2nd JOINT RUSSIAN-AMERICAN WORKSHOP

on Inverse Problems in Engineering

MOSCOW AVIATION INSTITUTE FINAL REFORT submitted to the Ministry of Science and Technical Policy of RUSSIA 2nd JOINT RUSSIAN-AMERICAN WORKSHOP on Inverse Problems in Engineering St.Petersburg, Russia August 21, 26, 27, 1994 Oleg M. Alifanov, Co-principal Organizer James V. Beck, Co-principal Organizer Aleksey V.Nenarokomov, Report Co-Editor Keith A.Woodbury, Report Co-Editor

TABLE OF CONTENTS INTRODUCTION................................................. 1. INTERNATIONAL COLLABORATION IN INVERSE PROBLEMS IN ENGINEERING......................................... 2. WORKSHOP PARTICIPANTS.................................... 3. PROGRAMME OF WORCKSHOP.................................... 4. RECOMMENDATIONS...........................................

The present study report is devoted to a summation of the 2nd Russian-American workshop "Inverse Problems in Engineering" held in St.Petersburg on 21-27 August, 1994.

In the past two decades the methodology of inverse problems is actively introduced in different branches of technology. Methods and algorithms for inverse problem solving set up a basis for optimal design of structures in aerospace and nuclear technology. They are used in simulation and diagnostics of processes in medicine, engine manufacturing, power engineering, metallurgy as well as in studying mechanical, thermoengineering and optical properties of new materials and in controlling transportation means, robots- manipulators and technological processes. The inverse problem methodology is refered to the most dinamically developing branch of modern science. It finds multivarious applications in technology. Big progress in ill-posed inverse problems solving was due to the achievements of country's mathematical schools and in the first run Academician Tikhonov's school. Scientists from Moscow Aviation Institute (MAI), Moscow State University (MGU), TsAGI, Bauman State Technical University (MGTU), Keldish Institute of Applied Mathematics (IPM) have recently developed unique and subject- oriented program complexes realizing the solution of inverse problems.

Similar work is being done front-wide in USA and France, in witness of which are the materials of representative scientific conferences of recent years: in Portsmooth, Great Britain, 1990; in Suzdal, USSR, 1990; in East-Lansing, USA, 1992; in Palm-Coast, USA, 1993.

At the first American-Russian workshop on Inverse Heat Transfer Problems held in East-Lansing ,Michigan, USA, in 1992 and financially supported by the USA National Scientific Foundation, the program was elaborated indicating the priorities in directions of scientific collaboration in the field of inverse problem solving.

The experience in co-operation with colleaques from foreign countries as USA, France, Japan et.al., shows that russian scientists can keep the high level in this field provided a sufficient support from the Government and Russian Academy of Sciences.

Under the auspices of the State Committee of High Education of Russia, the Ministry of Science and Technical Policy of Russia and the Russian Foundation for Fundamental Research (RFFI), 17 specialists in the area of inverse heat transfer (IHT) problems have met in August 21, 26, and 27 in St. Petersburg, Russia. The purpose of this workshop was to establish directions for research in the field and to foster collaboration between specialists from different countries.

The second Russian-American meeting was a new step in strengthening the international collaboration of scientists. The meeting has covered the broadened subjects having for its object the discussions of plans and results of joint scientific projects.

In order to set up directions for research in the field of inverse heat transfer (IHT) problems, the group reviewed the recommendations of the 1992 workshop. Twelve original recommendations were reaffirmed as areas in need of further research. The workshop participants revised 12 points and set forth the recommendations which follow. Specifically, three of the original points (1, 2, and 4) were combined into a single area, while a new fundamental recommendation was added.

The issues of discussion were as follows:

a) Development of methods and algorithms for multidimensional inverse problem solving.

b) Setting up the data base for numerical and physical experiments on verification of methods of inverse problem solving.

c) Investigation and improvement of mathematical models of heat- and-mass transfer including physical-and-chemical transformations in materials.

d) Identification of transfer parameters in the problems of radiative and radiative-conductive heat transfer.

e) Investigation of the measuring systems role in gaining results by means of identification algorithms.

f) Planning of physical experiments in the problems of parametric identification.

g) Development of experimental plants, instruments, sensors required for conducting investigations based on inverse problems.

