Glossary

13.2 Homotopy continuation methods

5 min read•july 30, 2024

methods are a powerful tool for solving complex polynomial systems. They work by gradually transforming a simple system with known solutions into the target system we want to solve. This approach can find all , including complex and multiple ones.

These methods are globally convergent and don't need a good initial guess. They're used in various fields like robotics, computer vision, and chemical engineering. By exploiting problem structure and using smart algorithms, we can efficiently solve systems that would be tough to crack otherwise.

Homotopy Continuation Methods

Principles and Basic Concepts

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  • Homotopy continuation is a numerical method for solving systems of polynomial equations by continuously deforming a simple system into the target system
  • The basic idea of homotopy continuation is to construct a H(x,t)H(x, t) that connects a start system with known solutions to the target system to be solved
  • The homotopy function H(x,t)H(x, t) satisfies H(x,0)=F(x)H(x, 0) = F(x) (start system) and H(x,1)=G(x)H(x, 1) = G(x) (target system), where F(x)F(x) and G(x)G(x) are polynomial systems
    • Example: A linear homotopy function H(x,t)=(1−t)∗F(x)+t∗G(x)H(x, t) = (1 - t) * F(x) + t * G(x)
  • The solution path is traced from the start system to the target system by varying the parameter tt from 0 to 1, typically using
    • Predictor-corrector methods take small steps in tt and correct the solution using Newton's method

Advantages and Applications

  • Homotopy continuation methods can find all isolated solutions of a polynomial system, including complex solutions and multiple solutions
    • This is an advantage over local optimization methods that may only find a single solution
  • Homotopy continuation methods are globally convergent, meaning they can find solutions starting from any initial point
    • In contrast, local optimization methods require a good initial guess close to the solution
  • Applications of homotopy continuation include solving systems of polynomial equations arising in various fields
    • Robotics (inverse kinematics problems)
    • Computer vision (structure from motion)
    • Chemical engineering (finding equilibrium states of reaction networks)
    • Physics (solving equations of motion for mechanical systems)

Constructing Homotopy Functions and Paths

Homotopy Function Selection

  • The choice of the homotopy function H(x,t)H(x, t) is crucial for the efficiency and convergence of the continuation method
  • The start system F(x)F(x) should be chosen such that its solutions are known or easily computed, and the solution paths from F(x)F(x) to G(x)G(x) are smooth and non-singular
    • Example: Choosing a start system with the same monomial structure as the target system
  • A common homotopy function is the linear homotopy: H(x,t)=(1−t)∗F(x)+t∗G(x)H(x, t) = (1 - t) * F(x) + t * G(x), where F(x)F(x) is the start system and G(x)G(x) is the target system
  • The can be used to avoid singularities in the solution path by adding a random complex number γγ to the homotopy function: H(x,t)=(1−t)∗F(x)−t∗γ∗G(x)H(x, t) = (1 - t) * F(x) - t * γ * G(x)
    • This ensures that the solution paths avoid singular points with probability one

Exploiting Problem Structure

  • exploit the structure of sparse polynomial systems by constructing homotopy functions based on the of the
    • The mixed volume provides a tighter bound on the number of solutions than the total degree
  • Exploiting problem structure can lead to more efficient homotopy continuation algorithms
    • Fewer solution paths need to be traced
    • The homotopy function can be chosen to have desirable properties (, )
  • Predictor-corrector methods, such as Euler's method or Runge-Kutta methods, are used to trace the solution path by taking small steps in the parameter tt and correcting the solution using Newton's method
    • The step size and order of the predictor-corrector method affect the accuracy and efficiency of

Convergence and Complexity of Algorithms

Convergence Analysis

  • The convergence of homotopy continuation methods depends on the regularity of the solution path and the step size used in the predictor-corrector scheme
  • The of the homotopy function plays a crucial role in the convergence analysis, as it determines the local behavior of the solution path
    • The of the Jacobian matrix affects the accuracy and stability of the numerical computations in the predictor-corrector steps
  • can be used to ensure convergence and efficiency
    • Smaller steps are taken when the solution path is highly curved or near singular points
    • Larger steps are taken when the solution path is relatively straight

