : This number is adjusted if FUNCTION_LENGTH or NUM_BF is specified.
108 : Additionally, some of the shifted basis functions will not have significant contributions because of their function values being close to zero over the full range of the bias.
109 : These 'tail wavelets' can be omitted by using the TAILS_THRESHOLD keyword.
110 : This omits all shifted functions that have only function values smaller than a fraction of their maximum value inside the bias range.
111 : Using a value of e.g. 0.01 will already reduce the number of basis functions significantly.
112 : The default setting will not omit any tail wavelets (i.e. TAILS_THRESHOLD=0).
113 :
114 : The number of basis functions is then not easily determinable a priori but will be given in the logfile.
115 : Additionally the starting point (leftmost defined point) of the individual basis functions is printed.
116 :
117 :
118 : With the PERIODIC keyword the basis set can also be used to bias periodic CVs.
119 : Then the shift between the functions will be chosen such that the function at the left border and right border coincide.
120 : If the FUNCTION_LENGTH keyword is used together with PERIODIC, a smaller length might be chosen to satisfy this requirement.
121 :
122 :
123 : \par Grid
124 :
125 : The values of the wavelet function are generated on a grid.
126 : Using the cascade algorithm results in doubling the grid values for each iteration.
127 : This means that the grid size will always be a power of two multiplied by the number of coefficients (\f$ 2*\text{ORDER} -1\f$) for the specified wavelet.
128 : Using the MIN_GRID_SIZE keyword a lower bound for the number of grid points can be specified.
129 : By default at least 1,000 grid points are used.
130 : Function values in between grid points are calculated by linear interpolation.
131 :
132 : \par Optimization notes
133 :
134 : To avoid 'blind' optimization of the basis functions outside the currently sampled area, it is often beneficial to use the OPTIMIZATION_THRESHOLD keyword of the \ref VES_LINEAR_EXPANSION (set it to a small value, e.g. 1e-6)
135 :
136 : \par Examples
137 :
138 :
139 : First a very simple example that relies on the default values.
140 : We want to bias some CV in the range of 0 to 4.
141 : The wavelets will therefore be scaled to match that range.
142 : Using Db8 wavelets this results in 30 basis functions (including the constant one), with their starting points given by \f$ -14*\frac{4}{15}, -13*\frac{4}{15}, \cdots , 0 , \cdots, 13*\frac{4}{15}, 14*\frac{4}{15} \f$.
143 : \plumedfile
144 : BF_WAVELETS ...
145 : ORDER=8
146 : TYPE=DAUBECHIES
147 : MINIMUM=0.0
148 : MAXIMUM=4.0
149 : LABEL=bf
150 : ... BF_WAVELETS
151 : \endplumedfile
152 :
153 :
154 : By omitting wavelets with only insignificant parts, we can reduce the number of basis functions. Using a threshold of 0.01 will in this example remove the 8 leftmost shifts, which we can check in the logfile.
155 : \plumedfile
156 : BF_WAVELETS ...
157 : ORDER=8
158 : TYPE=DAUBECHIES
159 : MINIMUM=0.0
160 : MAXIMUM=4.0
161 : TAILS_THRESHOLD=0.01
162 : LABEL=bf
163 : ... BF_WAVELETS
164 : \endplumedfile
165 :
166 :
167 : The length of the individual basis functions can also be adjusted to fit the specific problem.
168 : If for example the wavelets are instead scaled to length 3, there will be 35 basis functions, with leftmost points at \f$ -14*\frac{3}{15}, -13*\frac{3}{15}, \cdots, 0, \cdots, 18*\frac{3}{15}, 19*\frac{3}{15} \f$.
169 : \plumedfile
170 : BF_WAVELETS ...
171 : ORDER=8
172 : TYPE=DAUBECHIES
173 : MINIMUM=0.0
174 : MAXIMUM=4.0
175 : FUNCTION_LENGTH=3
176 : LABEL=bf
177 : ... BF_WAVELETS
178 : \endplumedfile
179 :
180 :
181 : Alternatively you can also specify the number of basis functions. Here we specify the usage of 40 Sym10 wavelet functions. We also used a custom minimum size for the grid and want it to be printed to a file with a specific numerical format.
182 : \plumedfile
183 : BF_WAVELETS ...
