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 PetscCall(PetscInitialize(&argc,&argv,(char*)0,help)); 216 PetscCall(PetscOptionsGetBool(NULL,NULL,"-forwardonly",&forwardonly,NULL)); 217 PetscCall(PetscOptionsGetBool(NULL,NULL,"-implicitform",&implicitform,NULL)); 218 PetscCall(PetscOptionsGetInt(NULL,NULL,"-perturbic",&perturbic,NULL)); 219 220 appctx.D1 = 8.0e-5; 221 appctx.D2 = 4.0e-5; 222 appctx.gamma = .024; 223 appctx.kappa = .06; 224 appctx.aijpc = PETSC_FALSE; 225 226 /* Create distributed array (DMDA) to manage parallel grid and vectors */ 227 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)); 228 PetscCall(DMSetFromOptions(da)); 229 PetscCall(DMSetUp(da)); 230 PetscCall(DMDASetFieldName(da,0,"u")); 231 PetscCall(DMDASetFieldName(da,1,"v")); 232 233 /* Extract global vectors from DMDA; then duplicate for remaining 234 vectors that are the same types */ 235 PetscCall(DMCreateGlobalVector(da,&appctx.U)); 236 237 /* Create timestepping solver context */ 238 PetscCall(TSCreate(PETSC_COMM_WORLD,&appctx.ts)); 239 PetscCall(TSSetType(appctx.ts,TSCN)); 240 PetscCall(TSSetDM(appctx.ts,da)); 241 PetscCall(TSSetProblemType(appctx.ts,TS_NONLINEAR)); 242 PetscCall(TSSetEquationType(appctx.ts,TS_EQ_ODE_EXPLICIT)); /* less Jacobian evaluations when adjoint BEuler is used, otherwise no effect */ 243 if (!implicitform) { 244 PetscCall(TSSetRHSFunction(appctx.ts,NULL,RHSFunction,&appctx)); 245 PetscCall(TSSetRHSJacobian(appctx.ts,NULL,NULL,RHSJacobian,&appctx)); 246 } else { 247 PetscCall(TSSetIFunction(appctx.ts,NULL,IFunction,&appctx)); 248 PetscCall(TSSetIJacobian(appctx.ts,NULL,NULL,IJacobian,&appctx)); 249 } 250 251 /* Set initial conditions */ 252 PetscCall(InitialConditions(da,appctx.U)); 253 PetscCall(TSSetSolution(appctx.ts,appctx.U)); 254 255 /* Set solver options */ 256 PetscCall(TSSetMaxTime(appctx.ts,2000.0)); 257 PetscCall(TSSetTimeStep(appctx.ts,0.5)); 258 PetscCall(TSSetExactFinalTime(appctx.ts,TS_EXACTFINALTIME_MATCHSTEP)); 259 PetscCall(TSSetFromOptions(appctx.ts)); 260 261 PetscCall(GenerateOBs(appctx.ts,appctx.U,&appctx)); 262 263 if (!forwardonly) { 264 Tao tao; 265 Vec P; 266 Vec lambda[1]; 267 #if defined(PETSC_USE_LOG) 268 PetscLogStage opt_stage; 269 #endif 270 271 PetscCall(PetscLogStageRegister("Optimization",&opt_stage)); 272 PetscCall(PetscLogStagePush(opt_stage)); 273 if (perturbic == 1) { 274 PetscCall(PerturbedInitialConditions(da,appctx.U)); 275 } else if (perturbic == 2) { 276 PetscCall(PerturbedInitialConditions2(da,appctx.U)); 277 } else if (perturbic == 3) { 278 PetscCall(PerturbedInitialConditions3(da,appctx.U)); 279 } 280 281 PetscCall(VecDuplicate(appctx.U,&lambda[0])); 282 PetscCall(TSSetCostGradients(appctx.ts,1,lambda,NULL)); 283 284 /* Have the TS save its trajectory needed by TSAdjointSolve() */ 285 PetscCall(TSSetSaveTrajectory(appctx.ts)); 286 287 /* Create TAO solver and set desired solution method */ 288 PetscCall(TaoCreate(PETSC_COMM_WORLD,&tao)); 289 PetscCall(TaoSetType(tao,TAOBLMVM)); 