1 static char help[] = "Demonstrates adjoint sensitivity analysis for Reaction-Diffusion Equations.\n"; 2 3 /* 4 See ex5.c for details on the equation. 5 This code demonestrates the TSAdjoint/TAO interface to solve an inverse initial value problem built on a system of 6 time-dependent partial differential equations. 7 In this problem, the initial value for the PDE is unknown, but the output (the final solution of the PDE) is known. 8 We want to determine the initial value that can produce the given output. 9 We formulate the problem as a nonlinear optimization problem that minimizes the discrepency between the simulated 10 result and given reference solution, calculate the gradient of the objective function with the discrete adjoint 11 solver, and solve the optimization problem with TAO. 12 13 Runtime options: 14 -forwardonly - run only the forward simulation 15 -implicitform - provide IFunction and IJacobian to TS, if not set, RHSFunction and RHSJacobian will be used 16 */ 17 18 #include "reaction_diffusion.h" 19 #include <petscdm.h> 20 #include <petscdmda.h> 21 #include <petsctao.h> 22 23 /* User-defined routines */ 24 extern PetscErrorCode FormFunctionAndGradient(Tao,Vec,PetscReal*,Vec,void*); 25 26 /* 27 Set terminal condition for the adjoint variable 28 */ 29 PetscErrorCode InitializeLambda(DM da,Vec lambda,Vec U,AppCtx *appctx) 30 { 31 char filename[PETSC_MAX_PATH_LEN]=""; 32 PetscViewer viewer; 33 Vec Uob; 34 35 PetscFunctionBegin; 36 PetscCall(VecDuplicate(U,&Uob)); 37 PetscCall(PetscSNPrintf(filename,sizeof filename,"ex5opt.ob")); 38 PetscCall(PetscViewerBinaryOpen(PETSC_COMM_WORLD,filename,FILE_MODE_READ,&viewer)); 39 PetscCall(VecLoad(Uob,viewer)); 40 PetscCall(PetscViewerDestroy(&viewer)); 41 PetscCall(VecAYPX(Uob,-1.,U)); 42 PetscCall(VecScale(Uob,2.0)); 43 PetscCall(VecAXPY(lambda,1.,Uob)); 44 PetscCall(VecDestroy(&Uob)); 45 PetscFunctionReturn(0); 46 } 47 48 /* 49 Set up a viewer that dumps data into a binary file 50 */ 51 PetscErrorCode OutputBIN(DM da, const char *filename, PetscViewer *viewer) 52 { 53 PetscFunctionBegin; 54 PetscCall(PetscViewerCreate(PetscObjectComm((PetscObject)da),viewer)); 55 PetscCall(PetscViewerSetType(*viewer,PETSCVIEWERBINARY)); 56 PetscCall(PetscViewerFileSetMode(*viewer,FILE_MODE_WRITE)); 57 PetscCall(PetscViewerFileSetName(*viewer,filename)); 58 PetscFunctionReturn(0); 59 } 60 61 /* 62 Generate a reference solution and save it to a binary file 63 */ 64 PetscErrorCode GenerateOBs(TS ts,Vec U,AppCtx *appctx) 65 { 66 char filename[PETSC_MAX_PATH_LEN] = ""; 67 PetscViewer viewer; 68 DM da; 69 70 PetscFunctionBegin; 71 PetscCall(TSGetDM(ts,&da)); 72 PetscCall(TSSolve(ts,U)); 73 PetscCall(PetscSNPrintf(filename,sizeof filename,"ex5opt.ob")); 74 PetscCall(OutputBIN(da,filename,&viewer)); 75 PetscCall(VecView(U,viewer)); 76 PetscCall(PetscViewerDestroy(&viewer)); 77 PetscFunctionReturn(0); 78 } 79 80 PetscErrorCode InitialConditions(DM da,Vec U) 81 { 82 PetscInt i,j,xs,ys,xm,ym,Mx,My; 83 Field **u; 84 PetscReal hx,hy,x,y; 85 86 PetscFunctionBegin; 87 PetscCall(DMDAGetInfo(da,PETSC_IGNORE,&Mx,&My,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE)); 88 89 hx = 2.5/(PetscReal)Mx; 90 hy = 2.5/(PetscReal)My; 91 92 PetscCall(DMDAVecGetArray(da,U,&u)); 93 /* Get local grid