184 : ORDER=10
185 : TYPE=SYMLETS
186 : MINIMUM=0.0
187 : MAXIMUM=4.0
188 : NUM_BF=40
189 : MIN_GRID_SIZE=500
190 : DUMP_WAVELET_GRID
191 : WAVELET_FILE_FMT=%11.4f
192 : LABEL=bf
193 : ... BF_WAVELETS
194 : \endplumedfile
195 :
196 : */
197 : //+ENDPLUMEDOC
198 :
199 :
200 : class BF_Wavelets : public BasisFunctions {
201 : private:
202 : void setupLabels() override;
203 : /// ptr to Grid that holds the Wavelet values and its derivative
204 : std::unique_ptr<Grid> waveletGrid_;
205 : /// calculate threshold for omitted tail wavelets
206 : std::vector<double> getCutoffPoints(const double& threshold);
207 : /// scale factor of the individual BFs to match specified length
208 : double scale_;
209 : /// shift of the individual BFs
210 : std::vector<double> shifts_;
211 : public:
212 : static void registerKeywords( Keywords&);
213 : explicit BF_Wavelets(const ActionOptions&);
214 : void getAllValues(const double, double&, bool&, std::vector<double>&, std::vector<double>&) const override;
215 : };
216 :
217 :
218 10466 : PLUMED_REGISTER_ACTION(BF_Wavelets,"BF_WAVELETS")
219 :
220 :
221 48 : void BF_Wavelets::registerKeywords(Keywords& keys) {
222 48 : BasisFunctions::registerKeywords(keys);
223 96 : keys.add("compulsory","TYPE","Specify the wavelet type. Currently available are \"DAUBECHIES\" Wavelets with minimum phase and the more symmetric \"SYMLETS\"");
224 96 : keys.add("optional","FUNCTION_LENGTH","The domain size of the individual basis functions (\"length\"). This is used to alter the scaling of the basis functions. By default it is set to the total size of the interval. This also influences the number of actually used basis functions, as all shifted functions that are partially supported in the CV space are used.");
225 96 : keys.add("optional","NUM_BF","The number of basis functions that should be used. Includes the constant one and N-1 shifted wavelets within the specified range. Cannot be used together with FUNCTION_LENGTH.");
226 96 : keys.add("optional","TAILS_THRESHOLD","The threshold for cutting off tail wavelets as a fraction of the maximum value. All shifted wavelet functions that only have values smaller than the threshold in the bias range will be excluded from the basis set. Defaults to 0 (include all).");
227 96 : keys.addFlag("MOTHER_WAVELET", false, "If this flag is set mother wavelets will be used instead of the scaling function (father wavelet). Makes only sense for multiresolution, which is at the moment not usable.");
228 96 : keys.add("optional","MIN_GRID_SIZE","The minimal number of grid bins of the Wavelet function. The true number depends also on the used wavelet type and will probably be larger. Defaults to 1000.");
229 96 : keys.addFlag("DUMP_WAVELET_GRID", false, "If this flag is set the grid with the wavelet values will be written to a file called \"wavelet_grid.data\".");
230 144 : keys.add("optional","WAVELET_FILE_FMT","The number format of the wavelet grid values and derivatives written to file. By default it is %15.8f.\n");
231 96 : keys.addFlag("PERIODIC", false, "Use periodic version of basis set.");
232 48 : keys.remove("NUMERICAL_INTEGRALS");
233 48 : }
234 :
235 :
236 47 : BF_Wavelets::BF_Wavelets(const ActionOptions& ao):
237 : PLUMED_VES_BASISFUNCTIONS_INIT(ao),
238 47 : waveletGrid_(nullptr),
239 47 : scale_(0.0)
240 : {
241 47 : log.printf(" Wavelet basis functions, see and cite ");
242 94 : log << plumed.cite("Pampel and Valsson, J. Chem. Theory Comput. 18, 4127-4141 (2022) - DOI:10.1021/acs.jctc.2c00197");
243 :
244 : // parse properties for waveletGrid and set it up
245 : bool use_mother_wavelet;
246 94 : parseFlag("MOTHER_WAVELET", use_mother_wavelet);
247 :
