Solving Ill-Posed Problems in Scientific Computing

By By J. William Bell

May 5, 2006

It's only fair that a puzzle have only one solution, right? That's the great thing about a crossword or one of those Sudoku number games that the lifestyle pages can't stop talking about right now. There's only one way to fill in the boxes and get a complete answer.

Well, nobody said science was always fair. In what are known as ill-posed problems, there is no unique solution. A slight change in the data fed into the system of functions that rule a given ill-posed problem can produce a large, unpredictable change in the results.

“In the late 1800s, a scientist named Hadamard proposed that ill-posed problems didn't exist [or those that did weren't scientifically significant]. He was totally wrong. They're everywhere,” says Rebecca Hartman-Baker, who recently completed her PhD in computer science at the University of Illinois at Urbana-Champaign.

They're found, for example, in medical imaging, financial modeling, environmental modeling, and astronomy. Though Hartman-Baker's PhD thesis focused on an ill-posed problem found in the field of geoprospecting, the approach applies to any of those fields.

“Any ill-posed problem for which you have an educated guess of where to start could use this,” says Hartman-Baker, now a post-doc at Oak Ridge National Laboratory. “As a scientific computing type, the thing that brings me the most joy is to contribute to real-world scientific problems through my little thing. I want to get people who didn't know they needed to be involved in computer science to be in it.”

Making hundreds of runs on NCSA's Platinum and Tungsten clusters, she developed a selection method for choosing the parameters that go into the problem and an optimization method for finding the ideal result among a sea of solutions.

Geoprospecting typically involves placing transmitters and receivers deep in the earth. The transmitter projects electromagnetic energy, or in some cases sound, that is picked up by the receiver miles away. The electromagnetic energy is altered in transit, based on the conductivity of the rock, water, oil, or other materials in the ground. From the data collected by the receiver, researchers can deduce what lies between it and the transmitter.

The challenge is — and this is where the ill-posed problem comes in — that different sizes, shapes, and orientations of underground deposits can produce the same data profile.

“Basically, you're trying to find the size and shape of an ellipse [that represents the deposit that a geoprospector might be targeting, such as oil]. Where the center is at. The rotation or orientation. How you do that is really indirect — kinda backhanded,” Hartman-Baker says.

The traditional method of solving this sort of ill-posed problem, known as Tikhonov regularization, gives a blurry picture of this ellipse. It stabilizes the problem around a single solution by adding things like smoothness constraints to the functions. But in situations like this, researchers tend to prefer distinct boundaries. To get these boundaries, Hartman-Baker proposed another class of stabilizing, known as selection methods. With selection methods, the solution is limited to some reasonable set of possible solutions, and the parameters fed into the problem are reduced to a manageable number (about 10 in the case of Hartman-Baker's work). These decisions limit the computational expense of solving the problem and provide a distinct ellipse.

“We're bringing selection methods back into the world. People have forgotten about them,” she says. In her numerous runs on NCSA's systems, she tested the viability of the approximate quasisolution method.

The selection method reduces the number of parameters and shrinks the size of the solution space, but the optimal solution still has to be found. To do so, researchers make an educated guess of the input that will produce the optimal solution, run a simulation using that input, and compare the output to real-world data. They repeat the process until they find the input that most closely resembles the real-world output. In the example used by Hartman-Baker, that reveals the shape of the ellipse that produces the output.

To find the output that most closely resembled the real world, Hartman-Baker's example required finding the global minimum for a system of mathematical functions. After again trying several different methods, she settled on the diffusion equation method (DEM), which finds local minima and then traces back through a series of those local minima to the global minimum. Hartman-Baker compares this process to finding the lowest point in Kentucky. It's easy to find the lowest point in a region; you just head downhill until you are forced to start uphill again. But there are a lot of regional low points spread about, so ensuring that you've found the lowest of the low is no small task. DEM blurs the function “so that at first you just see overall trends, but not all the details. Then as it progresses, you unblur the function more and more until the full features [of the global minimum that you homed in on] are visible,” she explains.

As part of her thesis work and again using time at NCSA, Hartman-Baker discretized this involved method so that it can be parallelized and can use multiple processors at the same time. Currently, the evaluation of a function can be run on eight processors and takes about a minute. This has to be done several thousand times to solve a single optimization problem in geoprospecting.

This research was supported by the National Computational Science Alliance and, more recently, the UIUC computer science department's Fulton Watson Copp Endowed Chair held by Professor Michael Heath.


Source: Access Online, NCSA

Subscribe to HPCwire's Weekly Update!

Be the most informed person in the room! Stay ahead of the tech trends with industy updates delivered to you every week!