The following directions of activity are closely related with the solution of the above problems:

h) Publishing projects. Issuing of monographs,training and reference aids, international journal "Inverse problems in engineering".

i) Organization and functioning of an international institute aimed at the development of inverse problem methodology.

Taking part in the conference were 17 scientists successfully working in the area of inverse problem theory and application including 9 from Russia, 4 from USA, 2 from Ukraine, 1 from France, 1 from Germany.

Characteristic of the present working meeting was the circumstance that within its frames the 2nd International Conference on Dynamic Systems Identification and Inverse Problems was organized and held in August 22-25, 1994, the majority of participants having been members of the International Programming Committee of the conference or from its Organizing Committee.

115 applications were submitted for participation in the conference. Taking part in the conference were over 60 persons from 45 organizations representing Russia, Ukraine, Belorussia, Kazakhstan, USA, France, Germany, England. 52 reports were heard and discussed at the plenary and sectional sessions.

Both the workshop and the conference were organized by Moscow Aviation Institute, Moscow State Technical University and Institute of Precise Mechanics and Optics (St.-Petersburg). The conduction of the conference and the workshop was financially supported by the Ministry of Science and Technical Policy of Russia, by the State Committee on Higher Education of Russia, and by the Russian Foundation on Fundamental Research.

Principal Organizer of the 1-st Joint American-Russian workshop Professor J.V.Beck has had numerous contacts, direct and indirect, with colleagues in the Former Soviet Union (FSU) dating back to 1982. Later he was asked to review Professor O.M.Alifanov's book "Identification of Heat Transfer Processes of Flying Vehicles" for possible translation and publication in English.

Early in 1988, Professor S.V.Rumyantsev made a short visit to Michigan State University. During the visit, an informal seminar was held at MSU in June 1988, with about 20 people (more than half from other universities) attending. Further seminars were requested by the group of researchers and seminars were held in 1989 (about 35 attending), in 1990 (about 50 attending), in 1991 (about 50 attending), and most recently in June 1992 (about 50 attending). The Russians researchers were invited, but were unable to attend until 1992, while the French, on the other hand, have been actively participating in the seminars since their inception.

Professor J.V.Beck was asked by the Russian Organizing Committee to organize a session and to invite some researchers to attend the International Conference on Dynamic Systems Identification and Inverse Problems at Suzdal, USSR (September 10 - 14, 1990). The main emphasis was almost the same as that at the MSU seminars, namely inverse heat transfer problems related to solids. Professor Beck led a delegation of seven professors to the USSR to the Conference (including Martin Raynaud of France), and more USA and Germany researchers attended by the invitation of the Organizing Committee. Also, in May of 1991, Professors E.A. Artyukhin and J.V.Beck were invited to the French Heat Transfer Conference by Professor J.P.Bardon and they visited the ISITEM at the University of Nantes.

Further collaboration of scientists was manifested at the 1st American- Russian workshop on inverse heat transfer problems held on 13-15 June, 1992. The USA National Scientific Foundation has sponsored the meeting inviting Professors O.M.Alifanov and E.A.Artyukhin from MAI, Professor S.V.Reznik from MGTU, Senior researcher V.M.Yudin from TsAGI, doctoral student A.V.Nenarokomov from MAI.

In addition, these scientists were able to participate in the 5th annual seminar on Inverse Heat Transfer Problems at the Michigan State University on June 11-12. Over 50 scientists from USA and 4 from France partipated in the working meeting and in the workshop.

This event being over, A.V.Nenarokomov, doctoral student from MAI was invited by Professor A.F.Emery (University of Washington) for 9 months to carry out research work, and Professor E.A.Artyukhin from MAI - to Nantes University for 30 months.

In 1993 Professor J.V.Beck, on the invitation of Professor J.P.Bardon, made a two-week visit to Nantes University.

The 1st international conference on Inverse Problems in Engineering (Theory and Practice) was held in June 1993 in Palm-Coast, Florida, USA. Taking part in it were scientists from USA, France, Germany, Japan, Czechia as well as two participants from Russia (Professor E.A.Artyukhin, MAI and Professor S.V.Reznuk, MSTU) and one from Ukraine (Y.M.Matsevity, Corresp. Member of the Ukrainian Academy of Sciences, Institute for Problems in Machinery). The conference was organized by American participants of the 1st American-Russian workshop.