248 : std::string wavelet_type_str;
249 47 : parse("TYPE", wavelet_type_str);
250 94 : addKeywordToList("TYPE", wavelet_type_str);
251 :
252 47 : unsigned min_grid_size = 1000;
253 47 : parse("MIN_GRID_SIZE", min_grid_size);
254 83 : if(min_grid_size != 1000) {addKeywordToList("MIN_GRID_SIZE",min_grid_size);}
255 :
256 94 : waveletGrid_ = WaveletGrid::setupGrid(getOrder(), min_grid_size, use_mother_wavelet, WaveletGrid::stringToType(wavelet_type_str));
257 47 : bool dump_wavelet_grid=false;
258 47 : parseFlag("DUMP_WAVELET_GRID", dump_wavelet_grid);
259 47 : if (dump_wavelet_grid) {
260 36 : OFile wavelet_gridfile;
261 36 : std::string fmt = "%13.6f";
262 72 : parse("WAVELET_FILE_FMT",fmt);
263 : waveletGrid_->setOutputFmt(fmt); // property of grid not OFile determines fmt
264 36 : wavelet_gridfile.link(*this);
265 36 : wavelet_gridfile.enforceBackup();
266 72 : wavelet_gridfile.open(getLabel()+".wavelet_grid.data");
267 36 : waveletGrid_->writeToFile(wavelet_gridfile);
268 36 : }
269 :
270 47 : bool periodic = false;
271 47 : parseFlag("PERIODIC",periodic);
272 51 : if (periodic) {addKeywordToList("PERIODIC",periodic);}
273 :
274 : // now set up properties of basis set
275 47 : unsigned intrinsic_length = 2*getOrder() - 1; // length of unscaled wavelet
276 47 : double bias_length = intervalMax() - intervalMin(); // intervalRange() is not yet set
277 :
278 : // parse threshold for tail wavelets and get respective cutoff points
279 47 : double threshold = 0.0;
280 47 : std::vector<double> cutoffpoints (2);
281 47 : parse("TAILS_THRESHOLD",threshold);
282 47 : plumed_massert(threshold < 1, "TAILS_THRESHOLD should be significantly smaller than 1.");
283 47 : if(threshold == 0.0) {
284 45 : cutoffpoints = {0.0, static_cast<double>(intrinsic_length)};
285 : }
286 : else {
287 2 : plumed_massert(!periodic, "TAILS_THRESHOLD can't be used with the periodic wavelet variant");
288 2 : addKeywordToList("TAILS_THRESHOLD",threshold);
289 4 : cutoffpoints = getCutoffPoints(threshold);
290 : };
291 :
292 47 : double function_length = bias_length;
293 47 : parse("FUNCTION_LENGTH",function_length);
294 47 : if(function_length != bias_length) {
295 4 : if (periodic) { // shifted functions need to fit into interval exactly -> reduce size if not
296 2 : unsigned num_shifts = ceil(bias_length * intrinsic_length / function_length);
297 2 : function_length = bias_length * intrinsic_length / num_shifts;
298 : }
299 8 : addKeywordToList("FUNCTION_LENGTH",function_length);
300 : }
301 :
302 : // determine number of BFs and needed scaling
303 47 : unsigned num_BFs = 0;
304 47 : parse("NUM_BF",num_BFs);
305 47 : if(num_BFs == 0) { // get from function length
306 43 : scale_ = intrinsic_length / function_length;
307 43 : if (periodic) {
308 : // this is the same value as num_shifts above + constant
309 2 : num_BFs = static_cast<unsigned>(bias_length * scale_) + 1;
310 : }
311 : else {
312 41 : num_BFs = 1; // constant one
313 : // left shifts (w/o left cutoff) + right shifts - right cutoff - 1
314 41 : num_BFs += static_cast<unsigned>(ceil(cutoffpoints[1] + (bias_length)*scale_ - cutoffpoints[0]) - 1);
315 : }
316 : }
317 : else {
318 : plumed_massert(num_BFs > 0, "The number of basis functions has to be positive (NUM_BF > 0)");
319 : // check does not work if function length was given as intrinsic length, but can't check for keyword use directly
320 4 : plumed_massert(function_length==bias_length,"The keywords \"NUM_BF\" and \"FUNCTION_LENGTH\" cannot be used at the same time");
321 4 : addKeywordToList("NUM_BF",num_BFs);
322 :
323 4 : if (periodic) { // inverted num_BFs calculation from where FUNCTION_LENGTH is specified
