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