Thus, while fruitful interactions between researchers of three countries existed before and have the background, they were reinforced during August 21-27, 1995, when the Russian participants of the 1st meeting have organized the 2nd Russian- American workshop on Inverse Problems in Engineering. Its running was combined with the work of the 2nd international conference on Dynamic Systems Identification and Inverse Problems which topic became the subject-matter of the present study report.

Name Affiliation Country Oleg M. Alifanov Moscow Aviation Institute Russia James V. Beck Michigan State University USA Ben Blackwell Sandia National Laboratory USA Sergei A. Budnik COSMOS - Moscow Aviation Russia Institute Alexandr M. Denisov Moscow State University Russia Yvon Jarny Nantes University, ISITEM France Leonid A.Kozdoba Institute of Technical Ukraine Thermal Physics Nicholai V. Kerov Moscow Aviation Institute Russia Yuri M.Matsevity Institute for Problems in Ukraine Machinery Valery V.Mikhailov Moscow Aviation Institute Russia Vladimir A. Morozov Moscow State University Russia Diego A. Murio University of Cincinnati USA Aleksey V. Moscow Aviation Institute Russia Nenarokomov Hans-Jurgen University of Siegen Germany Reinhardt Sergei V. Reznik Moscow State Technical Russia University Keith A. Woodbury University of Alabama USA Valery M. Yudin TsAGI Russia

Sunday, August 21, 1994 9.00 Opening and Welcome. Vice-Rector IPMO N.V.Vasil'ev 9.30 - 13.00 Session I 1. Presentation of O.M.Alifanov, A.V.Nenarokomov (Moscow Aviation Institute) 2. Presentation of V. Morozov (Moscow State University) 14.00 - 18.00 Session II 3. Presentation of Yu.M.Matsevitiy (IPM, Khar'kov, Ukrain) 4. Presentation of L.A.Kozdoba (ITTF, Kiev, Ukrain) 5. Presentation of S.V.Reznik (Moscow State Technical University) Friday, August 26, 1994 9.00 - 13.00 Session III 1. Presentation of J.V.Beck (MSU, USA) 2. Presentation of H.-J.Reinhardt (Siegen University, Germany) 3. Presentation of B.F.Blackwell (Sandia National Laboratory, USA) 14.00 - 18.00 Session IV 1. Presentation of D.A. Murio (UC, USA) 2. Presentation of Y.Jarny (ISITEM, France) Saturday, August 27, 1994 9.00 - 18.00 Session V Discussion about new steps in the field of inverse problems, based on the topics under consideration at the 1-st Workshop (MSU, East-Lansing, USA). Discussion about the future collaboration

Modern methods of solving ill-posed and inverse problems in heat and mass transfer require a rational combination of physics, mathematics, and engineering, including experimentation. This combination actually forms a new research paradigm. The same methodology can be applied to many research problems, in particular to design and control.

Characterization of a new or unknown process usually involves the following steps:

- Construction of a mathematical model of the process from basic physical laws. This is a mathematical statement of the direct problem governing the process.
- Development of inverse problem (IP) algorithms necessary to solve the corresponding inverse problem. The inverse problem naturally arises when parameters or functions in the mathematical model need to be determined.
- Mathematical validation of the IP algorithm by numerical experiments.
- Design of experiments to gather information about the process in order to solve the IP.
- Experimentation (data gathering).
- Use of the IP algorithms to identify the unknown information about the process. Analysis of these results, including statistical treatment, is crucial to verify the correctness of the results and ultimately, the adequacy of the description of the process.

The new research paradigm is the process of conducting these steps by a research team working closely together. This is in contrast to a more commonly observed research effort where these different steps are assigned to widely- separated, highly specialized research departments within a large organization.

Because most inverse heat transfer (IHT) problems are ill- posed or ill-conditioned, it is very important to construct special algorithms for their solution. These methods of solving ill-posed and inverse problems are extremely important because they hold the promise of describing processes and devices that cannot be analyzed in simpler ways. Examples include transient exothermic curing of composite materials, ablation of surfaces of advanced aerodynamic vehicles and various processes in manufacturing including quenching, casting, welding and machining. This field is positioned to make unique contributions for the understanding and advancement of many complex processes in heat and mass transfer. Furthermore, these methods can be applied widely to problems in many other fields including geo-physics, medicine, astrophysics.

