## Yogi Ahmad Erlangga on Multigrid Based Preconditioner for Heterogeneous Helmholtz Problems

#### Emerging Research Front Commentary, October 2010

Article: A novel multigrid based preconditioner for heterogeneous Helmholtz problems
Authors: |

Yogi Ahmad Erlangga talks with ScienceWatch.com and answers a few questions about this month's Emerging Research Front paper in the field of Mathematics.

**Why do you think your paper is highly
cited?**

Every year for the last five years, I've reviewed around two papers related to the work in the paper in question. There are many papers, which I became aware of over time. So, there are indeed citations. But, to answer your question, I first have to say that I had never thought that it would be highly cited.

One expert in iterative methods said that the problem I solved in the paper is an example that is often avoided because of its inherent difficulties. As it is true, it is so unfortunate because the underlying mathematical model actually has a wide range of important applications. These include, to mention a few, electromagnetic modeling, underwater acoustics, and seismic imaging. As of today, especially in the latter application, the wave equation-based seismic exploration will be dominating oil industries for the next 10 to 20 years. The interest from practical users is probably one reason.

"Despite new inventions and contributions from many people around the globe, the difficulties that I experienced in dealing with one important component in finding oil reserves taught me to be less dependent on oil..."

Another reason comes from the numerical linear algebra community, an active research community itself. If people know that one guy can solve a difficult problem, they may think of using their methods, with some modification, to solve the same problem. One person said that that paper is a seminal paper; others say the method described there now is the benchmark iterative method for the Helmholtz equation. If these are true, then this may also be a reason.

**Does it describe a new discovery, methodology, or
synthesis of knowledge?**

The paper itself does not describe a new discovery or methodology. It is best to say that the paper blends some existing methods with a proper glue. I say "proper" because some other types of glue have been proposed as far as in the early 1980s, in the work of Eli Turkel, Alvin Bayliss, and Charles Goldstein. I should give credit to them.

**Would you summarize the significance of your paper
in layman's terms?**

I will use an example from a flying bat. Everybody knows that a bat is blind—well, that is not completely true—and moreover, only flies in the cover of the darkness. A bat images its surroundings by sending ultrasound waves and processing the received echo, a similar process to radar or underwater sonar, in milliseconds. Now, consider yourself as a bat, who wants to know the structure of the Earth's subsurface. You send waves, and wait for a moment until you hear an echo. The question is: "what is the kind of rocks that produces that particular echo?"

In this case, a bat is smarter than humans. We simply do not know the exact answer. But we can predict as accurately as mathematics allows. The prediction goes as follows: we know that the wave travels according to some mathematical equation. We make an initial prediction of rocks, and then use it to solve the wave equation. We then compare the solution—called it the mathematical echo—with the actual echo.

If the two are not the same, then we change our initial prediction of rocks. Of course, the change could not be made arbitrarily; we should have a methodology that guarantees each time we change our prediction of rocks, the error between the two becomes smaller.

In practice, people send waves hundreds or thousands of times from different positions. You can imagine how huge the workload is: you have to solve the mathematical equation of waves so many times to get an acceptable image of Earth's subsurface. Now, if we want to get that image as fast as possible, like a bat, with a small computer, like the bat's small brain, one component that should be computed fast and efficiently is the mathematical echo, the solution of the wave equation. This is the contribution of that paper.

**How did you become involved in this research, and
how would you describe the particular challenges, setbacks, and
successes that you've encountered along the way?**

I became involved in this research when I was a Ph.D. student at the Institute of Applied Mathematics (DIAM), Delft University of Technology, The Netherlands, and I dealt with the problem from the first day of my doctoral studies. I have an MS background from the same university, but not in mathematics. Before finishing up my master's, my supervisor advised me to do a Ph.D. in mathematics, and referred me to her Ph.D. supervisor, who coincidently had an opening.

I probably took more math courses than non-math students of my batch, but to be a math student remained a big challenge. Trained as an engineer, I was always saved by the so-called "error margin" or "safety factor." You are not asked to provide an exact solution from any approach you use to attack a problem. But in mathematics, you have to be very precise, and this precision sometimes requires knowledge of mathematical truth from centuries ago, and often has to be shown and proved by yourself. And I was not used to such a paradigm.

Secondly, if you asked me "what is the Helmholtz equation" in 2001, my answer would be "I do not know." I did not know the problem at hand, let alone the difficulty to iteratively solve it. One day in the early phase of the research, in an email, my supervisor asked an expert in iterative methods about the Helmholtz equation. The reply was definitely negative. In Dutch they have an expression, "Een kat in de zag kopen," ("to buy a cat in a sack") and I did buy.

"If people know that one guy can solve a difficult problem, they may think of using their methods, with some modification, to solve the same problem."

I always see challenges as a means to progress. To progress, I had to work hard and learn hard. I brought to the table an idea to solve the problem after returning from a month-long vacation. Well, it seemed to me that every time I returned from my vacation, I got a new idea. One of my supervisors, Kees Oosterlee, once said that vacation always worked well for me.

I think challenges and a good balance between hard work and contemplation are the key to any success. You need some cleverness but it not is the sole thing you must rely on, and for a Ph.D. student good supervision is very important. I should thank my supervisors: Kees Vuik and Kees Oosterlee, who are also the co-authors of the paper.

**Where do you see your research leading in the
future?**

I learned that a US giant oil company will spend an incredibly large amount of money to re-direct their research towards a new methodology of predicting oil deposits, in which my research may become one of the key components. But we still have to wait and see.

As usual, development is one issue, dissemination is another issue, and science always requires time to mature and pay off. During this natural process, new and better methods may be discovered. And at the same time, I will probably have already switched gears to another research interest.

**Do you foresee any social or political
implications for your research?**

As far as I am concerned, I do not foresee any direct social or political implications. Referring again to oil industries, with the world's known reserves so much depleted amid high demands, any new method that is able to more accurately predict or find new oil deposits will always be welcome. We know how oil affects the world's politics, economy, and social life.

My educational background has made me a unique person: an aerospace engineer who does mathematics and digs into the Earth to find oil. Despite new inventions and contributions from many people around the globe, the difficulties that I experienced in dealing with one important component in finding oil reserves taught me to be less dependent on oil. Life should be greener than before.

**Yogi Ahmad Erlangga, Assistant Professor**

**Alfaisal** **University**

**College** **of Sciences** **and General
Studies**

**Riyadh****, Kingdom of Saudi Arabia**

KEYWORDS: HELMHOLTZ EQUATION; NONCONSTANT HIGH WAVENUMBER; COMPLEX MULTIGRID PRECONDITIONER; FOURIER ANALYSIS, ABSORBING BOUNDARY-CONDITIONS; NONSYMMETRIC LINEAR-SYSTEMS; FOURIER-ANALYSIS; EQUATION; SIMULATION; SOLVER; WAVES.