|
| 1 | +# Specialized Problem Types |
| 2 | + |
| 3 | +SciMLBase provides several specialized problem types that extend the basic problem interface for specific use cases. These problems offer enhanced functionality for particular mathematical structures or solution approaches. |
| 4 | + |
| 5 | +## Overview |
| 6 | + |
| 7 | +Specialized problem types in SciMLBase include: |
| 8 | +- Nonlinear least squares problems |
| 9 | +- Strongly connected component (SCC) nonlinear problems |
| 10 | +- Sampled integral problems |
| 11 | +- Incremental ODE problems |
| 12 | +- Two-point boundary value problems |
| 13 | +- Homotopy and interval nonlinear problems |
| 14 | + |
| 15 | +## Nonlinear Least Squares Problems |
| 16 | + |
| 17 | +### NonlinearLeastSquaresProblem |
| 18 | + |
| 19 | +```julia |
| 20 | +NonlinearLeastSquaresProblem(f, u0, p=nothing) |
| 21 | +``` |
| 22 | + |
| 23 | +A specialized nonlinear problem type for least squares optimization where the objective is to minimize `||f(u, p)||²`. |
| 24 | + |
| 25 | +**Arguments:** |
| 26 | +- `f`: A function that computes the residual vector |
| 27 | +- `u0`: Initial guess for the solution |
| 28 | +- `p`: Parameters (optional) |
| 29 | + |
| 30 | +**Use Cases:** |
| 31 | +- Parameter estimation and data fitting |
| 32 | +- Inverse problems where the model-data misfit is measured in L2 norm |
| 33 | +- Problems where the Jacobian structure can be exploited for efficiency |
| 34 | + |
| 35 | +**Example:** |
| 36 | +```julia |
| 37 | +using SciMLBase |
| 38 | + |
| 39 | +# Define residual function for curve fitting |
| 40 | +function residual!(r, u, p) |
| 41 | + # u = [a, b] are parameters to fit |
| 42 | + # p = (x_data, y_data) |
| 43 | + x_data, y_data = p |
| 44 | + for i in eachindex(x_data) |
| 45 | + r[i] = u[1] * exp(u[2] * x_data[i]) - y_data[i] |
| 46 | + end |
| 47 | +end |
| 48 | + |
| 49 | +# Create problem |
| 50 | +x_data = [0.0, 1.0, 2.0, 3.0] |
| 51 | +y_data = [1.0, 2.7, 7.4, 20.1] |
| 52 | +u0 = [1.0, 1.0] # Initial guess for [a, b] |
| 53 | +prob = NonlinearLeastSquaresProblem(residual!, u0, (x_data, y_data)) |
| 54 | +``` |
| 55 | + |
| 56 | +## Strongly Connected Component Problems |
| 57 | + |
| 58 | +### SCCNonlinearProblem |
| 59 | + |
| 60 | +```julia |
| 61 | +SCCNonlinearProblem(f, u0, p=nothing) |
| 62 | +``` |
| 63 | + |
| 64 | +A specialized nonlinear problem for systems that can be decomposed into strongly connected components, enabling more efficient solution strategies. |
| 65 | + |
| 66 | +**Use Cases:** |
| 67 | +- Large sparse nonlinear systems with block structure |
| 68 | +- Chemical reaction networks with distinct time scales |
| 69 | +- Circuit simulation with hierarchical organization |
| 70 | +- Systems where graph-based decomposition improves efficiency |
| 71 | + |
| 72 | +**Benefits:** |
| 73 | +- Exploits sparsity patterns for improved performance |
| 74 | +- Enables divide-and-conquer solution strategies |
| 75 | +- Reduces memory requirements for large systems |
| 76 | + |
| 77 | +## Integral Problems |
| 78 | + |
| 79 | +### SampledIntegralProblem |
| 80 | + |
| 81 | +```julia |
| 82 | +SampledIntegralProblem(y, x, p=nothing) |
| 83 | +``` |
| 84 | + |
| 85 | +A specialized integral problem for pre-sampled data where the integrand values are known at specific points. |
| 86 | + |
| 87 | +**Arguments:** |
| 88 | +- `y`: Function values at sample points |
| 89 | +- `x`: Sample points (quadrature nodes) |
| 90 | +- `p`: Parameters (optional) |
| 91 | + |
| 92 | +**Use Cases:** |
| 93 | +- Integration of experimental or simulation data |
| 94 | +- Monte Carlo integration with pre-computed samples |
| 95 | +- Adaptive quadrature where function evaluations are expensive |
| 96 | + |
| 97 | +**Example:** |
| 98 | +```julia |
| 99 | +using SciMLBase |
| 100 | + |
| 101 | +# Pre-sampled data points |
| 102 | +x = [0.0, 0.5, 1.0, 1.5, 2.0] |
| 103 | +y = sin.(x) # Function values at sample points |
| 104 | + |
| 105 | +# Create sampled integral problem |
| 106 | +prob = SampledIntegralProblem(y, x) |
| 107 | + |
| 108 | +# The integral can now be computed using various quadrature rules |
| 109 | +# without additional function evaluations |
| 110 | +``` |
| 111 | + |
| 112 | +## Incremental ODE Problems |
| 113 | + |
| 114 | +### IncrementingODEProblem |
| 115 | + |
| 116 | +```julia |
| 117 | +IncrementingODEProblem(f, u0, tspan, p=nothing) |
| 118 | +``` |
| 119 | + |
| 120 | +A specialized ODE problem type for systems where the right-hand side represents increments or rates that accumulate over time. |
| 121 | + |
| 122 | +**Use Cases:** |
| 123 | +- Population models with birth/death processes |
| 124 | +- Chemical reaction systems with mass conservation |
| 125 | +- Economic models with cumulative effects |
