Coverage for eminus/band_minimizer.py: 98.95%

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1# SPDX-FileCopyrightText: 2021 The eminus developers 

2# SPDX-License-Identifier: Apache-2.0 

3"""Minimization algorithms for fixed Hamiltonians. 

4 

5Similar to :mod:`eminus.minimizer` but for a fixed Hamiltonian the implementation can be simplified 

6and made more performant. 

7""" 

8 

9import copy 

10import logging 

11 

12import numpy as np 

13from scipy.linalg import inv, sqrtm 

14 

15from .dft import H, orth, orth_unocc 

16from .energies import get_Eband 

17from .logger import name 

18from .minimizer import cg_method, cg_test, check_convergence, linmin_test 

19from .utils import dotprod 

20 

21 

22def scf_step_occ(scf, W): 

23 """Perform one SCF step for an occupied band minimization calculation. 

24 

25 Args: 

26 scf: SCF object. 

27 W: Expansion coefficients of unconstrained wave functions in reciprocal space. 

28 

29 Returns: 

30 Band energy. 

31 """ 

32 atoms = scf.atoms 

33 scf.Y = orth(atoms, W) 

34 return get_Eband(scf, scf.Y, **scf._precomputed) 

35 

36 

37def scf_step_unocc(scf, Z): 

38 """Perform one SCF step for an unoccupied band minimization calculation. 

39 

40 Args: 

41 scf: SCF object. 

42 Z: Expansion coefficients of unconstrained wave functions in reciprocal space. 

43 

44 Returns: 

45 Band energy. 

46 """ 

47 atoms = scf.atoms 

48 scf.D = orth_unocc(atoms, scf.Y, Z) 

49 return get_Eband(scf, scf.D, **scf._precomputed) 

50 

51 

52def get_grad_occ(scf, ik, spin, W, **kwargs): 

53 """Calculate the occupied band energy gradient with respect to W. 

54 

55 Reference: Comput. Phys. Commun. 128, 1. 

56 

57 Args: 

58 scf: SCF object. 

59 ik: k-point index. 

60 spin: Spin variable to track whether to do the calculation for spin up or down. 

61 W: Expansion coefficients of unconstrained wave functions in reciprocal space. 

62 

63 Keyword Args: 

64 **kwargs: See :func:`H`. 

65 

66 Returns: 

67 Gradient. 

68 """ 

69 atoms = scf.atoms 

70 W = orth(atoms, W) 

71 HW = H(scf, ik, spin, W, **kwargs) 

72 WHW = W[ik][spin].conj().T @ HW 

73 OW = atoms.O(W[ik][spin]) 

74 U = W[ik][spin].conj().T @ OW 

75 invU = inv(U) 

76 U12 = sqrtm(invU) 

77 # grad E = (I - O(Y) Ydag) H(Y) U^-0.5 

78 return atoms.kpts.wk[ik] * ((HW - OW @ WHW) @ U12) 

79 

80 

81def get_grad_unocc(scf, ik, spin, Z, **kwargs): 

82 """Calculate the unoccupied band energy gradient with respect to Z. 

83 

84 Reference: Comput. Phys. Commun. 128, 1. 

85 

86 Args: 

87 scf: SCF object. 

88 ik: k-point index. 

89 spin: Spin variable to track whether to do the calculation for spin up or down. 

90 Z: Expansion coefficients of unconstrained wave functions in reciprocal space. 

91 

92 Keyword Args: 

93 **kwargs: See :func:`H`. 