boundaries */ 94 PetscCall(DMDAGetCorners(da,&xs,&ys,NULL,&xm,&ym,NULL)); 95 96 /* Compute function over the locally owned part of the grid */ 97 for (j=ys; j<ys+ym; j++) { 98 y = j*hy; 99 for (i=xs; i<xs+xm; i++) { 100 x = i*hx; 101 if ((1.0 <= x) && (x <= 1.5) && (1.0 <= y) && (y <= 1.5)) u[j][i].v = .25*PetscPowReal(PetscSinReal(4.0*PETSC_PI*x),2.0)*PetscPowReal(PetscSinReal(4.0*PETSC_PI*y),2.0); 102 else u[j][i].v = 0.0; 103 104 u[j][i].u = 1.0 - 2.0*u[j][i].v; 105 } 106 } 107 108 PetscCall(DMDAVecRestoreArray(da,U,&u)); 109 PetscFunctionReturn(0); 110 } 111 112 PetscErrorCode PerturbedInitialConditions(DM da,Vec U) 113 { 114 PetscInt i,j,xs,ys,xm,ym,Mx,My; 115 Field **u; 116 PetscReal hx,hy,x,y; 117 118 PetscFunctionBegin; 119 PetscCall(DMDAGetInfo(da,PETSC_IGNORE,&Mx,&My,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE)); 120 121 hx = 2.5/(PetscReal)Mx; 122 hy = 2.5/(PetscReal)My; 123 124 PetscCall(DMDAVecGetArray(da,U,&u)); 125 /* Get local grid boundaries */ 126 PetscCall(DMDAGetCorners(da,&xs,&ys,NULL,&xm,&ym,NULL)); 127 128 /* Compute function over the locally owned part of the grid */ 129 for (j=ys; j<ys+ym; j++) { 130 y = j*hy; 131 for (i=xs; i<xs+xm; i++) { 132 x = i*hx; 133 if ((1.0 <= x) && (x <= 1.5) && (1.0 <= y) && (y <= 1.5)) u[j][i].v = .25*PetscPowReal(PetscSinReal(4.0*PETSC_PI*0.5*x),2.0)*PetscPowReal(PetscSinReal(4.0*PETSC_PI*0.5*y),2.0); 134 else u[j][i].v = 0.0; 135 136 u[j][i].u = 1.0 - 2.0*u[j][i].v; 137 } 138 } 139 140 PetscCall(DMDAVecRestoreArray(da,U,&u)); 141 PetscFunctionReturn(0); 142 } 143 144 PetscErrorCode PerturbedInitialConditions2(DM da,Vec U) 145 { 146 PetscInt i,j,xs,ys,xm,ym,Mx,My; 147 Field **u; 148 PetscReal hx,hy,x,y; 149 150 PetscFunctionBegin; 151 PetscCall(DMDAGetInfo(da,PETSC_IGNORE,&Mx,&My,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE)); 152 153 hx = 2.5/(PetscReal)Mx; 154 hy = 2.5/(PetscReal)My; 155 156 PetscCall(DMDAVecGetArray(da,U,&u)); 157 /* Get local grid boundaries */ 158 PetscCall(DMDAGetCorners(da,&xs,&ys,NULL,&xm,&ym,NULL)); 159 160 /* Compute function over the locally owned part of the grid */ 161 for (j=ys; j<ys+ym; j++) { 162 y = j*hy; 163 for (i=xs; i<xs+xm; i++) { 164 x = i*hx; 165 if (PetscApproximateGTE(x,1.0) && PetscApproximateLTE(x,1.5) && PetscApproximateGTE(y,1.0) && PetscApproximateLTE(y,1.5)) u[j][i].v = PetscPowReal(PetscSinReal(4.0*PETSC_PI*(x-0.5)),2.0)*PetscPowReal(PetscSinReal(4.0*PETSC_PI*y),2.0)/4.0; 166 else u[j][i].v = 0.0; 167 168 u[j][i].u = 1.0 - 2.0*u[j][i].v; 169 } 170 } 171 172 PetscCall(DMDAVecRestoreArray(da,U,&u)); 173 PetscFunctionReturn(0); 174 } 175 176 PetscErrorCode PerturbedInitialConditions3(DM da,Vec U) 177 { 178 PetscInt i,j,xs,ys,xm,ym,Mx,My; 179 Field **u; 180 PetscReal hx,hy,x,y; 181 182 PetscFunctionBegin; 183 PetscCall(DMDAGetInfo(da,PETSC_IGNORE,&Mx,&My,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE)); 184 185 hx = 2.5/(PetscReal)Mx; 186 hy = 2.5/(PetscReal)My; 187 188 PetscCall(DMDAVecGetArray(da,U,&u)); 189 /* Get local grid boundaries */ 190 PetscCall(DMDAGetCorners(da,&xs,&ys,NULL,&xm,&ym,NULL)); 191 192 /* Compute function over the locally owned part of the grid */ 193 for (j=ys; j<ys+ym; j++) { 194 y = j*hy; 195 for (i=xs; i<xs+xm; i++) { 196 x = i*hx; 197 if ((0.5 <= x) && (x <= 2.0) && (0.5 <= y) && (y <= 2.0)) u[j][i].v = .25*PetscPowReal(PetscSinReal(4.0*PETSC_PI*x),2.0)*PetscPowReal(PetscSinReal(4.0*PETSC_PI*y),2.0); 198 else u[j][i].v = 0.0; 199 200 u[j][i].u = 1.0 - 2.0*u[j][i].v; 201 } 202 } 203 204 PetscCall(DMDAVecRestoreArray(da,U,&u)); 205 PetscFunctionReturn(0); 206 } 207 208 int main(int argc,char **argv) 209 { 210 DM da; 211 AppCtx appctx; 212 PetscBool forwardonly = PETSC_FALSE,implicitform = PETSC_FALSE; 213 PetscInt perturbic = 1; 214 215 PetscFunctionBeginUser; 216 PetscCall(PetscInitialize(&argc,&argv,(char*)0,help)); 217 PetscCall(PetscOptionsGetBool(NULL,NULL,"-forwardonly",&forwardonly,NULL)); 218 PetscCall(PetscOptionsGetBool(NULL,NULL,"-implicitform",&implicitform,NULL)); 219 PetscCall(PetscOptionsGetInt(NULL,NULL,"-perturbic",&perturbic,NULL)); 220 221 appctx.D1 = 8.0e-5; 222 appctx.D2 = 4.0e-5; 223 appctx.gamma = .024; 224 appctx.kappa = .06; 225 appctx.aijpc = PETSC_FALSE; 226 227 /* Create distributed array (DMDA) to manage parallel grid and vectors */ 228 PetscCall(DMDACreate2d(PETSC_COMM_WORLD,DM_BOUNDARY_PERIODIC,DM_BOUNDARY_PERIODIC,DMDA_STENCIL_STAR,64,64,PETSC_DECIDE,PETSC_DECIDE,2,1,NULL,NULL,&da)); 229 PetscCall(DMSetFromOptions(da)); 230 PetscCall(DMSetUp(da)); 231 PetscCall(DMDASetFieldName(da,0,"u")); 232 PetscCall(DMDASetFieldName(da,1,"v")); 233 234 /* Extract global vectors from DMDA; then duplicate for remaining 235 vectors that are the same types */ 236 PetscCall(DMCreateGlobalVector(da,&appctx.U)); 237 238 /* Create timestepping solver context */ 239 PetscCall(TSCreate(PETSC_COMM_WORLD,&appctx.ts)); 240 PetscCall(TSSetType(appctx.ts,TSCN)); 241 PetscCall(TSSetDM(appctx.ts,da)); 242 PetscCall(TSSetProblemType(appctx.ts,TS_NONLINEAR)); 243 PetscCall(TSSetEquationType(appctx.ts,TS_EQ_ODE_EXPLICIT)); /* less Jacobian evaluations when adjoint BEuler is used, otherwise no effect */ 244 if (!implicitform) { 245 PetscCall(TSSetRHSFunction(appctx.ts,NULL,RHSFunction,&appctx)); 246 PetscCall(TSSetRHSJacobian(appctx.ts,NULL,NULL,RHSJacobian,&appctx)); 247 } else { 248 PetscCall(TSSetIFunction(appctx.ts,NULL,IFunction,&appctx)); 249 PetscCall(TSSetIJacobian(appctx.ts,NULL,NULL,IJacobian,&appctx)); 250 } 251 252 /* Set initial conditions */ 253 PetscCall(InitialConditions(da,appctx.U)); 254 PetscCall(TSSetSolution(appctx.ts,appctx.U)); 255 256 /* Set solver options */ 257 PetscCall(TSSetMaxTime(appctx.ts,2000.0)); 258 PetscCall(TSSetTimeStep(appctx.ts,0.5)); 259 PetscCall(TSSetExactFinalTime(appctx.ts,TS_EXACTFINALTIME_MATCHSTEP)); 260 PetscCall(TSSetFromOptions(appctx.ts)); 261 262 PetscCall(GenerateOBs(appctx.ts,appctx.U,&appctx)); 263 264 if (!forwardonly) { 265 Tao tao; 266 Vec P; 267 Vec lambda[1]; 268 #if defined(PETSC_USE_LOG) 269 PetscLogStage opt_stage; 270 #endif 271 272 PetscCall(PetscLogStageRegister("Optimization",&opt_stage)); 273 PetscCall(PetscLogStagePush(opt_stage)); 274 if (perturbic == 1) { 275 PetscCall(PerturbedInitialConditions(da,appctx.U)); 276 } else if (perturbic == 2) { 277 PetscCall(PerturbedInitialConditions2(da,appctx.U)); 278 } else if (perturbic == 3) { 279 PetscCall(PerturbedInitialConditions3(da,appctx.U)); 280 } 281 282 PetscCall(VecDuplicate(appctx.U,&lambda[0])); 283 PetscCall(TSSetCostGradients(appctx.ts,1,lambda,NULL)); 