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DriverModifiedArnoldi.cpp
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2 //
3 // File DriverModifiedArnoldi.cpp
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9 // Copyright (c) 2006 Division of Applied Mathematics, Brown University (USA),
10 // Department of Aeronautics, Imperial College London (UK), and Scientific
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31 //
32 // Description: Driver for eigenvalue analysis using the modified Arnoldi
33 // method.
34 //
35 ///////////////////////////////////////////////////////////////////////////////
36 
37 #include <iomanip>
38 
39 #include <SolverUtils/DriverModifiedArnoldi.h>
40 
41 namespace Nektar
42 {
43  namespace SolverUtils
44  {
45 
46  string DriverModifiedArnoldi::className =
47  GetDriverFactory().RegisterCreatorFunction("ModifiedArnoldi",
48  DriverModifiedArnoldi::create);
49  string DriverModifiedArnoldi::driverLookupId =
50  LibUtilities::SessionReader::RegisterEnumValue("Driver",
51  "ModifiedArnoldi",0);
52 
53  /**
54  *
55  */
56  DriverModifiedArnoldi::DriverModifiedArnoldi(
57  const LibUtilities::SessionReaderSharedPtr pSession)
58  : DriverArnoldi(pSession)
59  {
60  }
61 
62 
63  /**
64  *
65  */
66  DriverModifiedArnoldi::~DriverModifiedArnoldi()
67  {
68  }
69 
70 
71  /**
72  *
73  */
74  void DriverModifiedArnoldi::v_InitObject(ostream &out)
75  {
76  DriverArnoldi::v_InitObject(out);
77 
78  m_equ[0]->PrintSummary(out);
79 
80  // Print session parameters
81  if (m_comm->GetRank() == 0)
82  {
83  out << "\tArnoldi solver type : Modified Arnoldi" << endl;
84  }
85 
86  DriverArnoldi::ArnoldiSummary(out);
87 
88  m_equ[m_nequ - 1]->DoInitialise();
89 
90  //FwdTrans Initial conditions to be in Coefficient Space
91  m_equ[m_nequ-1] ->TransPhysToCoeff();
92 
93  }
94 
95 
96  /**
97  *
98  */
99  void DriverModifiedArnoldi::v_Execute(ostream &out)
100  {
101  int i = 0;
102  int j = 0;
103  int nq = m_equ[0]->UpdateFields()[0]->GetNcoeffs();
104  int ntot = m_nfields*nq;
105  int converged = 0;
106  NekDouble resnorm = 0.0;
107  ofstream evlout;
108  std::string evlFile = m_session->GetSessionName() + ".evl";
109 
110  if (m_comm->GetRank() == 0)
111  {
112  evlout.open(evlFile.c_str());
113  }
114 
115  // Allocate memory
116  Array<OneD, NekDouble> alpha = Array<OneD, NekDouble> (m_kdim+1, 0.0);
117  Array<OneD, NekDouble> wr = Array<OneD, NekDouble> (m_kdim, 0.0);
118  Array<OneD, NekDouble> wi = Array<OneD, NekDouble> (m_kdim, 0.0);
119  Array<OneD, NekDouble> zvec = Array<OneD, NekDouble> (m_kdim*m_kdim, 0.0);
120 
121  Array<OneD, Array<OneD, NekDouble> > Kseq
122  = Array<OneD, Array<OneD, NekDouble> > (m_kdim + 1);
123  Array<OneD, Array<OneD, NekDouble> > Tseq
124  = Array<OneD, Array<OneD, NekDouble> > (m_kdim + 1);
125  for (i = 0; i < m_kdim + 1; ++i)
126  {
127  Kseq[i] = Array<OneD, NekDouble>(ntot, 0.0);
128  Tseq[i] = Array<OneD, NekDouble>(ntot, 0.0);
129  }
130 
131  // Copy starting vector into second sequence element (temporary).