| 126 | +- Systems where conservation laws must be maintained |
| 127 | + |
| 128 | +**Features:** |
| 129 | +- Built-in conservation checking |
| 130 | +- Specialized integrators that preserve invariants |
| 131 | +- Enhanced numerical stability for accumulative processes |
| 132 | + |
| 133 | +## Boundary Value Problems |
| 134 | + |
| 135 | +### TwoPointBVProblem |
| 136 | + |
| 137 | +```julia |
| 138 | +TwoPointBVProblem(f, bc, u0, tspan, p=nothing) |
| 139 | +``` |
| 140 | + |
| 141 | +A specialized boundary value problem for two-point boundary conditions. |
| 142 | + |
| 143 | +**Arguments:** |
| 144 | +- `f`: The differential equation function |
| 145 | +- `bc`: Boundary condition function |
| 146 | +- `u0`: Initial guess for the solution |
| 147 | +- `tspan`: Time interval |
| 148 | +- `p`: Parameters (optional) |
| 149 | + |
| 150 | +**Use Cases:** |
| 151 | +- Shooting methods for boundary value problems |
| 152 | +- Optimal control problems with fixed endpoints |
| 153 | +- Eigenvalue problems with specific boundary conditions |
| 154 | + |
| 155 | +### SecondOrderBVProblem and TwoPointSecondOrderBVProblem |
| 156 | + |
| 157 | +```julia |
| 158 | +SecondOrderBVProblem(f, bc, u0, du0, tspan, p=nothing) |
| 159 | +TwoPointSecondOrderBVProblem(f, bc, u0, du0, tspan, p=nothing) |
| 160 | +``` |
| 161 | + |
| 162 | +Specialized for second-order differential equations with boundary conditions. |
| 163 | + |
| 164 | +**Use Cases:** |
| 165 | +- Mechanical systems with position and velocity constraints |
| 166 | +- Beam deflection problems |
| 167 | +- Quantum mechanics eigenvalue problems |
| 168 | + |
| 169 | +## Interval and Homotopy Problems |
| 170 | + |
| 171 | +### IntervalNonlinearProblem |
| 172 | + |
| 173 | +```julia |
| 174 | +IntervalNonlinearProblem(f, u_interval, p=nothing) |
| 175 | +``` |
| 176 | + |
| 177 | +A nonlinear problem type for bracketing rootfinders where the solution is known to lie within a specific interval. |
| 178 | + |
| 179 | +**Arguments:** |
| 180 | +- `f`: Function to find roots of |
| 181 | +- `u_interval`: Interval `[a, b]` containing the root |
| 182 | +- `p`: Parameters (optional) |
| 183 | + |
| 184 | +**Use Cases:** |
| 185 | +- Guaranteed root finding with interval arithmetic |
| 186 | +- Robust rootfinding when derivative information is unreliable |
| 187 | +- Global optimization in one dimension |
| 188 | + |
| 189 | +### HomotopyNonlinearFunction |
| 190 | + |
| 191 | +A specialized function type for homotopy continuation methods where the problem is gradually deformed from a simple form to the target problem. |
| 192 | + |
| 193 | +**Use Cases:** |
| 194 | +- Polynomial root finding |
| 195 | +- Global optimization via continuation |
| 196 | +- Bifurcation analysis and parameter continuation |
| 197 | + |
| 198 | +## Performance Considerations |
| 199 | + |
| 200 | +### Memory and Computational Efficiency |
| 201 | + |
| 202 | +- **Specialized Storage**: Many specialized problems use tailored data structures that reduce memory overhead |
| 203 | +- **Exploiting Structure**: Problems like `SCCNonlinearProblem` exploit mathematical structure for faster solution |
| 204 | +- **Reduced Function Evaluations**: `SampledIntegralProblem` avoids repeated function calls |
| 205 | + |
| 206 | +### Algorithm Selection |
| 207 | + |
| 208 | +Different specialized problems work best with specific algorithm families: |
| 209 | + |
| 210 | +```julia |
| 211 | +# Example: Choosing algorithms based on problem type |
| 212 | +function select_algorithm(prob) |
| 213 | + if prob isa NonlinearLeastSquaresProblem |
| 214 | + return LevenbergMarquardt() # Exploits least squares structure |
| 215 | + elseif prob isa SCCNonlinearProblem |
| 216 | + return NewtonRaphson() # Good for sparse structured systems |
| 217 | + elseif prob isa IntervalNonlinearProblem |
| 218 | + return Bisection() # Guaranteed convergence in intervals |
| 219 | + else |
| 220 | + return TrustRegion() # General purpose fallback |
| 221 | + end |
| 222 | +end |
| 223 | +``` |
| 224 | + |
| 225 | +## Integration with the SciML Ecosystem |
| 226 | + |
| 227 | +Specialized problems integrate seamlessly with: |
| 228 | +- **Automatic Differentiation**: Support for forward and reverse mode AD |
| 229 | +- **Sensitivity Analysis**: Parameter sensitivity computation |
| 230 | +- **Ensemble Simulations**: Monte Carlo and parameter sweep studies |
| 231 | +- **Symbolic Computing**: Integration with ModelingToolkit.jl for symbolic problem setup |
| 232 | + |
| 233 | +These specialized problem types provide targeted solutions for specific mathematical structures while maintaining compatibility with the broader SciML ecosystem's tools and workflows. |
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