A very important aspect in the development of algorithms for ill-posed IHT problems is their validation by testing them against special selected model problems. Such computational experiments enable the efficiency (accuracy and speed of calculations) and range of applicability of the algorithms to be determined. However, researchers use different test examples making it difficult to render a true comparison of various algorithms. The organization of an international team of researchers (5-7 persons) is recommended to develop a system of test problems and input data as well as the criterion for comparison and certification of the algorithms.

Many applications (such as plate quenching, casting, forging, and many other industrial processes) require estimation of properties or boundary functions in two-and three-dimensional spaces These problems are geometrically complex and are probably best solved by unstructured grid or finite element based methods.

Since 1992, much work has been done in the area of multidimensional IHT algorithms, but mostly these works are applicable only to regular coordinate systems. The following multidimensional IHT problems are known to have been solved: 2-D and 3-D IHCP (boundary identification) in Cartesian, spherical, and cylindrical coordinates, and geometry identification in steady state phase change.

Of particular importance, due to the complexity of these problems, is the development of algorithms based on parallel computer architecture. Parallel computing algorithms are perceived to offer advantages for multi-dimensional inverse problems. However, at present there are few researchers working in this area.

The overall task of developing methods and algorithms for multi-dimensional problems is recognized as having two distinct components: development of methods for solving the corresponding direct problem, and development of the inverse algorithm.

Effective application of inverse problem methods requires development of proper algorithms and computer codes for solution of the corresponding direct heat transfer problems. In fact, the direct problem usually must be solved many times (for sensitivity coefficients, temperatures, etc.), so especially efficient direct problem solvers are needed.

Real heat-loaded structures usually have complex multi- dimensional and multi-connected forms. Heating or cooling of such structures takes place by conductive, radiative, and convective heat transfer, and will include contact heat transfer in various immovable and movable joints. As a result it is necessary to solve multi-dimensional mixed heat transfer problems for complex composite domains with one-, two- and three-dimensional components, such as rods, thin plates and massive bodies.

One approach is to use multi-domain solution methods to analyze each of the subcomponent portions of the geometry. Efficient implicit algorithms for solving the direct problems for these components and algorithms for non- iterative connection of the separate numerical solutions obtained for the components are necessary. Also, efficient algorithms for calculation of radiative heat transfer and geometric characteristics in an arbitrary system of specular-and-diffuse surfaces are needed for problems involving radiation.

Good commercial packages are available which can solve multi- dimensional problems which include one-, two-, and three- dimensional components such as rods, thin plates, and massive bodies. In spite of the availability of powerful commercial codes, IHT problem researchers need and should have access to source code. Some codes for unstructured grids have been developed at US National Labs, and the MODULEF library, the basis of complex two- and three- dimensional computations, is available from INRIA (France). Identification of these codes and their availability is important to the IHT community.

Many methods of solution exist, for example finite control volume, finite element, and boundary element methods. Some existing codes, such as TOPAZ, may be suitable for integration with inverse problem solvers. However, these methods and codes must be surveyed and classified for their suitability to different classes of problems. Of course, these computer codes must be capable of interfacing with modern pre- and post-processing software. Improvements in speed of the direct codes must be sought, particularly including solution on parallel machines.

o further promote development of practical applications of multidimensional algorithms it is necessary to modify available methods and algorithms for solving IHT problems. Additionally, efficient new methods of solution resulting in a reduction of computational time should be developed. It is essential to develop efficient gradient and non-gradient iterative methods for solution of inverse problems involving complex and multi-dimensional heat transfer.

Some inverse methodology exists which logically can be extended to higher dimensions, but some questions of correctness (existence and uniqueness) have not been adequately addressed. Among the approximation methods there are variational methods including iterative regularization using gradient methods, mollification methods including space marching schemes, and many others. The analysis of such methods should be directed toward existence and uniqueness questions as well as stability and error estimates including stability properties of the numerical schemes and/or an experimental investigation and verification of the methods.

The multidimensional applications will also require definition of experimental conditions, especially the location of sensors, which will ensure algorithm convergence and accuracy. Other questions, such as the choice of approximation of the unknown functions in higher dimensions, should be answered.

In the development of new methods and tools for practical application of inverse problems methods, it is very important to have databases for test examples to validate the methods. These databases are realized through application of modern computing technologies and inverse problem methods needed in practical use. A small committee to work on this should be formed.

