static char help[] = "Performs adjoint sensitivity analysis for the van der Pol equation.\n\ Input parameters include:\n\ -mu : stiffness parameter\n\n"; /* ------------------------------------------------------------------------ This program solves the van der Pol equation y'' - \mu (1-y^2)*y' + y = 0 (1) on the domain 0 <= x <= 1, with the boundary conditions y(0) = 2, y'(0) = 0, 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. Notes: This code demonstrates the TSForward interface to a system of ordinary differential equations (ODEs) in the form of u_t = f(u,t). (1) can be turned into a system of first order ODEs [ y' ] = [ z ] [ z' ] [ \mu (1 - y^2) z - y ] which then we can write as a vector equation [ u_1' ] = [ u_2 ] (2) [ u_2' ] [ \mu (1 - u_1^2) u_2 - u_1 ] which is now in the form of u_t = F(u,t). The user provides the right-hand-side function [ f(u,t) ] = [ u_2 ] [ \mu (1 - u_1^2) u_2 - u_1 ] the Jacobian function df [ 0 ; 1 ] -- = [ ] du [ -2 \mu u_1*u_2 - 1; \mu (1 - u_1^2) ] and the JacobainP (the Jacobian w.r.t. parameter) function df [ 0; 0; 0 ] --- = [ ] d\mu [ 0; 0; (1 - u_1^2) u_2 ] ------------------------------------------------------------------------- */ #include #include typedef struct _n_User *User; struct _n_User { PetscReal mu; PetscReal next_output; PetscReal tprev; }; /* User-defined routines */ static PetscErrorCode RHSFunction(TS ts, PetscReal t, Vec X, Vec F, PetscCtx ctx) { User user = (User)ctx; PetscScalar *f; const PetscScalar *x; PetscFunctionBeginUser; PetscCall(VecGetArrayRead(X, &x)); PetscCall(VecGetArray(F, &f)); f[0] = x[1]; f[1] = user->mu * (1. - x[0] * x[0]) * x[1] - x[0]; PetscCall(VecRestoreArrayRead(X, &x)); PetscCall(VecRestoreArray(F, &f)); PetscFunctionReturn(PETSC_SUCCESS); } static PetscErrorCode RHSJacobian(TS ts, PetscReal t, Vec X, Mat A, Mat B, PetscCtx ctx) { User user = (User)ctx; PetscReal mu = user->mu; PetscInt rowcol[] = {0, 1}; PetscScalar J[2][2]; const PetscScalar *x; PetscFunctionBeginUser; PetscCall(VecGetArrayRead(X, &x)); J[0][0] = 0; J[1][0] = -2. * mu * x[1] * x[0] - 1.; J[0][1] = 1.0; J[1][1] = mu * (1.0 - x[0] * x[0]); PetscCall(MatSetValues(A, 2, rowcol, 2, rowcol, &J[0][0], INSERT_VALUES)); PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY)); PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY)); if (A != B) { PetscCall(MatAssemblyBegin(B, MAT_FINAL_ASSEMBLY)); PetscCall(MatAssemblyEnd(B, MAT_FINAL_ASSEMBLY)); } PetscCall(VecRestoreArrayRead(X, &x)); PetscFunctionReturn(PETSC_SUCCESS); } static PetscErrorCode RHSJacobianP(TS ts, PetscReal t, Vec X, Mat A, PetscCtx ctx) { PetscInt row[] = {0, 1}, col[] = {2}; PetscScalar J[2][1]; const PetscScalar *x; PetscFunctionBeginUser; PetscCall(VecGetArrayRead(X, &x)); J[0][0] = 0; J[1][0] = (1. - x[0] * x[0]) * x[1]; PetscCall(VecRestoreArrayRead(X, &x)); PetscCall(MatSetValues(A, 2, row, 1, col, &J[0][0], INSERT_VALUES)); PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY)); PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY)); PetscFunctionReturn(PETSC_SUCCESS); } /* Monitor timesteps and use interpolation to output at integer multiples of 0.1 */ static PetscErrorCode Monitor(TS ts, PetscInt step, PetscReal t, Vec X, PetscCtx ctx) { const PetscScalar *x; PetscReal tfinal, dt, tprev; User user = (User)ctx; PetscFunctionBeginUser; PetscCall(TSGetTimeStep(ts, &dt)); PetscCall(TSGetMaxTime(ts, &tfinal)); PetscCall(TSGetPrevTime(ts, &tprev)); PetscCall(VecGetArrayRead(X, &x)); 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]))); PetscCall(PetscPrintf(PETSC_COMM_WORLD, "t %.6f (tprev = %.6f) \n", (double)t, (double)tprev)); PetscCall(VecRestoreArrayRead(X, &x)); PetscFunctionReturn(PETSC_SUCCESS); } int main(int argc, char **argv) { TS ts; /* nonlinear solver */ Vec x; /* solution, residual vectors */ Mat A; /* Jacobian matrix */ Mat Jacp; /* JacobianP matrix */ PetscInt steps; PetscReal ftime = 0.5; PetscBool monitor = PETSC_FALSE; PetscScalar *x_ptr; PetscMPIInt size; struct _n_User user; Mat sp; /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Initialize program - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ PetscFunctionBeginUser; PetscCall(PetscInitialize(&argc, &argv, NULL, help)); PetscCallMPI(MPI_Comm_size(PETSC_COMM_WORLD, &size)); PetscCheck(size == 1, PETSC_COMM_WORLD, PETSC_ERR_WRONG_MPI_SIZE, "This is a uniprocessor example only!"); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Set runtime options - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ user.mu = 1; user.next_output = 0.0; PetscCall(PetscOptionsGetReal(NULL, NULL, "-mu", &user.mu, NULL)); PetscCall(PetscOptionsGetBool(NULL, NULL, "-monitor", &monitor, NULL)); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Create necessary matrix and vectors, solve same ODE on every process - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ PetscCall(MatCreate(PETSC_COMM_WORLD, &A)); PetscCall(MatSetSizes(A, PETSC_DECIDE, PETSC_DECIDE, 2, 2)); PetscCall(MatSetFromOptions(A)); PetscCall(MatSetUp(A)); PetscCall(MatCreateVecs(A, &x, NULL)); PetscCall(MatCreate(PETSC_COMM_WORLD, &Jacp)); PetscCall(MatSetSizes(Jacp, PETSC_DECIDE, PETSC_DECIDE, 2, 3)); PetscCall(MatSetFromOptions(Jacp)); PetscCall(MatSetUp(Jacp)); PetscCall(MatCreateDense(PETSC_COMM_WORLD, PETSC_DECIDE, PETSC_DECIDE, 2, 3, NULL, &sp)); PetscCall(MatZeroEntries(sp)); PetscCall(MatShift(sp, 1.0)); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Create timestepping solver context - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ PetscCall(TSCreate(PETSC_COMM_WORLD, &ts)); PetscCall(TSSetType(ts, TSRK)); PetscCall(TSSetRHSFunction(ts, NULL, RHSFunction, &user)); /* Set RHS Jacobian for the adjoint integration */ PetscCall(TSSetRHSJacobian(ts, A, A, RHSJacobian, &user)); PetscCall(TSSetMaxTime(ts, ftime)); PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_MATCHSTEP)); if (monitor) PetscCall(TSMonitorSet(ts, Monitor, &user, NULL)); PetscCall(TSForwardSetSensitivities(ts, 3, sp)); PetscCall(TSSetRHSJacobianP(ts, Jacp, RHSJacobianP, &user)); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Set initial conditions - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ PetscCall(VecGetArray(x, &x_ptr)); x_ptr[0] = 2; x_ptr[1] = 0.66666654321; PetscCall(VecRestoreArray(x, &x_ptr)); PetscCall(TSSetTimeStep(ts, .001)); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Set runtime options - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ PetscCall(TSSetFromOptions(ts)); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Solve nonlinear system - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ PetscCall(TSSolve(ts, x)); PetscCall(TSGetSolveTime(ts, &ftime)); PetscCall(TSGetStepNumber(ts, &steps)); PetscCall(PetscPrintf(PETSC_COMM_WORLD, "mu %g, steps %" PetscInt_FMT ", ftime %g\n", (double)user.mu, steps, (double)ftime)); PetscCall(VecView(x, PETSC_VIEWER_STDOUT_WORLD)); PetscCall(PetscPrintf(PETSC_COMM_WORLD, "\n forward sensitivity: d[y(tf) z(tf)]/d[y0 z0 mu]\n")); PetscCall(MatView(sp, PETSC_VIEWER_STDOUT_WORLD)); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Free work space. All PETSc objects should be destroyed when they are no longer needed. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ PetscCall(MatDestroy(&A)); PetscCall(MatDestroy(&Jacp)); PetscCall(VecDestroy(&x)); PetscCall(MatDestroy(&sp)); PetscCall(TSDestroy(&ts)); PetscCall(PetscFinalize()); return 0; } /*TEST test: args: -monitor 0 -ts_adapt_type none TEST*/