1 static const char help[] = "Attempts to solve for root of a function with multiple local minima.\n\ 2 With the proper initial guess, a backtracking line-search fails because Newton's method gets\n\ 3 stuck in a local minimum. However, a critical point line-search or Newton's method without a\n\ 4 line search succeeds.\n"; 5 6 /* Solve 1D problem f(x) = 8 * exp(-4 * (x - 2)^2) * (x - 2) + 2 * x = 0 7 8 This problem is based on the example given here: https://scicomp.stackexchange.com/a/2446/24756 9 Originally an optimization problem to find the minimum of the function 10 11 g(x) = x^2 - exp(-4 * (x - 2)^2) 12 13 it has been reformulated to solve dg(x)/dx = f(x) = 0. The reformulated problem has several local 14 minima that can cause problems for some global Newton root-finding methods. In this particular 15 example, an initial guess of x0 = 2.5 generates an initial search direction (-df/dx is positive) 16 away from the root and towards a local minimum in which a back-tracking line search gets trapped. 17 However, omitting a line-search or using a critical point line search, the solve is successful. 18 19 The test outputs the final result for x and f(x). 20 21 Example usage: 22 23 Get help: 24 ./ex99 -help 25 26 Monitor run (with default back-tracking line search; solve fails): 27 ./ex99 -snes_converged_reason -snes_monitor -snes_linesearch_monitor -ksp_converged_reason -ksp_monitor 28 29 Run without a line search; solve succeeds: 30 ./ex99 -snes_linesearch_type basic 31 32 Run with a critical point line search; solve succeeds: 33 ./ex99 -snes_linesearch_type cp 34 */ 35 36 #include <math.h> 37 #include <petscsnes.h> 38 39 extern PetscErrorCode FormJacobian(SNES, Vec, Mat, Mat, void *); 40 extern PetscErrorCode FormFunction(SNES, Vec, Vec, void *); 41 42 int main(int argc, char **argv) 43 { 44 SNES snes; /* nonlinear solver context */ 45 KSP ksp; /* linear solver context */ 46 PC pc; /* preconditioner context */ 47 Vec x, r; /* solution, residual vectors */ 48 Mat J; /* Jacobian matrix */ 49 PetscMPIInt size; 50 51 PetscFunctionBeginUser; 52 PetscCall(PetscInitialize(&argc, &argv, (char *)0, help)); 53 PetscCallMPI(MPI_Comm_size(PETSC_COMM_WORLD, &size)); 54 PetscCheck(size == 1, PETSC_COMM_WORLD, PETSC_ERR_WRONG_MPI_SIZE, "Example is only for sequential runs"); 55 56 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 57 Create nonlinear solver context 58 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 59 PetscCall(SNESCreate(PETSC_COMM_WORLD, &snes)); 60 61 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 62 Create matrix and vector data structures; set corresponding routines 63 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 64 /* 65 Create vectors for solution and nonlinear function 66 */ 67 PetscCall(VecCreate(PETSC_COMM_WORLD, &x)); 68 PetscCall(VecSetSizes(x, PETSC_DECIDE, 1)); 69 PetscCall(VecSetFromOptions(x)); 70 PetscCall(VecDuplicate(x, &r)); 71 72 /* 73 Create Jacobian matrix data structure 74 */ 75 PetscCall(MatCreate(PETSC_COMM_WORLD, &J)); 76 PetscCall(MatSetSizes(J, PETSC_DECIDE, PETSC_DECIDE, 1, 1)); 77 PetscCall(MatSetFromOptions(J)); 78 PetscCall(MatSetUp(J)); 79 80 PetscCall(SNESSetFunction(snes, r, FormFunction, NULL)); 81 PetscCall(SNESSetJacobian(snes, J, J, FormJacobian, NULL)); 82 83 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 84 Customize nonlinear solver; set runtime options 85 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 86 /* 87 Set linear solver defaults for this problem. By extracting the 88 KSP and PC contexts from the SNES context, we can then 89 directly call any KSP and PC routines to set various options. 