Complexity Analysis

  • The complexity of homotopy continuation algorithms is determined by the number of solution paths that need to be traced and the cost of each predictor-corrector step
  • The has a complexity that depends on the product of the degrees of the polynomials in the system, which can be exponential in the number of variables
    • Example: A system of nn quadratic equations has a total degree of 2n2^n
  • The polyhedral homotopy has a complexity that depends on the mixed volume of the Newton polytopes, which is often much smaller than the total degree for sparse systems
    • The mixed volume can be computed efficiently using algorithms from computational geometry
  • Path tracking strategies, such as or , can be used to improve the efficiency of homotopy continuation algorithms
    • Parallel path tracking distributes the workload across multiple processors
    • Path merging avoids redundant computations by combining similar solution paths

Applications of Homotopy Continuation

Real-World Problems

  • Homotopy continuation methods have been successfully applied to solve polynomial systems arising in various fields
    • Robotics (inverse kinematics problems)
    • Computer vision (structure from motion)
    • Chemical engineering (finding equilibrium states of reaction networks)
    • Physics (solving equations of motion for mechanical systems)
  • The application of homotopy continuation to real-world problems often requires the formulation of the problem as a system of polynomial equations and the selection of appropriate homotopy functions and path tracking strategies

Problem Formulation and Solution Interpretation

  • Formulating real-world problems as systems of polynomial equations is a crucial step in applying homotopy continuation methods
    • This may involve introducing auxiliary variables or transforming the original equations
    • The choice of variables and equations affects the sparsity and structure of the polynomial system
  • The interpretation and validation of the solutions obtained by homotopy continuation methods are important steps in the application process
    • The solutions may represent physically meaningful states or configurations of the system under study
    • Domain knowledge is required to assess the feasibility and significance of the solutions
  • Visualization techniques can be used to gain insights into the solution space and the behavior of the system
    • Plotting the solution paths in the parameter space
    • Visualizing the solutions in the original problem space (joint angles, camera poses, chemical concentrations)

Key Terms to Review (27)