132  if(m_session->DefinesFunction("InitialConditions"))
133  {
134  if (m_comm->GetRank() == 0)
135  {
136  out << "\tInital vector : specified in input file " << endl;
137  }
138  m_equ[0]->SetInitialConditions(0.0,false);
139 
140  CopyFieldToArnoldiArray(Kseq[1]);
141  }
142  else
143  {
144  if (m_comm->GetRank() == 0)
145  {
146  out << "\tInital vector : random " << endl;
147  }
148 
149  NekDouble eps=1;
150  Vmath::FillWhiteNoise(ntot, eps , &Kseq[1][0], 1);
151  }
152 
153  // Perform one iteration to enforce boundary conditions.
154  // Set this as the initial value in the sequence.
155  EV_update(Kseq[1], Kseq[0]);
156  if (m_comm->GetRank() == 0)
157  {
158  out << "Iteration: " << 0 << endl;
159  }
160 
161  // Normalise first vector in sequence
162  alpha[0] = Blas::Ddot(ntot, &Kseq[0][0], 1, &Kseq[0][0], 1);
163  m_comm->AllReduce(alpha[0], Nektar::LibUtilities::ReduceSum);
164  alpha[0] = std::sqrt(alpha[0]);
165  Vmath::Smul(ntot, 1.0/alpha[0], Kseq[0], 1, Kseq[0], 1);
166 
167  // Fill initial krylov sequence
168  NekDouble resid0;
169  for (i = 1; !converged && i <= m_kdim; ++i)
170  {
171  // Compute next vector
172  EV_update(Kseq[i-1], Kseq[i]);
173 
174  // Normalise
175  alpha[i] = Blas::Ddot(ntot, &Kseq[i][0], 1, &Kseq[i][0], 1);
176  m_comm->AllReduce(alpha[i], Nektar::LibUtilities::ReduceSum);
177  alpha[i] = std::sqrt(alpha[i]);
178 
179  //alpha[i] = std::sqrt(alpha[i]);
180  Vmath::Smul(ntot, 1.0/alpha[i], Kseq[i], 1, Kseq[i], 1);
181 
182  // Copy Krylov sequence into temporary storage
183  for (int k = 0; k < i + 1; ++k)
184  {
185  Vmath::Vcopy(ntot, Kseq[k], 1, Tseq[k], 1);
186  }
187 
188  // Generate Hessenberg matrix and compute eigenvalues of it.
189  EV_small(Tseq, ntot, alpha, i, zvec, wr, wi, resnorm);
190 
191  // Test for convergence.
192  converged = EV_test(i, i, zvec, wr, wi, resnorm,
193  std::min(i, m_nvec), evlout, resid0);
194  converged = max (converged, 0);
195 
196  if (m_comm->GetRank() == 0)
197  {
198  out << "Iteration: " << i << " (residual : " << resid0
199  << ")" <<endl;
200  }
201  }
202 
203  // Continue with full sequence
204  if (!converged)
205  {
206  for (i = m_kdim + 1; !converged && i <= m_nits; ++i)
207  {
208  // Shift all the vectors in the sequence.
209  // First vector is removed.
210  for (int j = 1; j <= m_kdim; ++j)
211  {
212  alpha[j-1] = alpha[j];
213  Vmath::Vcopy(ntot, Kseq[j], 1, Kseq[j-1], 1);
214  }
215 
216  // Compute next vector
217  EV_update(Kseq[m_kdim - 1], Kseq[m_kdim]);
218 
219  // Compute new scale factor
220  alpha[m_kdim] = Blas::Ddot(ntot, &Kseq[m_kdim][0], 1,
221  &Kseq[m_kdim][0], 1);
222  m_comm->AllReduce(alpha[m_kdim],
223  Nektar::LibUtilities::ReduceSum);
224  alpha[m_kdim] = std::sqrt(alpha[m_kdim]);
225  Vmath::Smul(ntot, 1.0/alpha[m_kdim], Kseq[m_kdim], 1,
226  Kseq[m_kdim], 1);
227 
228  // Copy Krylov sequence into temporary storage
229  for (int k = 0; k < m_kdim + 1; ++k)
230  {
231  Vmath::Vcopy(ntot, Kseq[k], 1, Tseq[k], 1);
232  }
233 
234  // Generate Hessenberg matrix and compute eigenvalues of it
235  EV_small(Tseq, ntot, alpha, m_kdim, zvec, wr, wi, resnorm);
236 
237  // Test for convergence.
