static char help[] = "Performs adjoint sensitivity analysis for the van der Pol equation.\n\ Input parameters include:\n\ -mu : stiffness parameter\n\n"; /* Concepts: TS^time-dependent nonlinear problems Concepts: TS^van der Pol equation Concepts: TS^adjoint sensitivity analysis Processors: 1 */ /* ------------------------------------------------------------------------ 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,void *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(0); } static PetscErrorCode RHSJacobian(TS ts,PetscReal t,Vec X,Mat A,Mat B,void *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(0); } static PetscErrorCode RHSJacobianP(TS ts,PetscReal t,Vec X,Mat A,void *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(0); } /* Monitor timesteps and use interpolation to output at integer multiples of 0.1 */ static PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal t,Vec X,void *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] %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]))); PetscCall(PetscPrintf(PETSC_COMM_WORLD,"t %.6f (tprev = %.6f) \n",(double)t,(double)tprev)); PetscCall(VecRestoreArrayRead(X,&x)); PetscFunctionReturn(0); } 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 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 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 %D, 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*/