Adaptive step size control: Adaptive step size control is a numerical technique that adjusts the size of the steps taken during iterative computations based on the estimated local error of the solution. This method helps improve accuracy and efficiency by allowing smaller steps when the solution changes rapidly and larger steps when it is relatively stable. This balancing act ensures that the computational effort is optimized while maintaining the precision necessary for solving complex systems, particularly in polynomial system solving and homotopy continuation methods.
Bernd Sturmfels: Bernd Sturmfels is a prominent mathematician known for his contributions to algebraic geometry, particularly in the areas of polynomial systems and their applications. He has played a vital role in bridging the gap between symbolic and numerical methods, leading to significant advancements in hybrid algorithms and homotopy continuation techniques that solve complex polynomial equations. His work also extends into tropical geometry, influencing how these mathematical concepts are applied in various fields, including optimization and computational biology.
Chern classes: Chern classes are characteristic classes associated with complex vector bundles that provide a way to study the geometry and topology of manifolds. They capture important information about the bundle's curvature and are used in various mathematical areas, including algebraic geometry and topology. Chern classes play a significant role in understanding how different spaces can be transformed and connected through homotopy continuation methods.
Complex varieties: Complex varieties are geometric objects that arise from the study of solutions to polynomial equations with complex coefficients, making them central to both algebraic geometry and complex geometry. These varieties can be understood as sets of points in complex projective space that satisfy certain polynomial equations, and they exhibit rich structures that can be analyzed using tools from both algebra and topology. Their intricate properties make them vital for understanding various concepts in mathematics, including intersection theory and deformation theory.
Condition Number: The condition number is a measure of how sensitive a function or problem is to changes in its input. In the context of numerical analysis, it indicates how the output value will change in response to small perturbations in the input. A high condition number suggests that the problem is ill-conditioned, meaning that even tiny changes can lead to large variations in results, making numerical computations unstable or inaccurate.
Continuation path: A continuation path is a mathematical concept used in homotopy continuation methods, where one studies a family of problems that evolve smoothly from an initial problem to a target problem. This approach helps in finding solutions to complex equations by following a path in a parameter space, ensuring that solutions to the initial problem can be tracked as parameters change. It leverages the idea that if solutions exist for a simpler problem, they can be continuously transformed into solutions for more complicated ones.
David l. m. a. a. d. s. a. v. p. d. c.: David L. M. A. A. D. S. A. V. P. D. C. refers to a concept in computational algebraic geometry that relates to the study of homotopy continuation methods used for solving systems of polynomial equations. It provides a systematic framework for understanding how one can deform a system of equations into another while preserving the solutions, allowing for efficient numerical solutions and insights into the topology of solution spaces.
Gamma-trick: The gamma-trick is a technique used in homotopy continuation methods to simplify the computation of solutions to polynomial systems by transforming the problem into a more manageable form. This trick allows for a smooth path to be constructed between known solutions, often involving the introduction of a parameter that varies continuously, which helps in tracking the solutions as the system changes.
Global convergence: Global convergence refers to the property of a numerical algorithm that guarantees finding a solution from any starting point in the domain. This concept is crucial in understanding how various methods, such as homotopy continuation, can effectively trace paths in solving systems of equations. It highlights the reliability and robustness of these algorithms in yielding solutions regardless of the initial conditions.
Homotopy continuation: Homotopy continuation is a numerical method used to solve systems of polynomial equations by continuously deforming a simple system into a more complex one while tracking the solutions. This approach links the solutions of an easier system to those of the target system, allowing for a structured pathway to find solutions even in high-dimensional spaces. It connects concepts from algebra and geometry by illustrating how algebraic varieties can be represented and manipulated in a geometric context.
Homotopy function: A homotopy function is a continuous deformation between two functions, providing a way to relate solutions of different mathematical problems in a structured manner. This concept is foundational in homotopy continuation methods, as it allows for the gradual transformation of a system of equations into another while preserving their solutions, facilitating the exploration of solution spaces in algebraic geometry.
Isolated Solutions: Isolated solutions refer to specific solutions of a system of equations that do not have any nearby solutions in the solution space. This means that in the vicinity of an isolated solution, there are no other solutions within a certain neighborhood. Isolated solutions are particularly important in computational methods because they can simplify the process of finding solutions and allow for more efficient numerical techniques, such as homotopy continuation methods, to be applied.
Jacobian Matrix: The Jacobian matrix is a matrix of all first-order partial derivatives of a vector-valued function. It provides crucial information about the function's behavior, particularly in understanding how changes in input variables affect the output. This matrix is vital in many areas, including the study of projective varieties, numerical methods for solving polynomial systems, the geometric interpretation of algebraic sets, and homotopy continuation methods.
Mixed Volume: Mixed volume is a geometric concept that quantifies the volume of the Minkowski sum of several convex bodies, providing a measure of how these bodies interact in space. It plays a crucial role in understanding the geometry of polytopes and can be utilized in optimization problems, particularly in homotopy continuation methods where the solution paths of polynomial equations are tracked.
