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Discretization error, often referred to as discretization error or numerical error, is a type of error that occurs when continuous data or processes are approximated or represented using discrete values or methods. It is a common issue in various fields, including mathematics, computer science, and engineering, where continuous problems need to be solved using finite and discrete techniques.

Discretization error can manifest in several ways:

1. Quantization Error: This occurs when continuous data is represented with a limited number of discrete values. For example, when you round real numbers to a specific number of decimal places or represent continuous signals with digital values, you introduce quantization error. This error can lead to inaccuracies in calculations or measurements.

2. Integration Error: When numerical integration methods are used to approximate the definite integral of a continuous function, discretization error can occur. The choice of integration step size or method can affect the accuracy of the result. Smaller step sizes generally reduce the error but increase computational complexity.

3. Differential Equation Approximation: When solving differential equations numerically, such as using finite difference methods or finite element analysis, the discretization of time or space can introduce errors. These errors can propagate through the solution and affect the accuracy of the results.

4. Grid Discretization: In computational simulations and modeling, physical domains are often discretized into grids or meshes. Errors can arise from the choice of grid size, boundary conditions, and numerical methods used to solve partial differential equations on these grids.

5. Time Discretization: In time-dependent simulations, such as numerical simulations of dynamic systems, the time domain is discretized into time steps. Errors can occur if the time step is too large, causing important events to be missed, or if it's too small, leading to increased computational costs.

6. Numerical Methods: Different numerical methods, such as finite element analysis, finite difference methods, and numerical optimization techniques, can introduce discretization errors depending on their approximation schemes and convergence properties.

Managing discretization errors often involves a trade-off between computational efficiency and accuracy. Smaller discretization steps and more refined methods generally reduce error but increase computational costs. Engineers and scientists must carefully choose appropriate discretization techniques and parameters to balance these considerations and obtain accurate results in numerical simulations and analyses. Additionally, error analysis and convergence studies are often performed to assess and quantify the discretization errors in numerical computations.

Generalization error (see Probability Bounds Analysis (PBA) at page3947) sometimes can be given by.

          -------------------------------- [3929a]

The first term in the bracket is from the finite hypothesis case, while the second term is discretization error, (see page3929) coming from the K-Lipschitzness. Since the first term depends on log(1/ϵ) and increases very slowly as ϵ goes to 0, and the second term depends on ϵ, there is trading off between the two. Therefore, sometime, you can ignore the second term when ϵ is close to 0.

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