238  converged = EV_test(i, m_kdim, zvec, wr, wi, resnorm,
239  m_nvec, evlout, resid0);
240 
241  if (m_comm->GetRank() == 0)
242  {
243  out << "Iteration: " << i << " (residual : "
244  << resid0 << ")" <<endl;
245  }
246  }
247  }
248 
249  m_equ[0]->Output();
250 
251  // Evaluate and output computation time and solution accuracy.
252  // The specific format of the error output is essential for the
253  // regression tests to work.
254  // Evaluate L2 Error
255  for(j = 0; j < m_equ[0]->GetNvariables(); ++j)
256  {
257  NekDouble vL2Error = m_equ[0]->L2Error(j,false);
258  NekDouble vLinfError = m_equ[0]->LinfError(j);
259  if (m_comm->GetRank() == 0)
260  {
261  out << "L 2 error (variable " << m_equ[0]->GetVariable(j)
262  << ") : " << vL2Error << endl;
263  out << "L inf error (variable " << m_equ[0]->GetVariable(j)
264  << ") : " << vLinfError << endl;
265  }
266  }
267 
268  // Process eigenvectors and write out.
269  EV_post(Tseq, Kseq, ntot, min(--i, m_kdim), m_nvec, zvec, wr, wi,
270  converged);
271 
272  // store eigenvalues so they can be access from driver class
273  m_real_evl = wr;
274  m_imag_evl = wi;
275 
276  // Close the runtime info file.
277  if (m_comm->GetRank() == 0)
278  {
279  evlout.close();
280  }
281  }
282 
283 
284  /**
285  *
286  */
287  void DriverModifiedArnoldi::EV_update(
288  Array<OneD, NekDouble> &src,
289  Array<OneD, NekDouble> &tgt)
290  {
291  // Copy starting vector into first sequence element.
292  CopyArnoldiArrayToField(src);
293  m_equ[0]->TransCoeffToPhys();
294 
295  m_equ[0]->DoSolve();
296 
297  if(m_EvolutionOperator == eTransientGrowth)
298  {
299  Array<OneD, MultiRegions::ExpListSharedPtr> fields;
300  fields = m_equ[0]->UpdateFields();
301 
302  //start Adjoint with latest fields of direct
303  CopyFwdToAdj();
304  m_equ[1]->TransCoeffToPhys();
305 
306  m_equ[1]->DoSolve();
307  }
308 
309  // Copy starting vector into first sequence element.
310  CopyFieldToArnoldiArray(tgt);
311  }
312 
313 
314  /**
315  *
316  */
317  void DriverModifiedArnoldi::EV_small(
318  Array<OneD, Array<OneD, NekDouble> > &Kseq,
319  const int ntot,
320  const Array<OneD, NekDouble> &alpha,
321  const int kdim,
322  Array<OneD, NekDouble> &zvec,
323  Array<OneD, NekDouble> &wr,
324  Array<OneD, NekDouble> &wi,
325  NekDouble &resnorm)
326  {
327  int kdimp = kdim + 1;
328  int lwork = 10*kdim;
329  int ier;
330  Array<OneD, NekDouble> R(kdimp * kdimp, 0.0);
331  Array<OneD, NekDouble> H(kdimp * kdim, 0.0);
332  Array<OneD, NekDouble> rwork(lwork, 0.0);
333 
334  // Modified G-S orthonormalisation
335  for (int i = 0; i < kdimp; ++i)
336  {
337  NekDouble gsc = Blas::Ddot(ntot, &Kseq[i][0], 1, &Kseq[i][0], 1);
338  m_comm->AllReduce(gsc, Nektar::LibUtilities::ReduceSum);
339  gsc = std::sqrt(gsc);
340  ASSERTL0(gsc != 0.0, "Vectors are linearly independent.");
341 
342  R[i*kdimp+i] = gsc;
343  Vmath::Smul(ntot, 1.0/gsc, Kseq[i], 1, Kseq[i], 1);
344 