Supercomputer Research Reveals Star Cluster Born Outside Our Galaxy

July 11, 2020

The Milky Way is our galactic home, containing our solar system and continuing into a giant band of densely packed stars that stretches across clear night skies around the world – but, it turns out, not all of those st Read more…

By Oliver Peckham

Max Planck Society Begins Installation of Liquid-Cooled Supercomputer from Lenovo

July 9, 2020

Lenovo announced today that it is supplying a new high performance computer to the Max Planck Society, one of Germany's premier research organizations. Comprised of Intel Xeon processors and Nvidia A100 GPUs, and featuri Read more…

By Tiffany Trader

Xilinx Announces First Adaptive Computing Challenge

July 9, 2020

A new contest is challenging the computing world. Xilinx has announced the first Xilinx Adaptive Computing Challenge, a competition that will task developers and startups with finding creative workload acceleration solutions. Xilinx is running the Adaptive Computing Challenge in partnership with, a developing community... Read more…

By Staff report

Reviving Moore’s Law? LBNL Researchers See Promise in Heterostructure Oxides

July 9, 2020

The reality of Moore’s law’s decline is no longer doubted for good empirical reasons. That said, never say never. Recent work by Lawrence Berkeley National Laboratory researchers suggests heterostructure oxides may b Read more…

By John Russell

President’s Council Targets AI, Quantum, STEM; Recommends Spending Growth

July 9, 2020

Last week the President Council of Advisors on Science and Technology (PCAST) met (webinar) to review policy recommendations around three sub-committee reports: 1) Industries of the Future (IotF), chaired be Dario Gil (d Read more…

By John Russell

AWS Solution Channel

Best Practices for Running Computational Fluid Dynamics (CFD) Workloads on AWS

The scalable nature and variable demand of CFD workloads makes them well-suited for a cloud computing environment. Many of the AWS instance types, such as the compute family instance types, are designed to include support for this type of workload.  Read more…

Intel® HPC + AI Pavilion

Supercomputing the Pandemic: Scientific Community Tackles COVID-19 from Multiple Perspectives

Since their inception, supercomputers have taken on the biggest, most complex, and most data-intensive computing challenges—from confirming Einstein’s theories about gravitational waves to predicting the impacts of climate change. Read more…

Penguin Computing Brings Cascade Lake-AP to OCP Form Factor

July 7, 2020

Penguin Computing, a subsidiary of SMART Global Holdings, Inc., announced yesterday (July 6) a new Tundra server, Tundra AP, that is the first to implement the Intel Xeon Scalable 9200 series processors (codenamed Cascad Read more…

By Tiffany Trader

Max Planck Society Begins Installation of Liquid-Cooled Supercomputer from Lenovo

July 9, 2020

Lenovo announced today that it is supplying a new high performance computer to the Max Planck Society, one of Germany's premier research organizations. Comprise Read more…

By Tiffany Trader

President’s Council Targets AI, Quantum, STEM; Recommends Spending Growth

July 9, 2020

Last week the President Council of Advisors on Science and Technology (PCAST) met (webinar) to review policy recommendations around three sub-committee reports: Read more…

By John Russell

Google Cloud Debuts 16-GPU Ampere A100 Instances

July 7, 2020

On the heels of the Nvidia’s Ampere A100 GPU launch in May, Google Cloud is announcing alpha availability of the A100 “Accelerator Optimized” VM A2 instance family on Google Compute Engine. The instances are powered by the HGX A100 16-GPU platform, which combines two HGX A100 8-GPU baseboards using... Read more…

By Tiffany Trader

Q&A: HLRS’s Bastian Koller Tackles HPC and Industry in Germany and Europe

July 6, 2020

In this exclusive interview for HPCwire – sadly not face to face – Steve Conway, senior advisor for Hyperion Research, talks with Dr.-Ing Bastian Koller about the state of HPC and its collaboration with Industry in Europe. Koller is a familiar figure in HPC. He is the managing director at High Performance Computing Center Stuttgart (HLRS) and also serves... Read more…

By Steve Conway, Hyperion

OpenPOWER Reboot – New Director, New Silicon Partners, Leveraging Linux Foundation Connections

July 2, 2020

Earlier this week the OpenPOWER Foundation announced the contribution of IBM’s A21 Power processor core design to the open source community. Roughly this time Read more…

By John Russell

Hyperion Forecast – Headwinds in 2020 Won’t Stifle Cloud HPC Adoption or Arm’s Rise

June 30, 2020

The semiannual taking of HPC’s pulse by Hyperion Research – late fall at SC and early summer at ISC – is a much-watched indicator of things come. This yea Read more…

By John Russell

Racism and HPC: a Special Podcast

June 29, 2020

Promoting greater diversity in HPC is a much-discussed goal and ostensibly a long-sought goal in HPC. Yet it seems clear HPC is far from achieving this goal. Re Read more…

Top500 Trends: Movement on Top, but Record Low Turnover

June 25, 2020

The 55th installment of the Top500 list saw strong activity in the leadership segment with four new systems in the top ten and a crowning achievement from the f Read more…

