## Saturday, November 13, 2010

### What are inverse or ill-posed problems? Part 1.

First publications on inverse and ill-posed problems appeared in first half of 20th century. They were related to scientific research in physics (inverse problem of quantum theory of scattering, electrodynamics and acoustics), geophysics (inverse problems of electric sounding, seismology, theory of potential), astronomy and other areas of natural sciences. With the emergence of computers, inverse problems penetrated all scientific disciplines that use mathematical methods. Major application are geophysics, astronomy, data visualization, medical and industrial tomography, defect detection and distance sounding and many others.

In the direct problem of mathematical physics, researchers tend to find (explicitly or approximately) the functions describing various physical phenomena, such as the propagation of sound, heat and seismic waves, electromagnetic waves, and so on. The properties of the medium (the equation coefficients), as well as the initial state of the process (nonstationary case) or its properties on the boundary (in the case of a limited area and / or in the stationary case) are assumed to be known. However, it is the properties of the medium, in practice, are often unknown. This means that you need to formulate and solve the inverse problem, which is required to determine any coefficients of the equations, or unknown initial or boundary conditions or the location, boundaries and other properties of the region in which the process under study. These objectives are in most cases ill-posed (i.e., at least one of the three properties of well-posedness - a condition of existence, uniqueness and stability of the solution with respect to small variations in the data of the problem, is violated). And the unknown coefficients of the equations are as a rule, the density, electrical conductivity, thermal conductivity and other important properties of the medium. Also, very often in inverse problems is required to find the location, shape and structure of the inclusions, defects, sources (heat, vibration, stress, pollution) and so on. It is no wonder that with such a broad set of applications, the theory of inverse and ill-posed problems since its inception has been one of the most rapidly developing areas of modern science.

Everyone solves inverse and ill-posed problem every minute. Humans solve them, as a rule, quickly and efficiently (if they are in good health and a clear mind). Let's consider, for example, visual perception. It is known that our vision fixes only a finite number of points around us. Then how are we seeing everything? Brain (in this situation - a powerful personal computer) uses seen points (interpolates and extrapolates) to recover all that the eye did not percept. It is clear that to recover true picture (in the general case - the volume and color) using only several points can only be the case when it is already more or less familiar with (most of the subjects and images we have seen, and sometimes touched by hands). That is, despite the strong ill-posedness (non-uniqueness and instability of the solution) of the problem (to restore the observed object and all that surrounds it by several points), the brain solves this problem pretty quickly. Why? It uses a wealth of experience (a priori information). And anyway, if we want to understand something complex enough to solve the problem, the probability of error which is large enough, we tend to arrive at an unstable (ill-posed) problem.

We can say that people (especially inclined to seek unconventional solutions) are constantly facing inverse and ill-posed problems. In fact, everyone understands how easy to make a mistake when trying to restore the past, based on some facts of the present (trace the motives and details of the crime on the available evidence, understand the reasons for initiation and development stages of the disease on the survey results and so on). Or look into the future (anticipated development of the child, the direction of the country and another complicated processes). Or penetrate into the zone of inaccessibility and to understand what is happening there (to investigate the internal human organs, to detect mineral deposits, to learn something new about the universe and so on). In fact, any attempt to expand the boundaries of direct (sensory, visual, auditory, etc.) perception of the world leads to an ill-posed problems. It would seem that we can say that, having learned to solve the stable (well posed) problem, mathematicians have moved on to more complex unstable (inverse and ill-posed) problems. But historically this is not true, since all the ages man has been surrounded by ill-posed problems, and mathematicians tried to solve such problems, dispensing with the relevant terms.

Inverse and ill-posed problems are united by one important feature - the instability of solutions to small errors in measurement data. In most interesting cases, inverse problems are ill-posed and such problems, as a rule, can be formulated as inverse with respect to certain direct (well posed) problems. But since inverse and ill-posed problems are historically formulated and studied quite often independently and in parallel, nowadays both terms are used in the scientific literature.