Newton Polytopes: Newton polytopes are geometric objects that represent the relationships between the terms of a polynomial, specifically capturing the degrees of each variable in a multivariate polynomial. They serve as a powerful tool in algebraic geometry, providing insights into the structure of polynomial equations and their solutions. The vertices of a Newton polytope correspond to the monomials in the polynomial, while the edges illustrate how these monomials interact with each other.
Non-singularity: Non-singularity refers to the property of a mathematical object, such as a matrix or an algebraic variety, being well-defined and without any 'holes' or 'breaks' in its structure. In the context of homotopy continuation methods, non-singularity is crucial because it ensures that the solutions can be tracked continuously without encountering singular points where the behavior of the system can change unpredictably.
Numerical Algebraic Geometry: Numerical algebraic geometry is a branch of mathematics that combines computational methods with algebraic geometry to solve systems of polynomial equations numerically. It focuses on the geometric structures and their numerical properties, allowing for insights into the shape and solutions of algebraic varieties through algorithms and numerical techniques. This approach provides powerful tools for understanding intersections, dimensionality, and the behavior of solutions under perturbations.
Parallel Path Tracking: Parallel path tracking is a computational strategy used in homotopy continuation methods to efficiently follow multiple solution paths simultaneously. This technique leverages the idea of creating several paths from a common starting point, allowing for faster and more effective exploration of the solution space of a system of equations. By tracking these paths in parallel, it enhances the robustness and speed of finding all possible solutions, which is crucial when dealing with complex algebraic problems.
Path Merging: Path merging is a technique used in homotopy continuation methods to combine multiple solution paths into a single path when the solutions are close to each other. This approach helps improve computational efficiency and robustness in finding solutions to polynomial systems by effectively reducing the number of paths that need to be followed. By merging paths, one can navigate through complex solution spaces more effectively, especially in cases where paths intersect or converge.
Path Tracking: Path tracking refers to a numerical technique used to follow continuous solutions of polynomial systems as parameters change. This method is particularly useful in computational algebraic geometry, where it allows for the exploration of solution spaces and helps identify all possible solutions to polynomial equations. By moving along paths, one can gain insights into the behavior of these solutions as well as their connections to one another.
Polyhedral Homotopies: Polyhedral homotopies are a class of homotopy continuation methods that utilize polyhedral structures to find solutions to systems of polynomial equations. These methods are particularly useful for tracking solutions through changes in parameters and are based on the idea that solution paths can be studied via the combinatorial properties of polytopes. This approach leverages geometric insights to ensure that the paths taken through parameter space remain within the confines of the solution set.
Predictor-corrector methods: Predictor-corrector methods are numerical techniques used to solve differential equations and systems of equations, combining an initial estimate (the predictor) with a refinement process (the corrector). These methods are particularly useful for improving the accuracy of solutions in polynomial system solving and homotopy continuation, where finding precise roots is essential. By iteratively predicting a solution and then correcting it based on error estimates, these methods enhance convergence and stability in computations.
Projective Space: Projective space is a fundamental concept in algebraic geometry that extends the notion of Euclidean space by adding 'points at infinity' to allow for a more comprehensive study of geometric properties. This extension allows for the unification of various types of geometric objects, facilitating intersection theory, transformations, and various algebraic structures.
Real algebraic geometry: Real algebraic geometry is a branch of mathematics that studies the solutions to polynomial equations with real coefficients, focusing on the geometric properties of these solutions. It combines techniques from algebra, topology, and analysis to understand the structure of real algebraic sets, which are the zero sets of real polynomials. This area is significant because it connects various mathematical disciplines and has implications in optimization, robotics, and even economics.
Smoothness: Smoothness refers to a property of a geometric object where it has no singularities, meaning that it can be described by differentiable functions at every point. In algebraic geometry, smoothness indicates that the local structure of the variety behaves nicely, allowing for well-defined tangent spaces. This concept is crucial in understanding various algebraic constructs, as it ensures that methods applied to these objects, such as intersection theory or homotopy methods, are valid and effective.
Symbolic computation: Symbolic computation refers to the manipulation of mathematical expressions in a way that treats symbols as abstract entities rather than specific numerical values. This approach allows for the exact representation of mathematical objects and enables operations like simplification, differentiation, and solving equations in a symbolic form, providing powerful tools for tasks in both algebra and geometry.
Total Degree Homotopy: Total degree homotopy is a mathematical approach used in solving polynomial systems where the focus is on tracking solutions based on their total degree as parameters vary. This method connects the structure of polynomial equations with their geometric representations, enabling efficient navigation through solution spaces. By managing the total degree of the polynomials involved, one can effectively handle multiple paths in the solution space, which is crucial in the computation of roots and understanding solution continuity.
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