290 291 /* Set initial guess for TAO */ 292 PetscCall(VecDuplicate(appctx.U,&P)); 293 PetscCall(VecCopy(appctx.U,P)); 294 PetscCall(TaoSetSolution(tao,P)); 295 296 /* Set routine for function and gradient evaluation */ 297 PetscCall(TaoSetObjectiveAndGradient(tao,NULL,FormFunctionAndGradient,&appctx)); 298 299 /* Check for any TAO command line options */ 300 PetscCall(TaoSetFromOptions(tao)); 301 302 PetscCall(TaoSolve(tao)); 303 PetscCall(TaoDestroy(&tao)); 304 PetscCall(VecDestroy(&lambda[0])); 305 PetscCall(VecDestroy(&P)); 306 PetscCall(PetscLogStagePop()); 307 } 308 309 /* Free work space. All PETSc objects should be destroyed when they 310 are no longer needed. */ 311 PetscCall(VecDestroy(&appctx.U)); 312 PetscCall(TSDestroy(&appctx.ts)); 313 PetscCall(DMDestroy(&da)); 314 PetscCall(PetscFinalize()); 315 return 0; 316 } 317 318 /* ------------------------ TAO callbacks ---------------------------- */ 319 320 /* 321 FormFunctionAndGradient - Evaluates the function and corresponding gradient. 322 323 Input Parameters: 324 tao - the Tao context 325 P - the input vector 326 ctx - optional user-defined context, as set by TaoSetObjectiveAndGradient() 327 328 Output Parameters: 329 f - the newly evaluated function 330 G - the newly evaluated gradient 331 */ 332 PetscErrorCode FormFunctionAndGradient(Tao tao,Vec P,PetscReal *f,Vec G,void *ctx) 333 { 334 AppCtx *appctx = (AppCtx*)ctx; 335 PetscReal soberr,timestep; 336 Vec *lambda; 337 Vec SDiff; 338 DM da; 339 char filename[PETSC_MAX_PATH_LEN]=""; 340 PetscViewer viewer; 341 342 PetscFunctionBeginUser; 343 PetscCall(TSSetTime(appctx->ts,0.0)); 344 PetscCall(TSGetTimeStep(appctx->ts,×tep)); 345 if (timestep<0) { 346 PetscCall(TSSetTimeStep(appctx->ts,-timestep)); 347 } 348 PetscCall(TSSetStepNumber(appctx->ts,0)); 349 PetscCall(TSSetFromOptions(appctx->ts)); 350 351 PetscCall(VecDuplicate(P,&SDiff)); 352 PetscCall(VecCopy(P,appctx->U)); 353 PetscCall(TSGetDM(appctx->ts,&da)); 354 *f = 0; 355 356 PetscCall(TSSolve(appctx->ts,appctx->U)); 357 PetscCall(PetscSNPrintf(filename,sizeof filename,"ex5opt.ob")); 358 PetscCall(PetscViewerBinaryOpen(PETSC_COMM_WORLD,filename,FILE_MODE_READ,&viewer)); 359 PetscCall(VecLoad(SDiff,viewer)); 360 PetscCall(PetscViewerDestroy(&viewer)); 361 PetscCall(VecAYPX(SDiff,-1.,appctx->U)); 362 PetscCall(VecDot(SDiff,SDiff,&soberr)); 363 *f += soberr; 364 365 PetscCall(TSGetCostGradients(appctx->ts,NULL,&lambda,NULL)); 366 PetscCall(VecSet(lambda[0],0.0)); 367 PetscCall(InitializeLambda(da,lambda[0],appctx->U,appctx)); 368 PetscCall(TSAdjointSolve(appctx->ts)); 369 370 PetscCall(VecCopy(lambda[0],G)); 371 372 PetscCall(VecDestroy(&SDiff)); 373 PetscFunctionReturn(0); 374 } 375 376 /*TEST 377 378 build: 379 depends: reaction_diffusion.c 380 requires: !complex !single 381 382 test: 383 args: -ts_max_steps 5 -ts_type rk -ts_rk_type 3 -ts_trajectory_type memory -tao_monitor -tao_view -tao_gatol 1e-6 384 output_file: output/ex5opt_ic_1.out 385 386 TEST*/ 387