Summarizing, we can say that experts in the inverse and ill-posed problems
are involved in the study of properties and methods for the regularization of unstable problems. In other words, mathematicians are trying to create and explore sustainable methods of approximation of unstable maps. From the viewpoint of linear algebra this is a search of approximate methods for finding the normal pseudosolutions of systems of linear algebraic equations with rectangular, singular or poorly conditioned matrices. In functional analysis, a prime example of ill-posed problem is an operator equation Aq = f, where A is compact (completely continuous). Recently published studies interpret some problems of mathematical statistics as the inverse problem of probability theory. From the viewpoint of information theory, experts in the inverse and ill-posed problems investigate the properties of maps of compacts with a high epsilon-entropy in the table with a small epsilon-entropy.

In this introductory article we have learned about definitions of inverse and ill-posed problems, examples of their practical application and possible areas of development.
Next part will cover historical background of inverse and ill-posed problems.

## Thursday, September 9, 2010

### Amazon AWS Micro Instances

At last Amazon AWS came up with cheaper versions of their instances, as they call ‘micro instances’.

This change is pretty logical from economical standpoint, as in most cases clients need less resources than usually given by cloud provider. It is also important to note, that less usage implies less money spent on cloud resources. This was one of the reasons of my migration from Amazon AWS to Rackspace CloudServers product line, as it turned out to be cheaper and more powerful alternative.

Introduction of lower rate, however, still won’t make me move all of my deployments back to Amazon AWS, as besides economical part of the equation we still observe disparity in critical subsystems such as CPU cores. See below an output of cat /proc/cpuinfo ran on one of my Rackspace instances:

root@myhost:~# cat /proc/cpuinfo
processor : 0
vendor_id : AuthenticAMD
cpu family : 16
model : 2
model name : Quad-Core AMD Opteron(tm) Processor 2350 HE
stepping : 3
cpu MHz : 1995.002
cache size : 512 KB
fpu : yes
fpu_exception : yes
cpuid level : 5
wp : yes
flags : fpu de tsc msr pae cx8 apic cmov pat clflush mmx fxsr sse sse2 ht syscall nx mmxext fxsr_opt lm 3dnowext 3dnow constant_tsc rep_good nonstop_tsc pni cx16 popcnt lahf_lm cmp_legacy extapic cr8_legacy abm sse4a misalignsse 3dnowprefetch
bogomips : 3994.49
TLB size : 1024 4K pages
clflush size : 64
cache_alignment : 64
address sizes : 48 bits physical, 48 bits virtual
power management: ts ttp tm stc 100mhzsteps hwpstate

processor : 1
vendor_id : AuthenticAMD
cpu family : 16
model : 2
model name : Quad-Core AMD Opteron(tm) Processor 2350 HE
stepping : 3
cpu MHz : 1995.002
cache size : 512 KB
fpu : yes
fpu_exception : yes
cpuid level : 5
wp : yes
flags : fpu de tsc msr pae cx8 apic cmov pat clflush mmx fxsr sse sse2 ht syscall nx mmxext fxsr_opt lm 3dnowext 3dnow constant_tsc rep_good nonstop_tsc pni cx16 popcnt lahf_lm cmp_legacy extapic cr8_legacy abm sse4a misalignsse 3dnowprefetch
bogomips : 3994.49
TLB size : 1024 4K pages
clflush size : 64
cache_alignment : 64
address sizes : 48 bits physical, 48 bits virtual
power management: ts ttp tm stc 100mhzsteps hwpstate

processor : 2
vendor_id : AuthenticAMD
cpu family : 16
model : 2
model name : Quad-Core AMD Opteron(tm) Processor 2350 HE
stepping : 3
cpu MHz : 1995.002
cache size : 512 KB
fpu : yes
fpu_exception : yes
cpuid level : 5
wp : yes
flags : fpu de tsc msr pae cx8 apic cmov pat clflush mmx fxsr sse sse2 ht syscall nx mmxext fxsr_opt lm 3dnowext 3dnow constant_tsc rep_good nonstop_tsc pni cx16 popcnt lahf_lm cmp_legacy extapic cr8_legacy abm sse4a misalignsse 3dnowprefetch
bogomips : 3994.49
TLB size : 1024 4K pages
clflush size : 64
cache_alignment : 64
address sizes : 48 bits physical, 48 bits virtual
power management: ts ttp tm stc 100mhzsteps hwpstate