94 

95 Returns: 

96 Gradient. 

97 """ 

98 atoms = scf.atoms 

99 Y = scf.Y[ik][spin][:, scf.atoms.occ.f[ik][spin] > 0] 

100 Ydag = Y.conj().T 

101 # We need X12 later, so orthogonalize in-place and only the current state 

102 rhoZ = Z[ik][spin] - Y @ Ydag @ atoms.O(Z[ik][spin]) 

103 X12 = inv(sqrtm(rhoZ.conj().T @ atoms.O(rhoZ))) 

104 D = rhoZ @ X12 

105 # Create the correct input shape for the Hamiltonian 

106 D_tmp = [None] * len(Z) 

107 D_tmp[ik] = np.empty_like(Z[ik]) 

108 D_tmp[ik][spin] = D 

109 HD = H(scf, ik, spin, D_tmp, **kwargs) 

110 DHD = D.conj().T @ HD 

111 I = np.eye(Z[ik].shape[1]) 

112 # grad E = (I - O(Y) Ydag) (I - O(D) Ddag) H(D) X^-0.5 

113 return atoms.kpts.wk[ik] * ((I - atoms.O(Y) @ Ydag) @ (HD - atoms.O(D) @ DHD) @ X12) 

114 

115 

116@name("steepest descent minimization") 

117def sd( 

118 scf, 

119 W, 

120 Nit, 

121 cost=scf_step_occ, 

122 grad=get_grad_occ, 

123 condition=check_convergence, 

124 betat=3e-5, 

125 **kwargs, 

126): 

127 """Steepest descent minimization algorithm for a fixed Hamiltonian. 

128 

129 Args: 

130 scf: SCF object. 

131 W: Expansion coefficients of unconstrained wave functions in reciprocal space. 

132 Nit: Maximum number of SCF steps. 

133 

134 Keyword Args: 

135 cost: Function that will run every SCF step. 

136 grad: Function that calculates the respective gradient. 

137 condition: Function to check and log the convergence condition. 

138 betat: Step size. 

139 **kwargs: Throwaway arguments. 

140 

141 Returns: 

142 Band energies per SCF cycle and optimized expansion coefficients. 

143 """ 

144 atoms = scf.atoms 

145 costs = [] 

146 

147 for _ in range(Nit): 

148 c = cost(scf, W) 

149 costs.append(c) 

150 if condition(scf, "sd", costs): 

151 break 

152 for ik in range(atoms.kpts.Nk): 

153 for spin in range(atoms.occ.Nspin): 

154 g = grad(scf, ik, spin, W, **scf._precomputed) 

155 W[ik][spin] = W[ik][spin] - betat * g 

156 return costs, W 

157 

158 

159@name("preconditioned line minimization") 

160def pclm( 

161 scf, 

162 W, 

163 Nit, 

164 cost=scf_step_occ, 

165 grad=get_grad_occ, 

166 condition=check_convergence, 

167 betat=3e-5, 

168 precondition=True, 

169 **kwargs, 

170): 

171 """Preconditioned line minimization algorithm for a fixed Hamiltonian. 

172 

173 Args: 

174 scf: SCF object. 

175 W: Expansion coefficients of unconstrained wave functions in reciprocal space. 

176 Nit: Maximum number of SCF steps. 

177 

178 Keyword Args: 

179 cost: Function that will run every SCF step. 

180 grad: Function that calculates the respective gradient. 

181 condition: Function to check and log the convergence condition. 

182 betat: Step size. 

183 precondition: Whether to use a preconditioner. 

184 **kwargs: Throwaway arguments. 

185 

186 Returns: 

187 Band energies per SCF cycle and optimized expansion coefficients. 

188 """ 

189 atoms = scf.atoms 

190 costs = [] 

191 

192 linmin = np.empty((atoms.kpts.Nk, atoms.occ.Nspin)) 

193 d = [np.empty_like(Wk) for Wk in W] 

194 

195 if precondition: 

196 method = "pclm" 

197 else: 

198 method = "lm" 

199 

200 for i in range(Nit): 

201 c = cost(scf, W) 

202 costs.append(c) 

203 if condition(scf, method, costs, linmin): 

204 break 

205 for ik in range(atoms.kpts.Nk): 

206 for spin in range(atoms.occ.Nspin): 

207 g = grad(scf, ik, spin, W, **scf._precomputed) 

208 if scf._log.level <= logging.DEBUG and i > 0: 

209 linmin[ik][spin] = linmin_test(g, d[ik][spin]) 

