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How can you use error estimation to reduce computational costs for numerical solutions?

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Numerical solutions are often used to approximate the behavior of complex systems that cannot be solved analytically. However, numerical methods can be computationally expensive and require a trade-off between accuracy and efficiency. How can you use error estimation to reduce computational costs for numerical solutions? In this article, you will learn about some basic concepts and techniques of error analysis and convergence for numerical solutions, and how they can help you optimize your computational resources.

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1 Error sources and types

Numerical solutions are affected by various sources of error, such as rounding, truncation, discretization, and iteration. Rounding error occurs when the computer uses a finite number of digits to represent a real number. Truncation error occurs when a mathematical operation or a series expansion is cut off at a certain point. Discretization error occurs when a continuous domain or function is approximated by a finite set of points or values. Iteration error occurs when a numerical method uses an iterative process to converge to a solution, and the process is stopped before reaching the exact solution. Each type of error has a different impact and magnitude on the numerical solution, and can be estimated using different methods.

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Andre Lima da Silva

Linkedin Top Voice | Electrical Engineer | Health and Safety Engineer | Energy Industry | Tech | Analist | PwC | PCD
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The numerical methods commonly used to solve mathematical problems on computers are subject to various sources of error, each with its own nature and impact on the solution. Some important considerations about these errors include: Round-off error, truncation error, discretization error, and iteration error.

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Muhammad Maaz

PEC Registered Electrical Engineer | Major in Computer Engineering |
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Understanding the various sources of error in numerical solutions is crucial for electrical engineers, especially when designing algorithms for system simulations or control systems. By estimating errors, engineers can optimize computational resources, balancing the trade-off between accuracy and efficiency. For instance, in power system simulations, discretization error is critical due to the continuous nature of electrical signals, and error control methods can significantly improve the reliability of the results without unnecessary computational expense.

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Omogbai Martins

Incident Management Specialist || Telecommunications Generalist || Engineering Professional || Fiber Optics Specialist || Research and Technical Writing Consultant || RF Optimization (Planning) Engineer
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Adaptive Mesh Refinement: Implement adaptive mesh refinement techniques to concentrate computational resources in regions where the error is high, allowing for more efficient allocation of resources. Error-Controlled Iterative Methods: Utilize iterative methods that dynamically adjust the solution based on error estimates, focusing computational efforts where accuracy improvements are needed the most. Optimal Time Step Adjustment: Employ algorithms that adaptively adjust the time step during numerical simulations based on error estimates, optimizing the balance between accuracy and computational cost.

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sohrab ghafari

Director of Electrical and Instrumentation Group
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Sharing known error contributes to the error reduction or management. In addition, for specific problems, one can prepare multiple programs with different rounding methods and or different precisions, and can compare the results among the programs generated for a specific problem

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2 Error estimation methods

Error estimation methods are techniques that can help you quantify and control the error in your numerical solution. For instance, absolute and relative error measure the difference between the numerical solution and the exact or reference solution, while local and global error measure the error at a single point or over the whole domain. Error propagation and stability are measures of how the error changes or accumulates as the numerical method progresses. All these methods can help you assess the accuracy of your numerical solution, identify regions or points with higher error, evaluate the reliability and robustness of your numerical method, and choose a suitable step size or stopping criterion.

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3 Convergence and convergence rate

Convergence is the property of a numerical method that ensures its numerical solution approaches the exact solution as some parameter, such as the step size or the number of iterations, is reduced. The measure of how fast this happens is known as convergence rate. Both are important indicators of the efficiency and performance of a numerical method, and can be estimated using graphical or algebraic methods. Graphical methods involve visualizing the convergence behavior with plots or graphs, while algebraic methods involve using formulas or equations to calculate the convergence and convergence rate. Graphical methods may provide a qualitative understanding of the convergence and convergence rate, while algebraic methods may provide a quantitative understanding, though they may require assumptions or approximations.

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Andre Lima da Silva

Linkedin Top Voice | Electrical Engineer | Health and Safety Engineer | Energy Industry | Tech | Analist | PwC | PCD
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Convergence in numerical methods is a fundamental property that indicates the method's ability to approach the exact solution as certain parameters are adjusted. The convergence rate, in turn, measures how quickly this approximation occurs. Here are some key points related to these concepts: convergence, convergence rate, graphical methods, and algebraic methods.

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4 Error control and adaptive methods

Error control and adaptive methods are techniques that can help reduce computational costs for numerical solutions by adjusting parameters or the numerical method based on error estimation. Common techniques include error tolerance and stopping criterion, which involve using a predefined or user-specified level of error to determine when to stop the numerical method or iterative process. Step size control and grid refinement are other techniques that use a variable or adaptive parameter to optimize the numerical method or discretization. These methods can save computational time and resources, as well as improve accuracy and efficiency by using a smaller parameter where the error is larger, and a larger parameter where the error is smaller.

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5 Error estimation tools and examples

Error estimation tools and examples are software or applications that can help you implement and apply the error estimation methods and techniques for numerical solutions. For instance, MATLAB is a programming language and environment that offers various functions and tools for error estimation, such as eps, norm, cond, ode45, pdepe, and fzero. It can be used to analyze and test numerical solutions of ordinary and partial differential equations, optimization problems, and other problems. Wolfram Alpha is another computational knowledge engine that can answer queries and perform calculations for error estimation. It can be used to check or verify your error estimation results or explore different scenarios or examples. Numerical Methods for Engineers is a textbook that covers various topics and examples of error estimation for numerical solutions. It can be used to learn more about the theory and practice of error estimation for numerical solutions.

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6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?

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How can you use error estimation to reduce computational costs for numerical solutions?

1

2

3

4

5

6

1 Error sources and types

2 Error estimation methods

3 Convergence and convergence rate

4 Error control and adaptive methods

5 Error estimation tools and examples

6 Here’s what else to consider

Electrical Engineering

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How can you use error estimation to reduce computational costs for numerical solutions?

1

2

3

4

5

6

1 Error sources and types

2 Error estimation methods

3 Convergence and convergence rate

4 Error control and adaptive methods

5 Error estimation tools and examples

6 Here’s what else to consider

Electrical Engineering

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