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 or bisection line search, the solve is 18 successful. 19 20 The test outputs the final result for x and f(x). 21 22 Example usage: 23 24 Get help: 25 ./ex99 -help 26 27 Monitor run (with default back-tracking line search; solve fails): 28 ./ex99 -snes_converged_reason -snes_monitor -snes_linesearch_monitor -ksp_converged_reason -ksp_monitor 29 30 Run without a line search; solve succeeds: 31 ./ex99 -snes_linesearch_type basic 32 33 Run with a critical point line search; solve succeeds: 34 ./ex99 -snes_linesearch_type cp 35 */ 36 37 #include <math.h> 38 #include <petscsnes.h> 39 40 extern PetscErrorCode FormJacobian(SNES, Vec, Mat, Mat, void *); 41 extern PetscErrorCode FormFunction(SNES, Vec, Vec, void *); 42 43 int main(int argc, char **argv) 44 { 45 SNES snes; /* nonlinear solver context */ 46 KSP ksp; /* linear solver context */ 47 PC pc; /* preconditioner context */ 48 Vec x, r; /* solution, residual vectors */ 49 Mat J; /* Jacobian matrix */ 50 PetscMPIInt size; 51 52 PetscFunctionBeginUser; 53 PetscCall(PetscInitialize(&argc, &argv, NULL, help)); 54 PetscCallMPI(MPI_Comm_size(PETSC_COMM_WORLD, &size)); 55 PetscCheck(size == 1, PETSC_COMM_WORLD, PETSC_ERR_WRONG_MPI_SIZE, "Example is only for sequential runs"); 56 57 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 58 Create nonlinear solver context 59 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 60 PetscCall(SNESCreate(PETSC_COMM_WORLD, &snes)); 61 62 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 63 Create matrix and vector data structures; set corresponding routines 64 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 65 /* 66 Create vectors for solution and nonlinear function 67 */ 68 PetscCall(VecCreate(PETSC_COMM_WORLD, &x)); 69 PetscCall(VecSetSizes(x, PETSC_DECIDE, 1)); 70 PetscCall(VecSetFromOptions(x)); 71 PetscCall(VecDuplicate(x, &r)); 72 73 /* 74 Create Jacobian matrix data structure 75 */ 76 PetscCall(MatCreate(PETSC_COMM_WORLD, &J)); 77 PetscCall(MatSetSizes(J, PETSC_DECIDE, PETSC_DECIDE, 1, 1)); 78 PetscCall(MatSetFromOptions(J)); 79 PetscCall(MatSetUp(J)); 80 81 PetscCall(SNESSetFunction(snes, r, FormFunction, NULL)); 82 PetscCall(SNESSetJacobian(snes, J, J, FormJacobian, NULL)); 83 84 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 85 Customize nonlinear solver; set runtime options 86 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 87 /* 88 Set linear solver defaults for this problem. By extracting the 89 KSP and PC contexts from the SNES context, we can then 90 directly call any KSP and PC routines to set various options. 91 */ 92 PetscCall(SNESGetKSP(snes, &ksp)); 93 PetscCall(KSPGetPC(ksp, &pc)); 94 PetscCall(PCSetType(pc, PCNONE)); 95 PetscCall(KSPSetTolerances(ksp, 1.e-4, PETSC_CURRENT, PETSC_CURRENT, 20)); 96 97 /* 98 Set SNES/KSP/KSP/PC runtime options, e.g., 99 -snes_view -snes_monitor -ksp_type <ksp> -pc_type <pc> 100 These options will override those specified above as long as 101 SNESSetFromOptions() is called _after_ any other customization 102 routines. 103 */ 104 PetscCall(SNESSetFromOptions(snes)); 105 106 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 107 Evaluate initial guess; then solve nonlinear system 108 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 109 PetscCall(VecSet(x, 2.5)); 110 111 PetscCall(SNESSolve(snes, NULL, x)); 112 113 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 114 Output x and f(x) 115 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 116 PetscCall(VecView(x, PETSC_VIEWER_STDOUT_WORLD)); 117 PetscCall(VecView(r, PETSC_VIEWER_STDOUT_WORLD)); 118 119 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 120 Free work space. All PETSc objects should be destroyed when they 121 are no longer needed. 122 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 123 124 PetscCall(VecDestroy(&x)); 125 PetscCall(VecDestroy(&r)); 126 PetscCall(MatDestroy(&J)); 127 PetscCall(SNESDestroy(&snes)); 128 PetscCall(PetscFinalize()); 129 return 0; 130 } 131 132 PetscErrorCode FormFunction(SNES snes, Vec x, Vec f, void *ctx) 133 { 134 const PetscScalar *xx; 135 PetscScalar *ff; 136 137 PetscFunctionBeginUser; 138 /* 139 Get pointers to vector data. 140 - For default PETSc vectors, VecGetArray() returns a pointer to 141 the data array. Otherwise, the routine is implementation dependent. 142 - You MUST call VecRestoreArray() when you no longer need access to 143 the array. 144 */ 145 PetscCall(VecGetArrayRead(x, &xx)); 146 PetscCall(VecGetArray(f, &ff)); 147 148 /* Compute function */ 149 ff[0] = 8. * PetscExpScalar(-4. * (xx[0] - 2.) * (xx[0] - 2.)) * (xx[0] - 2.) + 2. * xx[0]; 150 151 /* Restore vectors */ 152 PetscCall(VecRestoreArrayRead(x, &xx)); 153 PetscCall(VecRestoreArray(f, &ff)); 154 PetscFunctionReturn(PETSC_SUCCESS); 155 } 156 157 PetscErrorCode FormJacobian(SNES snes, Vec x, Mat jac, Mat B, void *dummy) 158 { 159 const PetscScalar *xx; 160 PetscScalar A[1]; 161 PetscInt idx[1] = {0}; 162 163 PetscFunctionBeginUser; 164 /* 165 Get pointer to vector data 166 */ 167 PetscCall(VecGetArrayRead(x, &xx)); 168 169 /* 170 Compute Jacobian entries and insert into matrix. 171 - Since this is such a small problem, we set all entries for 172 the matrix at once. 173 */ 174 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.; 175 176 PetscCall(MatSetValues(B, 1, idx, 1, idx, A, INSERT_VALUES)); 177 178 /* 179 Restore vector 180 */ 181 PetscCall(VecRestoreArrayRead(x, &xx)); 182 183 /* 184 Assemble matrix 185 */ 186 PetscCall(MatAssemblyBegin(B, MAT_FINAL_ASSEMBLY)); 187 PetscCall(MatAssemblyEnd(B, MAT_FINAL_ASSEMBLY)); 188 if (jac != B) { 189 PetscCall(MatAssemblyBegin(jac, MAT_FINAL_ASSEMBLY)); 190 PetscCall(MatAssemblyEnd(jac, MAT_FINAL_ASSEMBLY)); 191 } 192 PetscFunctionReturn(PETSC_SUCCESS); 193 } 194 195 /*TEST 196 197 test: 198 suffix: 1 199 args: -snes_linesearch_type cp 200 test: 201 suffix: 2 202 args: -snes_linesearch_type basic 203 test: 204 suffix: 3 205 test: 206 suffix: 4 207 args: -snes_type newtontrdc 208 test: 209 suffix: 5 210 args: -snes_type newtontrdc -snes_trdc_use_cauchy false 211 test: 212 suffix: 6 213 args: -snes_linesearch_type bisection 214 215 TEST*/ 216