210 if precondition: 

211 d[ik][spin] = -atoms.K(g, ik) 

212 else: 

213 d[ik][spin] = -g 

214 W[ik][spin] = W[ik][spin] + betat * d[ik][spin] 

215 gt = grad(scf, ik, spin, W, **scf._precomputed) 

216 beta = abs(betat * dotprod(g, d[ik][spin]) / dotprod(g - gt, d[ik][spin])) 

217 W[ik][spin] = W[ik][spin] + beta * d[ik][spin] 

218 return costs, W 

219 

220 

221@name("line minimization") 

222def lm( 

223 scf, 

224 W, 

225 Nit, 

226 cost=scf_step_occ, 

227 grad=get_grad_occ, 

228 condition=check_convergence, 

229 betat=3e-5, 

230 **kwargs, 

231): 

232 """Line minimization algorithm for a fixed Hamiltonian. 

233 

234 Args: 

235 scf: SCF object. 

236 W: Expansion coefficients of unconstrained wave functions in reciprocal space. 

237 Nit: Maximum number of SCF steps. 

238 

239 Keyword Args: 

240 cost: Function that will run every SCF step. 

241 grad: Function that calculates the respective gradient. 

242 condition: Function to check and log the convergence condition. 

243 betat: Step size. 

244 **kwargs: Throwaway arguments. 

245 

246 Returns: 

247 Band energies per SCF cycle and optimized expansion coefficients. 

248 """ 

249 return pclm(scf, W, Nit, cost, grad, condition, betat, precondition=False) 

250 

251 

252@name("preconditioned conjugate-gradient minimization") 

253def pccg( 

254 scf, 

255 W, 

256 Nit, 

257 cost=scf_step_occ, 

258 grad=get_grad_occ, 

259 condition=check_convergence, 

260 betat=3e-5, 

261 cgform=1, 

262 precondition=True, 

263): 

264 """Preconditioned conjugate-gradient minimization algorithm for a fixed Hamiltonian. 

265 

266 Args: 

267 scf: SCF object. 

268 W: Expansion coefficients of unconstrained wave functions in reciprocal space. 

269 Nit: Maximum number of SCF steps. 

270 

271 Keyword Args: 

272 cost: Function that will run every SCF step. 

273 grad: Function that calculates the respective gradient. 

274 condition: Function to check and log the convergence condition. 

275 betat: Step size. 

276 cgform: Conjugate gradient form. 

277 precondition: Whether to use a preconditioner. 

278 

279 Returns: 

280 Band energies per SCF cycle and optimized expansion coefficients. 

281 """ 

282 atoms = scf.atoms 

283 costs = [] 

284 

285 linmin = np.empty((atoms.kpts.Nk, atoms.occ.Nspin)) 

286 cg = np.empty((atoms.kpts.Nk, atoms.occ.Nspin)) 

287 norm_g = np.empty((atoms.kpts.Nk, atoms.occ.Nspin)) 

288 d = [np.empty_like(Wk) for Wk in W] 

289 d_old = [np.empty_like(Wk) for Wk in W] 

290 g_old = [np.empty_like(Wk) for Wk in W] 

291 

292 if precondition: 

293 method = "pccg" 

294 else: 

295 method = "cg" 

296 

297 c = cost(scf, W) 

298 costs.append(c) 

299 condition(scf, method, costs) 

300 # Do the first step without the linmin and cg tests, and without the cg_method 

301 for ik in range(atoms.kpts.Nk): 

302 for spin in range(atoms.occ.Nspin): 

303 g = grad(scf, ik, spin, W, **scf._precomputed) 

304 if precondition: 

305 d[ik][spin] = -atoms.K(g, ik) 

306 else: 

307 d[ik][spin] = -g 

308 W[ik][spin] = W[ik][spin] + betat * d[ik][spin] 

309 gt = grad(scf, ik, spin, W, **scf._precomputed) 