90 */ 91 PetscCall(SNESGetKSP(snes, &ksp)); 92 PetscCall(KSPGetPC(ksp, &pc)); 93 PetscCall(PCSetType(pc, PCNONE)); 94 PetscCall(KSPSetTolerances(ksp, 1.e-4, PETSC_DEFAULT, PETSC_DEFAULT, 20)); 95 96 /* 97 Set SNES/KSP/KSP/PC runtime options, e.g., 98 -snes_view -snes_monitor -ksp_type <ksp> -pc_type <pc> 99 These options will override those specified above as long as 100 SNESSetFromOptions() is called _after_ any other customization 101 routines. 102 */ 103 PetscCall(SNESSetFromOptions(snes)); 104 105 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 106 Evaluate initial guess; then solve nonlinear system 107 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 108 PetscCall(VecSet(x, 2.5)); 109 110 PetscCall(SNESSolve(snes, NULL, x)); 111 112 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 113 Output x and f(x) 114 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 115 PetscCall(VecView(x, PETSC_VIEWER_STDOUT_WORLD)); 116 PetscCall(VecView(r, PETSC_VIEWER_STDOUT_WORLD)); 117 118 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 119 Free work space. All PETSc objects should be destroyed when they 120 are no longer needed. 121 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 122 123 PetscCall(VecDestroy(&x)); 124 PetscCall(VecDestroy(&r)); 125 PetscCall(MatDestroy(&J)); 126 PetscCall(SNESDestroy(&snes)); 127 PetscCall(PetscFinalize()); 128 return 0; 129 } 130 131 PetscErrorCode FormFunction(SNES snes, Vec x, Vec f, void *ctx) 132 { 133 const PetscScalar *xx; 134 PetscScalar *ff; 135 136 /* 137 Get pointers to vector data. 138 - For default PETSc vectors, VecGetArray() returns a pointer to 139 the data array. Otherwise, the routine is implementation dependent. 140 - You MUST call VecRestoreArray() when you no longer need access to 141 the array. 142 */ 143 PetscCall(VecGetArrayRead(x, &xx)); 144 PetscCall(VecGetArray(f, &ff)); 145 146 /* Compute function */ 147 ff[0] = 8. * PetscExpScalar(-4. * (xx[0] - 2.) * (xx[0] - 2.)) * (xx[0] - 2.) + 2. * xx[0]; 148 149 /* Restore vectors */ 150 PetscCall(VecRestoreArrayRead(x, &xx)); 151 PetscCall(VecRestoreArray(f, &ff)); 152 return 0; 153 } 154 155 PetscErrorCode FormJacobian(SNES snes, Vec x, Mat jac, Mat B, void *dummy) 156 { 157 const PetscScalar *xx; 158 PetscScalar A[1]; 159 PetscInt idx[1] = {0}; 160 161 /* 162 Get pointer to vector data 163 */ 164 PetscCall(VecGetArrayRead(x, &xx)); 165 166 /* 167 Compute Jacobian entries and insert into matrix. 168 - Since this is such a small problem, we set all entries for 169 the matrix at once. 170 */ 171 A[0] = 8. * ((xx[0] - 2.) * (PetscExpScalar(-4. * (xx[0] - 2.) * (xx[0] - 2.)) * -8. * (xx[0] - 2.)) + PetscExpScalar(-4. * (xx[0] - 2.) * (xx[0] - 2.))) + 2.; 172 173 PetscCall(MatSetValues(B, 1, idx, 1, idx, A, INSERT_VALUES)); 174 175 /* 176 Restore vector 177 */ 178 PetscCall(VecRestoreArrayRead(x, &xx)); 179 180 /* 181 Assemble matrix 182 */ 183 PetscCall(MatAssemblyBegin(B, MAT_FINAL_ASSEMBLY)); 184 PetscCall(MatAssemblyEnd(B, MAT_FINAL_ASSEMBLY)); 185 if (jac != B) { 186 PetscCall(MatAssemblyBegin(jac, MAT_FINAL_ASSEMBLY)); 187 PetscCall(MatAssemblyEnd(jac, MAT_FINAL_ASSEMBLY)); 188 } 189 return 0; 190 } 191 192 /*TEST 193 194 test: 195 suffix: 1 196 args: -snes_linesearch_type cp 197 test: 198 suffix: 2 199 args: -snes_linesearch_type basic 200 test: 201 suffix: 3 202 test: 203 suffix: 4 204 args: -snes_type newtontrdc 205 test: 206 suffix: 5 207 args: -snes_type newtontrdc -snes_trdc_use_cauchy false 208 209 TEST*/ 210