It is recommended that two forms of databases be generated, one numerical and the other experimental. The numerical database can be utilized by investigators who create new algorithms for inverse problems as benchmark cases to test their new methods. Any such numerically generated data should include simulated noise (errors). The efficiency and accuracy of the different inverse methods can then be investigated using the numerical data. Methodologies can also be validated using experimental problems for which the "unknown" characteristics to be estimated are measured previously. Both databases should be extended beyond heat conduction to radiation, convection and other problems.

Equally important are the identification of new reference materials for experimental researchers. Due to the availability of new thermostable materials, it is reasonable to carry out some study of new standard specimens (reference objects) for heat conduction. These should be established for test measurements in a wide range of temperatures, including the materials with low (less 0.1W/m-K) and high (over 50W/m-K) conductivities.

Another important aspect of experimental studies is the investigation and characterization of real measurement errors. (correction schemes may be needed.) Real errors tend to have characteristics not present in simulated errors such as being correlated and biased. One experimental approach uses multiple sensors (even of different sizes and kinds) to measure the same temperature. This problem is also addressed in section 4.5.

The principal objective of this recommendation is the development and improvement of mathematical models for complex heat transfer processes that can be solved by the use of inverse algorithms. The processes of interest include:

- ablation
- irradiation from opaque surfaces
- gas injection through porous media into boundary layers
- convection in thin layers or gaps
- internal charges/destruction as a result of chemical reactions or combustion
- phase change including solidification/melting
- radiation and convective heat fluxes from boundary layer thermo-diffusion

The above models have application in the design of thermal protection systems, spacecraft, aircraft, engines, nuclear reactors, chemical reactors, food production equipment, gas turbines and other systems.

Inverse heat transfer problems involving complex heat transfer models require supplemental data beyond temperature data. For example, in a curing process, it is necessary to measure the degree of cure in addition to temperature. The list above is certainly not all-inclusive, as other processes involving other (possibly coupled) physical phenomenal are also important.

Problems in this area which have been solved include the curing of rubber.

The objective of this recommendation is to systematically use inverse methods for the determination of radiative properties of opaque and semi-transparent materials. These properties mostly have been determined by direct measurements through the use of very expensive experiments. The new approach will first focus on using the same experiments but with the use of inverse methods in order to improve the results. In addition to these laboratory experiments, numerically-simulated experiments must be performed to assess new algorithms and techniques needed to improve the interpretation of the laboratory experiments.

There are many situations where radiative measurements are needed to describe surface properties. Ordinary opaque engineering materials typically reflect radiation with a specular component plus an angle-dependent diffuse component. The angle-dependence of the surface reflectivity is needed to correctly compute the heat transfer into the medium. The measurement of the reflection coefficient for semi-transparent materials is even more difficult because of the influence of the underlying material (i.e. substrate). A better knowledge of reflection coefficients will improve the fabrication of engineering materials.

New techniques to assess the absorption, scattering, and other properties of bulk materials are needed. This is because new engineering materials often are anisotropic, so traditional methods of solving even the direct problem no longer are adequate, and such solutions are necessary for iterative inverse methods.

The measurement of the temperature field within a medium is important, especially for semi-transparent materials. This is because the production of glasses and ceramics that undergo a transformation from an amorphous state to a crystalline state are highly dependent on the temperature.

Theoretical substantiation and experimental verification are still important in the effort of simultaneous determination of thermophysical and optical properties of partially transparent materials.

Problems which have been addressed in this area include volumetric parameter estimation and initial state estimation.

This is an important area. Observation equations, which are used in control theory to relate measurements to system state variables, should be used in IHT problems as well.

All contact thermal sensors provide values of their own temperatures, rather than the desired temperatures of the undisturbed surrounding media. Contact sensors in solids have errors caused by imperfect contact with the medium, by dissimilar thermal properties, and by other factors, such as electrical fields. Detailed thermal models are needed to account for these factors. If the thermal characteristics of the sensors are not considered, then large errors can occur in the estimated values of heat fluxes, thermal properties and other quantities.

To address this problem appropriate sensor models must be formulated for different sensors, including the sheathed thermocouple, the bare thermocouple, and the welded thermocouple. Often the parameters of the measurement model will depend upon the peculiarities of the sensor's installation. Hence, further modification of conventional inverse methods will be necessary to characterize the parameters of these measurement models for in situ sensors.