processor : 3
vendor_id : AuthenticAMD
cpu family : 16
model : 2
model name : Quad-Core AMD Opteron(tm) Processor 2350 HE
stepping : 3
cpu MHz : 1995.002
cache size : 512 KB
fpu : yes
fpu_exception : yes
cpuid level : 5
wp : yes
flags : fpu de tsc msr pae cx8 apic cmov pat clflush mmx fxsr sse sse2 ht syscall nx mmxext fxsr_opt lm 3dnowext 3dnow constant_tsc rep_good nonstop_tsc pni cx16 popcnt lahf_lm cmp_legacy extapic cr8_legacy abm sse4a misalignsse 3dnowprefetch
bogomips : 3994.49
TLB size : 1024 4K pages
clflush size : 64
cache_alignment : 64
address sizes : 48 bits physical, 48 bits virtual
power management: ts ttp tm stc 100mhzsteps hwpstate

root@myhost:~#

Amazon AWS Micro Instance provides 613 MB of memory, up to 2 ECUs (for short periodic bursts), EBS storage only, 32-bit or 64-bit platform

It is easy to see that CPU-wise Amazon has to add at least 2 cores to be comparable. We should also take into account other critical subsystems like disk and network throughput. Rackspace declares 20Mbit throughput for their 512Mb instance, while I couldn’t find official numbers from Amazon. Perhaps this could be a good exercise for yourself during weekends.

## Thursday, August 5, 2010

### UP-START Cloud Awards

Cloudcor and PricewaterhouseCoopers announce the 2010 UP-START Cloud Computing Awards recognition event to be held November 15, 2010 on Day 1 of the UP 2010 Cloud Computing Conference.

Are you redefining the next big thing in Cloud Computing?

Is your company defining cloud technology or is your product going to change the future of the Information Communications Technology (ICT) landscape?

Cloudcor Inc, seeks to broaden the end-user community’s understanding of where and how Cloud Computing will influence our future life, within both public and private sector environments.

We are looking for disruptive and next generation Cloud Computing companies, as well as innovative stealth mode / emerging solution providers, to step up to the plate and showcase their capabilities!

Here is the opportunity for you to showcase your products and solutions at the UP START Cloud Computing Awards.

Distinguished delegates to the inaugural UP START Cloud Awards will comprise of:

- Venture Capitalists,
- Expert Industry analysts
- Tech / Cloud evangelists,
- Innovators in the Cloud Computing space.

The UP START Cloud Awards, provide the ideal platform for emerging Cloud providers; (or those newly entering the Cloud Space), providing a springboard to success in this rapidly shifting and fast paced environment.

Winners in each category will receive a coveted industry recognition, as well as a tangible prize; consisting of an all expenses paid “Gold Partner Status” sponsorship, for the forthcoming “world renowned” CloudSlam’11 conference in Early 2011. (\$20,000+ equivalent value)

The judges comprising of recognized Cloud Computing industry experts, will award the most promising UP and coming business cases and unique creativity / differentiators, of the product / solution being profiled.

This will consist of the following award categories:

- Fastest Growing Cloud Computing Company Award
- Overall most innovative Cloud Computing Provider Award
- Most promising start-up cloud provider Award
- Efficiency in Technology Award

The judges on the day, will be announced shortly before the conference start date.

This ensures that the short listing process (and subsequent finalists), are determined solely on individual submission merit.

Members of the broader Cloud Computing, end-user and ICT community can also have their say by providing comments and feedback on finalists presentations.

If you have a viable Cloud Computing solution, driving innovation, or have the next big thing in Cloud Computing this is the event for you!

We invite stealth mode, emerging, first / second round financed, and more established companies, with innovative and cutting edge Cloud Computing solutions to get involved!

You can register for the UP START Cloud Awards by filling in an online form here.

The deadline for nomination submissions is September 30, 2010.

The finalists in each class will be selected from the submitted applications and will be invited to demonstrate in front of the judges and the conference audience on Monday,15th November 2010. (free access to all finalists)