1 static char help[] = "Performs adjoint sensitivity analysis for the van der Pol equation.\n\ 2 Input parameters include:\n\ 3 -mu : stiffness parameter\n\n"; 4 5 /* 6 Concepts: TS^time-dependent nonlinear problems 7 Concepts: TS^van der Pol equation 8 Concepts: TS^adjoint sensitivity analysis 9 Processors: 1 10 */ 11 /* ------------------------------------------------------------------------ 12 13 This program solves the van der Pol equation 14 y'' - \mu (1-y^2)*y' + y = 0 (1) 15 on the domain 0 <= x <= 1, with the boundary conditions 16 y(0) = 2, y'(0) = 0, 17 and computes the sensitivities of the final solution w.r.t. initial conditions and parameter \mu with an explicit Runge-Kutta method and its discrete tangent linear model. 18 19 Notes: 20 This code demonstrates the TSForward interface to a system of ordinary differential equations (ODEs) in the form of u_t = F(u,t). 21 22 (1) can be turned into a system of first order ODEs 23 [ y' ] = [ z ] 24 [ z' ] [ \mu (1 - y^2) z - y ] 25 26 which then we can write as a vector equation 27 28 [ u_1' ] = [ u_2 ] (2) 29 [ u_2' ] [ \mu (1 - u_1^2) u_2 - u_1 ] 30 31 which is now in the form of u_t = F(u,t). 32 33 The user provides the right-hand-side function 34 35 [ F(u,t) ] = [ u_2 ] 36 [ \mu (1 - u_1^2) u_2 - u_1 ] 37 38 the Jacobian function 39 40 dF [ 0 ; 1 ] 41 -- = [ ] 42 du [ -2 \mu u_1*u_2 - 1; \mu (1 - u_1^2) ] 43 44 and the JacobainP (the Jacobian w.r.t. parameter) function 45 46 dF [ 0; 0; 0 ] 47 --- = [ ] 48 d\mu [ 0; 0; (1 - u_1^2) u_2 ] 49 50 51 ------------------------------------------------------------------------- */ 52 53 #include <petscts.h> 54 #include <petscmat.h> 55 typedef struct _n_User *User; 56 struct _n_User { 57 PetscReal mu; 58 PetscReal next_output; 59 PetscReal tprev; 60 }; 61 62 /* 63 * User-defined routines 64 */ 65 static PetscErrorCode RHSFunction(TS ts,PetscReal t,Vec X,Vec F,void *ctx) 66 { 67 PetscErrorCode ierr; 68 User user = (User)ctx; 69 PetscScalar *f; 70 const PetscScalar *x; 71 72 PetscFunctionBeginUser; 73 ierr = VecGetArrayRead(X,&x);CHKERRQ(ierr); 74 ierr = VecGetArray(F,&f);CHKERRQ(ierr); 75 f[0] = x[1]; 76 f[1] = user->mu*(1.-x[0]*x[0])*x[1]-x[0]; 77 ierr = VecRestoreArrayRead(X,&x);CHKERRQ(ierr); 78 ierr = VecRestoreArray(F,&f);CHKERRQ(ierr); 79 PetscFunctionReturn(0); 80 } 81 82 static PetscErrorCode RHSJacobian(TS ts,PetscReal t,Vec X,Mat A,Mat B,void *ctx) 83 { 84 PetscErrorCode ierr; 85 User user = (User)ctx; 86 PetscReal mu = user->mu; 87 PetscInt rowcol[] = {0,1}; 88 PetscScalar J[2][2]; 89 const PetscScalar *x; 90 91 PetscFunctionBeginUser; 92 ierr = VecGetArrayRead(X,&x);CHKERRQ(ierr); 93 J[0][0] = 0; 94 J[1][0] = -2.*mu*x[1]*x[0]-1.; 95 J[0][1] = 1.0; 96 J[1][1] = mu*(1.0-x[0]*x[0]); 97 ierr = MatSetValues(A,2,rowcol,2,rowcol,&J[0][0],INSERT_VALUES);CHKERRQ(ierr); 98 ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr); 99 ierr = MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr); 100 if (A != B) { 101 ierr = MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr); 102 ierr = MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr); 103 } 104 ierr = VecRestoreArrayRead(X,&x);CHKERRQ(ierr); 105 PetscFunctionReturn(0); 106 } 107 108 static PetscErrorCode RHSJacobianP(TS ts,PetscReal t,Vec X,Mat A,void *ctx) 109 { 110 PetscErrorCode ierr; 111 PetscInt row[] = {0,1},col[]={2}; 112 PetscScalar J[2][1]; 113 const PetscScalar *x; 114 115 PetscFunctionBeginUser; 116 ierr = VecGetArrayRead(X,&x);CHKERRQ(ierr); 117 J[0][0] = 0; 118 J[1][0] = (1.