Two approaches can be used for the analysis. The first one is to consider this problem separately from an inverse problem and to analyze different errors (statistical and systematic) of thermosensors including their dynamic properties and the connection between a body material and thermosensors. The second approach is to consider the measurement model as part of a general (combined) inverse problem. From the point of view of correctness of the inverse methodology the second one is more accurate but presently this approach is often difficult to apply because of computational difficulties.

The above comments are valid not only for temperature sensors (thermocouple, resistance thermometers, etc.), but also for other heat transfer measuring devices such as heat flux transducers, infrared thermography, fiber optics and others.

The nature of inverse problems in heat transfer demands the application of special mathematical and statistical techniques. In particular, the utilization of powerful numerical methods is required. As problems become more complex, it is necessary to develop with greater rigor and understanding the most important aspects of inverse problems, in particular the study of the questions of existence and uniqueness of the solutions, the stability analysis of the numerical methods, and the errors associated with the numerical and statistical aspects of measurements.

Results of a solution of an inverse problem should include mean values for desired parameters/functions and confidence regions. By using different methods for inverse problems one can get both unbiased and biased estimates. Error analyses are needed for such estimates, with all biased cases requiring more study.

Statistical errors are important but are not being adequately addressed. Error characteristics of measuring devices should be reported and compared with residuals. Confidence intervals (or uncertainty estimates) should also be given. It must be recognized that there are at least three types of errors: modeling errors, measurement errors, and numerical errors.

One of the most important problems in IHT problems is the design of heating apparatuses and instruments for measuring temperature fields and heat fluxes. These are needed to study temperature-dependent thermophysical and optical properties, to employ one-dimensional inverse algorithms for isotropic material, and to use multi-dimensional inverse problem algorithms for orthotropic materials. Such apparatuses will facilitate measurement of thermophysical and optical properties from cryogenic up to extreme temperatures and from low to high pressures. Among candidate methods for high temperature heating are arc-jets, arc- lamps, solar concentrators, lasers, and other high energy systems. The design of new instruments for measuring temperature and radiation intensity fields is also very important. A related important need is the development of sensors to measure boundary heat fluxes for commercial dissemination, e.g. miniature thermocouples with elastic electrical insulation.

It is necessary to widely introduce computational technologies both for experiment planning and data reduction. The present theory for design of experiments (e.g., D-optimality) should be expanded and applied to many problems. Optimal experiments should be designed and run. Sensors with appropriate frequency response should be designed and used.

Many thermal processes require new modeling and acquisition of process behavior in order to achieve control of these processes. In effect, these applications cry out for application of the "new research paradigm". Very nice work which typifies this idea has been done on control of curing of elastomers.

The principle objective of this recommendation is the development of methods for the optimization of thermal processes. Some control and design problems of interest are:

- Control of phase-change problems, including solidification/melting front motion and interface heat fluxes, especially for alloyed materials
- Control of volumetric heating of materials (such as inductive heating and microwave heating)
- Control of residual stresses in solids (optimal heating or cooling)
- Optimal design of multi-layer heat protection systems (including porous media).

The theory of inverse methods facilitates the collection of information in situations where conventional methods fail. The potential for use of these methods in applications should be disseminated widely so that a typical researcher will be aware of them. This can be accomplished by the publication of journals, textbooks, handbooks, and encyclopedias, and through the development of short courses in the field of inverse problems in engineering.

The literature on inverse problems presently is dispersed throughout numerous journals dealing with specific engineering applications. Inverse problems also are investigated in the fields of geo-sciences (such as oceanography and atmospheric sciences) and physics, for example. Only one journal, Inverse Problems, currently exists, and the focus primarily is on mathematical methods and proofs of less interest to those designing inverse experiments in engineering.

The journal aimed at inverse problem applications and methods for engineering analysis may soon be realized. Inverse Problems in Engineering is the journal described in the 1992 Workshop report, and its publication is now imminent.

To further advance the field of inverse heat transfer, it is also important that researchers begin to think about preparing an edited collection of longer-length review articles. Also needed are handbooks of results on heat and mass transfer parameter estimation, including effects of radiation, chemical reactions, and ablation. There are no such handbooks for specialists. As a result they encounter considerable difficulties in their investigations. For example, it would be very useful to prepare and publish a special encyclopedia about problem statements, mathematical studies, solutions, and experimental application of inverse heat transfer problems. The handbooks would involve the main mathematical aspects, statements of the problems, methods and algorithms of their solution, test examples, and other topics.