345  for (int j = i + 1; j < kdimp; ++j)
346  {
347  gsc = Blas::Ddot(ntot, &Kseq[i][0], 1, &Kseq[j][0], 1);
348  m_comm->AllReduce(gsc, Nektar::LibUtilities::ReduceSum);
349  Vmath::Svtvp(ntot, -gsc, Kseq[i], 1, Kseq[j], 1, Kseq[j], 1);
350  R[j*kdimp+i] = gsc;
351  }
352  }
353 
354  // Compute H matrix
355  for (int i = 0; i < kdim; ++i)
356  {
357  for (int j = 0; j < kdim; ++j)
358  {
359  H[j*kdim+i] = alpha[j+1] * R[(j+1)*kdimp+i]
360  - Vmath::Dot(j, &H[0] + i, kdim, &R[0] + j*kdimp, 1);
361  H[j*kdim+i] /= R[j*kdimp+j];
362  }
363  }
364 
365  H[(kdim-1)*kdim+kdim] = alpha[kdim]
366  * std::fabs(R[kdim*kdimp+kdim] / R[(kdim-1)*kdimp + kdim-1]);
367 
368  Lapack::dgeev_('N', 'V', kdim, &H[0], kdim, &wr[0], &wi[0], 0, 1,
369  &zvec[0], kdim, &rwork[0], lwork, ier);
370 
371  ASSERTL0(!ier, "Error with dgeev");
372 
373  resnorm = H[(kdim-1)*kdim + kdim];
374  }
375 
376 
377  /**
378  *
379  */
380  int DriverModifiedArnoldi::EV_test(
381  const int itrn,
382  const int kdim,
383  Array<OneD, NekDouble> &zvec,
384  Array<OneD, NekDouble> &wr,
385  Array<OneD, NekDouble> &wi,
386  const NekDouble resnorm,
387  const int nvec,
388  ofstream &evlout,
389  NekDouble &resid0)
390  {
391  int idone = 0;
392  // NekDouble period = 0.1;
393 
394  Array<OneD, NekDouble> resid(kdim);
395  for (int i = 0; i < kdim; ++i)
396  {
397  NekDouble tmp = std::sqrt(Vmath::Dot(kdim, &zvec[0] + i*kdim, 1,
398  &zvec[0] + i*kdim, 1));
399  resid[i] = resnorm * std::fabs(zvec[kdim - 1 + i*kdim]) / tmp;
400  if (wi[i] < 0.0)
401  {
402  resid[i-1] = resid[i] = hypot(resid[i-1], resid[i]);
403  }
404  }
405 
406  EV_sort(zvec, wr, wi, resid, kdim);
407 
408  if (resid[nvec-1] < m_evtol)
409  {
410  idone = nvec;
411  }
412 
413  if (m_comm->GetRank() == 0)
414  {
415  evlout << "-- Iteration = " << itrn << ", H(k+1, k) = "
416  << resnorm << endl;
417  evlout.precision(4);
418  evlout.setf(ios::scientific, ios::floatfield);
419  if(m_timeSteppingAlgorithm)
420  {
421  evlout << "EV Magnitude Angle Growth "
422  << "Frequency Residual" << endl;
423  }
424  else
425  {
426  evlout << "EV Real Imaginary inverse real "
427  << "inverse imag Residual" << endl;
428  }
429 
430  for (int i = 0; i < kdim; i++)
431  {
432  WriteEvs(evlout,i,wr[i],wi[i],resid[i]);
433  }
434  }
435 
436  resid0 = resid[nvec-1];
437  return idone;
438  }
439 
440 
441  /**
442  *
443  */
444  void DriverModifiedArnoldi::EV_sort(
445  Array<OneD, NekDouble> &evec,
446  Array<OneD, NekDouble> &wr,
447  Array<OneD, NekDouble> &wi,
448  Array<OneD, NekDouble> &test,
449  const int dim)
450  {
451  Array<OneD, NekDouble> z_tmp(dim,0.0);
452  NekDouble wr_tmp, wi_tmp, te_tmp;
453  for (int j = 1; j < dim; ++j)
454  {
455  wr_tmp = wr[j];
456  wi_tmp = wi[j];
457  te_tmp = test[j];
458  Vmath::Vcopy(dim, &evec[0] + j*dim, 1, &z_tmp[0], 1);
459  int i = j - 1;
460  while (i >= 0 && test[i] > te_tmp)
461  {
462  wr[i+1] = wr[i];
463  wi[i+1] = wi[i];