310 beta = abs(betat * dotprod(g, d[ik][spin]) / dotprod(g - gt, d[ik][spin])) 

311 g_old[ik][spin], d_old[ik][spin] = g, d[ik][spin] 

312 W[ik][spin] = W[ik][spin] + beta * d[ik][spin] 

313 

314 for _ in range(1, Nit): 

315 c = cost(scf, W) 

316 costs.append(c) 

317 if condition(scf, method, costs, linmin, cg, norm_g): 

318 break 

319 for ik in range(atoms.kpts.Nk): 

320 for spin in range(atoms.occ.Nspin): 

321 g = grad(scf, ik, spin, W, **scf._precomputed) 

322 # Calculate linmin and cg for each spin separately 

323 if scf._log.level <= logging.DEBUG: 

324 linmin[ik][spin] = linmin_test(g, d[ik][spin]) 

325 cg[ik][spin] = cg_test(atoms, ik, g, g_old[ik][spin], precondition) 

326 beta, norm_g[ik][spin] = cg_method( 

327 scf, ik, cgform, g, g_old[ik][spin], d_old[ik][spin], precondition 

328 ) 

329 if precondition: 

330 d[ik][spin] = -atoms.K(g, ik) + beta * d_old[ik][spin] 

331 else: 

332 d[ik][spin] = -g + beta * d_old[ik][spin] 

333 W[ik][spin] = W[ik][spin] + betat * d[ik][spin] 

334 gt = grad(scf, ik, spin, W, **scf._precomputed) 

335 beta = abs(betat * dotprod(g, d[ik][spin]) / dotprod(g - gt, d[ik][spin])) 

336 g_old[ik][spin], d_old[ik][spin] = g, d[ik][spin] 

337 W[ik][spin] = W[ik][spin] + beta * d[ik][spin] 

338 return costs, W 

339 

340 

341@name("conjugate-gradient minimization") 

342def cg( 

343 scf, 

344 W, 

345 Nit, 

346 cost=scf_step_occ, 

347 grad=get_grad_occ, 

348 condition=check_convergence, 

349 betat=3e-5, 

350 cgform=1, 

351): 

352 """Conjugate-gradient minimization algorithm for a fixed Hamiltonian. 

353 

354 Args: 

355 scf: SCF object. 

356 W: Expansion coefficients of unconstrained wave functions in reciprocal space. 

357 Nit: Maximum number of SCF steps. 

358 

359 Keyword Args: 

360 cost: Function that will run every SCF step. 

361 grad: Function that calculates the respective gradient. 

362 condition: Function to check and log the convergence condition. 

363 betat: Step size. 

364 cgform: Conjugate gradient form. 

365 

366 Returns: 

367 Band energies per SCF cycle and optimized expansion coefficients. 

368 """ 

369 return pccg(scf, W, Nit, cost, grad, condition, betat, cgform, precondition=False) 

370 

371 

372@name("auto minimization") 

373def auto( 

374 scf, 

375 W, 

376 Nit, 

377 cost=scf_step_occ, 

378 grad=get_grad_occ, 

379 condition=check_convergence, 

380 betat=3e-5, 

381 cgform=1, 

382): 

383 """Automatic precond. conjugate-gradient minimization algorithm for a fixed Hamiltonian. 

384 

385 This function chooses an sd step over the pccg step if the energy goes up. 

386 

387 Args: 

388 scf: SCF object. 

389 W: Expansion coefficients of unconstrained wave functions in reciprocal space. 

390 Nit: Maximum number of SCF steps. 

391 

392 Keyword Args: 

393 cost: Function that will run every SCF step. 

394 grad: Function that calculates the respective gradient. 

395 condition: Function to check and log the convergence condition. 

396 betat: Step size. 

397 cgform: Conjugate gradient form. 