324 2 : scale_ = (num_BFs - 1) / bias_length ;
325 : }
326 : else {
327 2 : double cutoff_length = cutoffpoints[1] - cutoffpoints [0];
328 2 : double intrinsic_bias_length = num_BFs - cutoff_length + 1; // length of bias in intrinsic scale of wavelets
329 2 : scale_ = intrinsic_bias_length / bias_length;
330 : }
331 : }
332 :
333 47 : setNumberOfBasisFunctions(num_BFs);
334 :
335 : // now set up the starting points of the basis functions
336 47 : shifts_.push_back(0.0); // constant BF – never used, just for clearer notation
337 1908 : for(unsigned int i = 1; i < getNumberOfBasisFunctions(); ++i) {
338 1861 : shifts_.push_back(-intervalMin()*scale_ + cutoffpoints[1] - i);
339 : }
340 :
341 : // set some properties
342 47 : setIntrinsicInterval(0.0,intrinsic_length);
343 47 : periodic ? setPeriodic() : setNonPeriodic();
344 : setIntervalBounded();
345 : setType(wavelet_type_str);
346 47 : setDescription("Wavelets as localized basis functions");
347 47 : setupBF();
348 47 : checkRead();
349 :
350 47 : log.printf(" Each basisfunction spans %f in CV space\n", intrinsic_length/scale_);
351 47 : }
352 :
353 :
354 62249 : void BF_Wavelets::getAllValues(const double arg, double& argT, bool& inside_range, std::vector<double>& values, std::vector<double>& derivs) const {
355 62249 : argT=checkIfArgumentInsideInterval(arg,inside_range);
356 : //
357 62249 : values[0]=1.0;
358 62249 : derivs[0]=0.0;
359 2315762 : for(unsigned int i = 1; i < getNumberOfBasisFunctions(); ++i) {
360 : // scale and shift argument to match current wavelet
361 2253513 : double x = shifts_[i] + argT*scale_;
362 2253513 : if (arePeriodic()) { // periodic interval [0,intervalRange*scale]
363 171766 : x = x - floor(x/(intervalRange()*scale_))*intervalRange()*scale_;
364 : }
365 :
366 2253513 : if (x < 0 || x >= intrinsicIntervalMax()) { // Wavelets are 0 outside the defined range
367 989659 : values[i] = 0.0; derivs[i] = 0.0;
368 : }
369 : else {
370 1263854 : std::vector<double> temp_deriv (1);
371 2527708 : values[i] = GridLinearInterpolation::getGridValueAndDerivativesWithLinearInterpolation(waveletGrid_.get(), {x}, temp_deriv);
372 1263854 : derivs[i] = temp_deriv[0] * scale_; // scale derivative
373 : }
374 : }
375 67885 : if(!inside_range) {for(auto& deriv : derivs) {deriv=0.0;}}
376 62249 : }
377 :
378 :
379 : // returns left and right cutoff point of Wavelet
380 : // threshold is a percent value of maximum
381 2 : std::vector<double> BF_Wavelets::getCutoffPoints(const double& threshold) {
382 2 : double threshold_value = threshold * waveletGrid_->getMaxValue();
383 : std::vector<double> cutoffpoints;
384 :
385 475 : for (size_t i = 0; i < waveletGrid_->getSize(); ++i) {
386 475 : if (fabs(waveletGrid_->getValue(i)) >= threshold_value) {
387 2 : cutoffpoints.push_back(waveletGrid_->getPoint(i)[0]);
388 2 : break;
389 : }
390 : }
391 :
392 1073 : for (int i = waveletGrid_->getSize() - 1; i >= 0; --i) {
393 1073 : if (fabs(waveletGrid_->getValue(i)) >= threshold_value) {
394 2 : cutoffpoints.push_back(waveletGrid_->getPoint(i)[0]);
395 2 : break;
396 : }
397 : }
398 :
399 2 : return cutoffpoints;
400 : }
401 :
402 :
403 : // labels according to minimum position in CV space
404 47 : void BF_Wavelets::setupLabels() {
405 47 : setLabel(0,"const");
406 1908 : for(unsigned int i=1; i < getNumberOfBasisFunctions(); i++) {
407 1861 : double pos = -shifts_[i]/scale_;
408 1861 : if (arePeriodic()) {
409 88 : pos = pos - floor((pos-intervalMin())/intervalRange())*intervalRange();
410 : }
411 1861 : std::string is; Tools::convert(pos, is);
412 3722 : setLabel(i,"i="+is);
413 : }
414 47 : }
415 :
416 :
417 : }
418 : }
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