Image Denoising and Other Multidimensional Variational Problems

Temesgen Kindo September 21, 2018

We previously discussed how to solve 1D variational problems with the COMSOL Multiphysics® software and implement complex domain and boundary conditions using a unified constraint enforcement framework. Here, we extend the discussion to multiple dimensions, higher-order derivatives, and multiple unknowns with what we hope will be an enjoyable example: variational image denoising. We conclude this blog series on variational problems with some recommendations for further study.

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Temesgen Kindo September 17, 2018

How do you find the shortest overland distance between two points across a lake? Such obstacles and bounds on solutions are often called inequality constraints. Requirements for nonnegativity of gaps between objects in contact mechanics, species concentrations in chemistry, and population in ecology are some examples of inequality constraints. Previously in this series, we discussed equality constraints on variational problems. Today, we will show you how to implement inequality constraints when using equation-based modeling in COMSOL Multiphysics®.

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Temesgen Kindo September 11, 2018

In the first part of this blog series, we discussed how to solve variational problems with simple boundary conditions. Next, we proceeded to more sophisticated constraints and used Lagrange multipliers to set up equivalent unconstrained problems. Today, we focus on the numerical aspects of constraint enforcement. The method of Lagrange multipliers is theoretically exact, yet its use in numerical solutions poses some challenges. We will go over these challenges and show two mitigation strategies: the penalty and augmented Lagrangian methods.

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Temesgen Kindo September 7, 2018

In the first part of this blog series, we discussed variational problems and demonstrated how to solve them using the COMSOL Multiphysics® software. In that case, we used simple built-in boundary conditions. Today, we will discuss more general boundary conditions and constraints. We will also show how to implement these boundary conditions and constraints in the COMSOL® software using the same variational problem from Part 1: (the soap film) — and just as much math.

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Temesgen Kindo September 4, 2018

What do soap films, catenary cables, and light beams have in common? They behave in ways that minimize certain quantities. Such problems are prevalent in science and engineering fields such as biology, economics, elasticity theory, material science, and image processing. You can simulate many such problems using the built-in physics interfaces in the COMSOL Multiphysics® software, but in this blog series, we will show you how to solve variational problems using the equation-based modeling features.

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Christian Wollblad June 13, 2018

We have already discussed the factors that make a high-quality mesh and how to prepare a CFD model geometry for meshing. In this follow-up blog post, learn about physics-controlled meshing, adaptive mesh refinement, and how to use a variety of meshing tools in the COMSOL Multiphysics® software for your fluid flow simulations.

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Pär Persson Mattsson June 12, 2018

In a previous blog post, we explained how to run a job from the COMSOL Multiphysics® software on clusters directly from the COMSOL Desktop® environment, without any interaction with a Linux® operating system terminal. Since this terminal is sometimes treated with excessive respect, the ability to start a cluster job directly from the graphical user interface is one of the most useful features in the COMSOL® software. Plus, there’s more to it… Enter the Cluster Sweep node.

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Christian Wollblad June 11, 2018

The quality of a computational fluid dynamics (CFD) model is often determined by the quality of the mesh used to solve the problem. A good mesh facilitates convergence, reduces memory requirements, and results in accurate solutions. It is therefore worthwhile to invest time and thought into creating the mesh when solving a CFD problem. In this blog post, we describe the factors of a quality mesh and how to prepare a fluid flow model geometry for meshing.

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Mranal Jain June 4, 2018

If you want to create an animation from your simulation results, the COMSOL Multiphysics® software offers a variety of powerful and flexible options. In this blog post, we will explore the idea of creating a nontrivial animation by combining slices along the azimuthal direction for a 3D model.

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Andrew Griesmer May 10, 2018

The Application Libraries, accessed directly in the COMSOL Multiphysics® software, contain an ever-growing number of examples filled with valuable information on how to model certain types of problems, use certain features, and employ specific modeling techniques. To get the most out of the Application Libraries, you should be able to easily find the information you need.

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Brianne Costa May 9, 2018

During his life, John Scott Russell chased his passion for science — literally. While watching horses pull a boat through a shallow canal, he noticed a wave behaving strangely and followed it for one or two miles on horseback. For the rest of his life, he continued to chase this wave (which he called the “wave of translation”) figuratively, persevering even when his theories were ridiculed by scientists. Did Scott Russell ever catch up to his wave?

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