464  test[i+1] = test[i];
465  Vmath::Vcopy(dim, &evec[0] + i*dim, 1, &evec[0] + (i+1)*dim, 1);
466  i--;
467  }
468  wr[i+1] = wr_tmp;
469  wi[i+1] = wi_tmp;
470  test[i+1] = te_tmp;
471  Vmath::Vcopy(dim, &z_tmp[0], 1, &evec[0] + (i+1)*dim, 1);
472  }
473  }
474 
475 
476  /**
477  *
478  */
479  void DriverModifiedArnoldi::EV_post(
480  Array<OneD, Array<OneD, NekDouble> > &Tseq,
481  Array<OneD, Array<OneD, NekDouble> > &Kseq,
482  const int ntot,
483  const int kdim,
484  const int nvec,
485  Array<OneD, NekDouble> &zvec,
486  Array<OneD, NekDouble> &wr,
487  Array<OneD, NekDouble> &wi,
488  const int icon)
489  {
490  if (icon == 0)
491  {
492  // Not converged, write final Krylov vector
493  ASSERTL0(false, "Convergence was not achieved within the "
494  "prescribed number of iterations.");
495  }
496  else if (icon < 0)
497  {
498  // Minimum residual reached
499  ASSERTL0(false, "Minimum residual reached.");
500  }
501  else if (icon == nvec)
502  {
503  // Converged, write out eigenvectors
504  EV_big(Tseq, Kseq, ntot, kdim, icon, zvec, wr, wi);
505  Array<OneD, MultiRegions::ExpListSharedPtr> fields
506  = m_equ[0]->UpdateFields();
507 
508  for (int j = 0; j < icon; ++j)
509  {
510  std::string file = m_session->GetSessionName() + "_eig_"
511  + boost::lexical_cast<std::string>(j);
512 
513  WriteFld(file,Kseq[j]);
514  }
515  }
516  else
517  {
518  // Not recognised
519  ASSERTL0(false, "Unrecognised value.");
520  }
521  }
522 
523 
524  /**
525  *
526  */
527  void DriverModifiedArnoldi::EV_big(
528  Array<OneD, Array<OneD, NekDouble> > &bvecs,
529  Array<OneD, Array<OneD, NekDouble> > &evecs,
530  const int ntot,
531  const int kdim,
532  const int nvec,
533  Array<OneD, NekDouble> &zvec,
534  Array<OneD, NekDouble> &wr,
535  Array<OneD, NekDouble> &wi)
536  {
537  NekDouble wgt, norm;
538 
539  // Generate big eigenvectors
540  for (int j = 0; j < nvec; ++j)
541  {
542  Vmath::Zero(ntot, evecs[j], 1);
543  for (int i = 0; i < kdim; ++i)
544  {
545  wgt = zvec[i + j*kdim];
546  Vmath::Svtvp(ntot, wgt, bvecs[i], 1, evecs[j], 1, evecs[j], 1);
547  }
548  }
549 
550  // Normalise the big eigenvectors
551  for (int i = 0; i < nvec; ++i)
552  {
553  if (wi[i] == 0.0) // Real mode
554  {
555  norm = Blas::Ddot(ntot, &evecs[i][0], 1, &evecs[i][0], 1);
556  m_comm->AllReduce(norm, Nektar::LibUtilities::ReduceSum);
557  norm = std::sqrt(norm);
558  Vmath::Smul(ntot, 1.0/norm, evecs[i], 1, evecs[i], 1);
559  }
560  else
561  {
562  norm = Blas::Ddot(ntot, &evecs[i][0], 1, &evecs[i][0], 1);
563  norm += Blas::Ddot(ntot, &evecs[i+1][0], 1, &evecs[i+1][0], 1);
564  m_comm->AllReduce(norm, Nektar::LibUtilities::ReduceSum);
565  norm = std::sqrt(norm);
566 
567  Vmath::Smul(ntot, 1.0/norm, evecs[i], 1, evecs[i], 1);
568  Vmath::Smul(ntot, 1.0/norm, evecs[i+1], 1, evecs[i+1], 1);
569 
570  i++;
571  }
572  }
573  }
574 
575  }
576 }
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