284 285 /* Have the TS save its trajectory needed by TSAdjointSolve() */ 286 PetscCall(TSSetSaveTrajectory(appctx.ts)); 287 288 /* Create TAO solver and set desired solution method */ 289 PetscCall(TaoCreate(PETSC_COMM_WORLD,&tao)); 290 PetscCall(TaoSetType(tao,TAOBLMVM)); 291 292 /* Set initial guess for TAO */ 293 PetscCall(VecDuplicate(appctx.U,&P)); 294 PetscCall(VecCopy(appctx.U,P)); 295 PetscCall(TaoSetSolution(tao,P)); 296 297 /* Set routine for function and gradient evaluation */ 298 PetscCall(TaoSetObjectiveAndGradient(tao,NULL,FormFunctionAndGradient,&appctx)); 299 300 /* Check for any TAO command line options */ 301 PetscCall(TaoSetFromOptions(tao)); 302 303 PetscCall(TaoSolve(tao)); 304 PetscCall(TaoDestroy(&tao)); 305 PetscCall(VecDestroy(&lambda[0])); 306 PetscCall(VecDestroy(&P)); 307 PetscCall(PetscLogStagePop()); 308 } 309 310 /* Free work space. All PETSc objects should be destroyed when they 311 are no longer needed. */ 312 PetscCall(VecDestroy(&appctx.U)); 313 PetscCall(TSDestroy(&appctx.ts)); 314 PetscCall(DMDestroy(&da)); 315 PetscCall(PetscFinalize()); 316 return 0; 317 } 318 319 /* ------------------------ TAO callbacks ---------------------------- */ 320 321 /* 322 FormFunctionAndGradient - Evaluates the function and corresponding gradient. 323 324 Input Parameters: 325 tao - the Tao context 326 P - the input vector 327 ctx - optional user-defined context, as set by TaoSetObjectiveAndGradient() 328 329 Output Parameters: 330 f - the newly evaluated function 331 G - the newly evaluated gradient 332 */ 333 PetscErrorCode FormFunctionAndGradient(Tao tao,Vec P,PetscReal *f,Vec G,void *ctx) 334 { 335 AppCtx *appctx = (AppCtx*)ctx; 336 PetscReal soberr,timestep; 337 Vec *lambda; 338 Vec SDiff; 339 DM da; 340 char filename[PETSC_MAX_PATH_LEN]=""; 341 PetscViewer viewer; 342 343 PetscFunctionBeginUser; 344 PetscCall(TSSetTime(appctx->ts,0.0)); 345 PetscCall(TSGetTimeStep(appctx->ts,×tep)); 346 if (timestep<0) { 347 PetscCall(TSSetTimeStep(appctx->ts,-timestep)); 348 } 349 PetscCall(TSSetStepNumber(appctx->ts,0)); 350 PetscCall(TSSetFromOptions(appctx->ts)); 351 352 PetscCall(VecDuplicate(P,&SDiff)); 353 PetscCall(VecCopy(P,appctx->U)); 354 PetscCall(TSGetDM(appctx->ts,&da)); 355 *f = 0; 356 357 PetscCall(TSSolve(appctx->ts,appctx->U)); 358 PetscCall(PetscSNPrintf(filename,sizeof filename,"ex5opt.ob")); 359 PetscCall(PetscViewerBinaryOpen(PETSC_COMM_WORLD,filename,FILE_MODE_READ,&viewer)); 360 PetscCall(VecLoad(SDiff,viewer)); 361 PetscCall(PetscViewerDestroy(&viewer)); 362 PetscCall(VecAYPX(SDiff,-1.,appctx->U)); 363 PetscCall(VecDot(SDiff,SDiff,&soberr)); 364 *f += soberr; 365 366 PetscCall(TSGetCostGradients(appctx->ts,NULL,&lambda,NULL)); 367 PetscCall(VecSet(lambda[0],0.0)); 368 PetscCall(InitializeLambda(da,lambda[0],appctx->U,appctx)); 369 PetscCall(TSAdjointSolve(appctx->ts)); 370 371 PetscCall(VecCopy(lambda[0],G)); 372 373 PetscCall(VecDestroy(&SDiff)); 374 PetscFunctionReturn(0); 375 } 376 377 /*TEST 378 379 build: 380 depends: reaction_diffusion.c 381 requires: !complex !single 382 383 test: 384 args: -ts_max_steps 5 -ts_type rk -ts_rk_type 3 -ts_trajectory_type memory -tao_monitor -tao_view -tao_gatol 1e-6 385 output_file: output/ex5opt_ic_1.out 386 387 TEST*/ 388