By Tiffany Trader

Supercomputer Modeling Tests How COVID-19 Spreads in Grocery Stores

April 8, 2020

In the COVID-19 era, many people are treating simple activities like getting gas or groceries with caution as they try to heed social distancing mandates and protect their own health. Still, significant uncertainty surrounds the relative risk of different activities, and conflicting information is prevalent. A team of Finnish researchers set out to address some of these uncertainties by... Read more…

By Oliver Peckham

[email protected] Turns Its Massive Crowdsourced Computer Network Against COVID-19

March 16, 2020

For gamers, fighting against a global crisis is usually pure fantasy – but now, it’s looking more like a reality. As supercomputers around the world spin up Read more…

By Oliver Peckham

[email protected] Rallies a Legion of Computers Against the Coronavirus

March 24, 2020

Last week, we highlighted [email protected], a massive, crowdsourced computer network that has turned its resources against the coronavirus pandemic sweeping the globe – but [email protected] isn’t the only game in town. The internet is buzzing with crowdsourced computing... Read more…

By Oliver Peckham

Supercomputer Simulations Reveal the Fate of the Neanderthals

May 25, 2020

For hundreds of thousands of years, neanderthals roamed the planet, eventually (almost 50,000 years ago) giving way to homo sapiens, which quickly became the do Read more…

By Oliver Peckham

DoE Expands on Role of COVID-19 Supercomputing Consortium

March 25, 2020

After announcing the launch of the COVID-19 High Performance Computing Consortium on Sunday, the Department of Energy yesterday provided more details on its sco Read more…

By John Russell

Honeywell’s Big Bet on Trapped Ion Quantum Computing

April 7, 2020

Honeywell doesn’t spring to mind when thinking of quantum computing pioneers, but a decade ago the high-tech conglomerate better known for its control systems waded deliberately into the then calmer quantum computing (QC) waters. Fast forward to March when Honeywell announced plans to introduce an ion trap-based quantum computer whose ‘performance’ would... Read more…

By John Russell

Neocortex Will Be First-of-Its-Kind 800,000-Core AI Supercomputer

June 9, 2020

Pittsburgh Supercomputing Center (PSC - a joint research organization of Carnegie Mellon University and the University of Pittsburgh) has won a $5 million award Read more…

By Tiffany Trader

Global Supercomputing Is Mobilizing Against COVID-19

March 12, 2020

Tech has been taking some heavy losses from the coronavirus pandemic. Global supply chains have been disrupted, virtually every major tech conference taking place over the next few months has been canceled... Read more…

By Oliver Peckham

Leading Solution Providers


10nm, 7nm, 5nm…. Should the Chip Nanometer Metric Be Replaced?

June 1, 2020

The biggest cool factor in server chips is the nanometer. AMD beating Intel to a CPU built on a 7nm process node* – with 5nm and 3nm on the way – has been i Read more…

By Doug Black

Nvidia’s Ampere A100 GPU: Up to 2.5X the HPC, 20X the AI

May 14, 2020

Nvidia's first Ampere-based graphics card, the A100 GPU, packs a whopping 54 billion transistors on 826mm2 of silicon, making it the world's largest seven-nanom Read more…

By Tiffany Trader

‘Billion Molecules Against COVID-19’ Challenge to Launch with Massive Supercomputing Support

April 22, 2020

Around the world, supercomputing centers have spun up and opened their doors for COVID-19 research in what may be the most unified supercomputing effort in hist Read more…

By Oliver Peckham

Australian Researchers Break All-Time Internet Speed Record

May 26, 2020

If you’ve been stuck at home for the last few months, you’ve probably become more attuned to the quality (or lack thereof) of your internet connection. Even Read more…

By Oliver Peckham

15 Slides on Programming Aurora and Exascale Systems

May 7, 2020

Sometime in 2021, Aurora, the first planned U.S. exascale system, is scheduled to be fired up at Argonne National Laboratory. Cray (now HPE) and Intel are the k Read more…

By John Russell

Summit Supercomputer is Already Making its Mark on Science

September 20, 2018

Summit, now the fastest supercomputer in the world, is quickly making its mark in science – five of the six finalists just announced for the prestigious 2018 Read more…

By John Russell

TACC Supercomputers Run Simulations Illuminating COVID-19, DNA Replication

March 19, 2020

As supercomputers around the world spin up to combat the coronavirus, the Texas Advanced Computing Center (TACC) is announcing results that may help to illumina Read more…

By Staff report

$100B Plan Submitted for Massive Remake and Expansion of NSF

May 27, 2020

Legislation to reshape, expand - and rename - the National Science Foundation has been submitted in both the U.S. House and Senate. The proposal, which seems to Read more…

By John Russell

  • arrow
  • Click Here for More Headlines
  • arrow
Do NOT follow this link or you will be banned from the site!
Share This