398 

399 Returns: 

400 Band energies per SCF cycle and optimized expansion coefficients. 

401 """ 

402 atoms = scf.atoms 

403 costs = [] 

404 

405 linmin = np.empty((atoms.kpts.Nk, atoms.occ.Nspin)) 

406 cg = np.empty((atoms.kpts.Nk, atoms.occ.Nspin)) 

407 norm_g = np.empty((atoms.kpts.Nk, atoms.occ.Nspin)) 

408 g = [np.empty_like(Wk) for Wk in W] 

409 d = [np.empty_like(Wk) for Wk in W] 

410 d_old = [np.empty_like(Wk) for Wk in W] 

411 g_old = [np.empty_like(Wk) for Wk in W] 

412 

413 # Do the first step without the linmin and cg tests, and without the cg_method 

414 for ik in range(atoms.kpts.Nk): 

415 for spin in range(atoms.occ.Nspin): 

416 g[ik][spin] = grad(scf, ik, spin, W, **scf._precomputed) 

417 d[ik][spin] = -atoms.K(g[ik][spin], ik) 

418 W[ik][spin] = W[ik][spin] + betat * d[ik][spin] 

419 gt = grad(scf, ik, spin, W, **scf._precomputed) 

420 beta = abs( 

421 betat * dotprod(g[ik][spin], d[ik][spin]) / dotprod(g[ik][spin] - gt, d[ik][spin]) 

422 ) 

423 g_old[ik][spin], d_old[ik][spin] = g[ik][spin], d[ik][spin] 

424 W[ik][spin] = W[ik][spin] + beta * d[ik][spin] 

425 

426 c = cost(scf, W) 

427 costs.append(c) 

428 if condition(scf, "pccg", costs): 

429 return costs 

430 

431 for _ in range(1, Nit): 

432 W_old = copy.deepcopy(W) 

433 for ik in range(atoms.kpts.Nk): 

434 for spin in range(atoms.occ.Nspin): 

435 g[ik][spin] = grad(scf, ik, spin, W, **scf._precomputed) 

436 # Calculate linmin and cg for each spin separately 

437 if scf._log.level <= logging.DEBUG: 

438 linmin[ik][spin] = linmin_test(g[ik][spin], d[ik][spin]) 

439 cg[ik][spin] = cg_test(atoms, ik, g[ik][spin], g_old[ik][spin]) 

440 beta, norm_g[ik][spin] = cg_method( 

441 scf, ik, cgform, g[ik][spin], g_old[ik][spin], d_old[ik][spin] 

442 ) 

443 d[ik][spin] = -atoms.K(g[ik][spin], ik) + beta * d_old[ik][spin] 

444 W[ik][spin] = W[ik][spin] + betat * d[ik][spin] 

445 gt = grad(scf, ik, spin, W, **scf._precomputed) 

446 beta = abs( 

447 betat 

448 * dotprod(g[ik][spin], d[ik][spin]) 

449 / dotprod(g[ik][spin] - gt, d[ik][spin]) 

450 ) 

451 g_old[ik][spin], d_old[ik][spin] = g[ik][spin], d[ik][spin] 

452 W[ik][spin] = W[ik][spin] + beta * d[ik][spin] 

453 

454 c = cost(scf, W) 

455 # If the energy does not go down use the steepest descent step and recalculate the energy 

456 if c > costs[-1]: 

457 W = W_old 

458 for ik in range(atoms.kpts.Nk): 

459 for spin in range(atoms.occ.Nspin): 

460 W[ik][spin] = W[ik][spin] - betat * g[ik][spin] 

461 c = cost(scf, W) 

462 costs.append(c) 

463 # Do not print cg and linmin if we do the sd step 

464 if condition(scf, "sd", costs, norm_g=norm_g): 

465 break 

466 else: 

467 costs.append(c) 

468 if condition(scf, "pccg", costs, linmin, cg, norm_g): 

469 break 

470 return costs, W 

471 

472 

473#: Map minimizer names with their respective implementation. 

474IMPLEMENTED = { 

475 "sd": sd, 

476 "lm": lm, 

477 "pclm": pclm, 

478 "cg": cg, 

479 "pccg": pccg, 

480 "auto": auto, 

481}