-x[0]*x[0])*x[1]; 119 ierr = VecRestoreArrayRead(X,&x);CHKERRQ(ierr); 120 ierr = MatSetValues(A,2,row,1,col,&J[0][0],INSERT_VALUES);CHKERRQ(ierr); 121 122 ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr); 123 ierr = MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr); 124 PetscFunctionReturn(0); 125 } 126 127 /* Monitor timesteps and use interpolation to output at integer multiples of 0.1 */ 128 static PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal t,Vec X,void *ctx) 129 { 130 PetscErrorCode ierr; 131 const PetscScalar *x; 132 PetscReal tfinal, dt, tprev; 133 User user = (User)ctx; 134 135 PetscFunctionBeginUser; 136 ierr = TSGetTimeStep(ts,&dt);CHKERRQ(ierr); 137 ierr = TSGetMaxTime(ts,&tfinal);CHKERRQ(ierr); 138 ierr = TSGetPrevTime(ts,&tprev);CHKERRQ(ierr); 139 ierr = VecGetArrayRead(X,&x);CHKERRQ(ierr); 140 ierr = PetscPrintf(PETSC_COMM_WORLD,"[%.1f] %D TS %.6f (dt = %.6f) X % 12.6e % 12.6e\n",(double)user->next_output,step,(double)t,(double)dt,(double)PetscRealPart(x[0]),(double)PetscRealPart(x[1]));CHKERRQ(ierr); 141 ierr = PetscPrintf(PETSC_COMM_WORLD,"t %.6f (tprev = %.6f) \n",(double)t,(double)tprev);CHKERRQ(ierr); 142 ierr = VecRestoreArrayRead(X,&x);CHKERRQ(ierr); 143 PetscFunctionReturn(0); 144 } 145 146 int main(int argc,char **argv) 147 { 148 TS ts; /* nonlinear solver */ 149 Vec x; /* solution, residual vectors */ 150 Mat A; /* Jacobian matrix */ 151 Mat Jacp; /* JacobianP matrix */ 152 PetscInt steps; 153 PetscReal ftime =0.5; 154 PetscBool monitor = PETSC_FALSE; 155 PetscScalar *x_ptr; 156 PetscMPIInt size; 157 struct _n_User user; 158 PetscErrorCode ierr; 159 Mat sp; 160 161 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 162 Initialize program 163 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 164 ierr = PetscInitialize(&argc,&argv,NULL,help);if (ierr) return ierr; 165 ierr = MPI_Comm_size(PETSC_COMM_WORLD,&size);CHKERRQ(ierr); 166 if (size != 1) SETERRQ(PETSC_COMM_WORLD,PETSC_ERR_WRONG_MPI_SIZE,"This is a uniprocessor example only!"); 167 168 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 169 Set runtime options 170 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 171 user.mu = 1; 172 user.next_output = 0.0; 173 174 175 ierr = PetscOptionsGetReal(NULL,NULL,"-mu",&user.mu,NULL);CHKERRQ(ierr); 176 ierr = PetscOptionsGetBool(NULL,NULL,"-monitor",&monitor,NULL);CHKERRQ(ierr); 177 178 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 179 Create necessary matrix and vectors, solve same ODE on every process 180 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 181 ierr = MatCreate(PETSC_COMM_WORLD,&A);CHKERRQ(ierr); 182 ierr = MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,2,2);CHKERRQ(ierr); 183 ierr = MatSetFromOptions(A);CHKERRQ(ierr); 184 ierr = MatSetUp(A);CHKERRQ(ierr); 185 ierr = MatCreateVecs(A,&x,NULL);CHKERRQ(ierr); 186 187 ierr = MatCreate(PETSC_COMM_WORLD,&Jacp);CHKERRQ(ierr); 188 ierr = MatSetSizes(Jacp,PETSC_DECIDE,PETSC_DECIDE,2,3);CHKERRQ(ierr); 189 ierr = MatSetFromOptions(Jacp);CHKERRQ(ierr); 190 ierr = MatSetUp(Jacp);CHKERRQ(ierr); 191 192 ierr = MatCreateDense(PETSC_COMM_WORLD,PETSC_DECIDE,PETSC_DECIDE,2,3,NULL,&sp);CHKERRQ(ierr); 193 ierr = MatZeroEntries(sp);CHKERRQ(ierr); 194 ierr = MatShift(sp,1.0);CHKERRQ(ierr); 195 196 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 197 Create timestepping solver context 198 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 199 ierr = TSCreate(PETSC_COMM_WORLD,&ts);CHKERRQ(ierr); 200 ierr = TSSetType(ts,TSRK);CHKERRQ(ierr); 201 ierr = TSSetRHSFunction(ts,NULL,RHSFunction,&user);CHKERRQ(ierr); 202 /* Set RHS Jacobian for the adjoint integration */ 203 ierr = TSSetRHSJacobian(ts,A,A,RHSJacobian,&user);CHKERRQ(ierr); 204 ierr = TSSetMaxTime(ts,ftime);CHKERRQ(ierr); 205 ierr = TSSetExactFinalTime(ts,TS_EXACTFINALTIME_MATCHSTEP);CHKERRQ(ierr); 206 if (monitor) { 207 ierr = TSMonitorSet(ts,Monitor,&user,NULL);CHKERRQ(ierr); 208 } 209 ierr = TSForwardSetSensitivities(ts,3,sp);CHKERRQ(ierr); 210 ierr = TSSetRHSJacobianP(ts,Jacp,RHSJacobianP,&user);CHKERRQ(ierr); 211 212 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 213 Set initial conditions 214 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 215 ierr = VecGetArray(x,&x_ptr);CHKERRQ(ierr); 216 217 x_ptr[0] = 2; x_ptr[1] = 0.66666654321; 218 ierr = VecRestoreArray(x,&x_ptr);CHKERRQ(ierr); 219 ierr = TSSetTimeStep(ts,.001);CHKERRQ(ierr); 220 221 222 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 223 Set runtime options 224 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 225 ierr = TSSetFromOptions(ts);CHKERRQ(ierr); 226 227 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 228 Solve nonlinear system 229 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 230 ierr = TSSolve(ts,x);CHKERRQ(ierr); 231 ierr = TSGetSolveTime(ts,&ftime);CHKERRQ(ierr); 232 ierr = TSGetStepNumber(ts,&steps);CHKERRQ(ierr); 233 ierr = PetscPrintf(PETSC_COMM_WORLD,"mu %g, steps %D, ftime %g\n",(double)user.mu,steps,(double)ftime);CHKERRQ(ierr); 234 ierr = VecView(x,PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr); 235 236 ierr = PetscPrintf(PETSC_COMM_WORLD,"\n forward sensitivity: d[y(tf) z(tf)]/d[y0 z0 mu]\n");CHKERRQ(ierr); 237 ierr = MatView(sp,PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr); 238 239 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 240 Free work space. All PETSc objects should be destroyed when they 241 are no longer needed. 242 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 243 ierr = MatDestroy(&A);CHKERRQ(ierr); 244 ierr = MatDestroy(&Jacp);CHKERRQ(ierr); 245 ierr = VecDestroy(&x);CHKERRQ(ierr); 246 ierr = MatDestroy(&sp);CHKERRQ(ierr); 247 ierr = TSDestroy(&ts);CHKERRQ(ierr); 248 ierr = PetscFinalize(); 249 return ierr; 250 } 251 252 /*TEST 253 254 test: 255 args: -monitor 0 -ts_adapt_type none 256 257 TEST*/ 258