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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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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Guest René Christensen February 28, 2018

Today, guest blogger René Christensen of GN Hearing discusses including thermoviscous losses in the topology optimization of microacoustic devices. Topology optimization helps engineers design applications in an optimized manner with respect to certain a priori objectives. Mainly used in structural mechanics, topology optimization is also used for thermal, electromagnetics, and acoustics applications. One physics that was missing from this list until last year is microacoustics. This blog post describes a new method for including thermoviscous losses for microacoustics topology optimization.

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Caty Fairclough December 20, 2017

Creating new physics interfaces that you can save and share, modifying the underlying equations of a model, and simulating a wider variety of devices and processes: These are just a few ways you can benefit from the equation-based modeling capabilities of the COMSOL Multiphysics® software.

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Bjorn Sjodin June 20, 2017

You can generate and visualize randomized material data with specified statistical properties determined by a spectral density distribution by using the tools available under the Results node in the COMSOL Multiphysics® software. In this blog post, we show examples that are quite general and have potential uses in many application areas, including heat transfer, structural mechanics, subsurface flow, and more.

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Bjorn Sjodin June 2, 2017

To easily generate random-looking geometric surfaces, the COMSOL Multiphysics® software provides a powerful set of built-in functions and operators, such as functions for uniform and Gaussian random distributions and a very useful sum operator. In this blog post, we show you how to generate a randomized surface with what amounts to a “one liner” expression with detailed control of the constituent spatial frequency components that determine the nature of the surface’s roughness.

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Temesgen Kindo May 9, 2017

When your simulations consume significant memory, do you buy a bigger computer? When they take too long to solve, do you just run them overnight? Often, you don’t have another option. But sometimes, if you have the right tools, you can find a better approach by exploiting the mathematical structure. Today, we will show you how to use the so-called maximum principles to save computational resources and time in the COMSOL Multiphysics® software.

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