Preparation of such edited collection of longer-length review articles and handbooks would best be done only by efforts of special international teams of scientists and specialists. At present, the situation is such that countries such as Russia, USA, and France possess unique developments and experience in certain areas, and only as a result of cooperation will it be possible to publish the best possible reference books.

Of more immediate need now, however, is the development of materials to educate new people in the field. This could be promoted by the NSF and/or RFFI, for example, through grants for authors preparing textbooks or monographs, or for the translation of existing monographs (especially from Russian into English). While a few books exist at present (some only in Russian), these works are primarily on theoretical aspects, so the greatest need is for textbooks on methods for experiments to measure parameters and functions with inverse methods.

Short courses are another mechanism for transmitting information about inverse problems. The objective of a short course could be to disseminate to experimentalists the methodologies that have been developed, so that they could improve the quality of their experiments and data processing.

Since 1992, several short courses have been given in USA. Two, sponsored by NSF, were for college teachers. At least five books [1-5] have appeared, and the journal Inverse Problems in Engineering is nearing production. An International Conference Inverse Problems in Engineering: Theory and Practice, sponsored by the Engineering Foundation, was held in USA. Informal seminars have continued to be held in USA, and two have been held in FRANCE. Also, special lectures have been given in China and Colombia, as well as the USA.

An ECMI/SIAM international conference Inverse Problems and Optimal Design in Industry was held in Philadelphia, July 1993; a GAMM/SIAM international conference Inverse Problems in Diffusion Processes was held in Austria, June 1994. (ECMI = European Consortium for Mathematics in Industry; GAMM = Gesellschaft fr Angewandte Mathematik und Mechanik; SIAM = Society for Industrial and Applied Mathematics)

[1] Murio, D. "The Mollification Method and the Numerical Solution of Ill-Posed Problems", Wiley - Interscience, 1993.

[2] Groetsch, C.W., "Inverse problems in the Mathematical Sciences", VIEWEG, 1993

[3] Morozov, V. A., "Regularization Methods for Ill- posed Problems", CRC Press, 1993

[4] Alifanov O. M., "Inverse Heat Transfer Problems", Springer-Verlag, 1994

[5] Denisov A. M., "Introtuction in Inverse Problems Theory", MGU Publ., 1994 (in Russian)

At present there are many different methods for the solution of realistic problems using inverse methods. These problems include those in heat transfer and other branches in engineering science and mathematics. Some methods and tools are quite different, but some of them have common features. Often these inverse methods have been developed separately by different researchers in different countries and some methods are more general than others. Each country would benefit from cooperative efforts to analyze the features of different methods. An expected result of joint international efforts of scientists and engineers would be improved investigations, significantly increased effectiveness of practical applications, and more rapid dissemination of information relative to inverse heat transfer problems.

To carry out and coordinate cooperative investigations, it is highly desirable to establish the International Institute for Advanced Studies in Inverse Methods. Founders of this Institute could be the National Science Foundation (USA), the Russian Foundation for Fundamental Research (Russia) and the National Center of Scientific Research (France). The activities of the Institute could include the following:

- formulation of the most reasonable directions in the further development of the theory and methods for solution and application of inverse problems,
- generalization and systemization of the most important theoretical and practical results,
- development for carrying out experimental-and- computational studies as well as validation of methods and algorithms,
- conduction of joint investigations and projects, preparation and organization of international seminars and workshops on different subjects in the field, etc.

While heat transfer applications would logically form the basis of a large part of the Institute, efforts should be made to disseminate the techniques to other branches of engineering.

Although establishment of an international institute is an excellent long term goal, it has been seen since 1992 that national groups are forming. A formal group (METTI) has formed already in France, and a similar group is emerging in Russia. An informal group exists in the USA. An activity group of the GAMM on Inverse Problems exists in Germany. It must be noted that SIAM supports conferences on inverse problems as it was mentioned above.

Therefore, the spirit of this recommendation is to first foster informal cooperation between these groups, and to hold as a goal their unification under an international umbrella.

woodbury@me.ua.edu

May, 1995