Ill-posed problem - Factbites

Ill-posed problem - Factbites
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Topic: Ill-posed problem

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	Well-posed problem - Wikipedia, the free encyclopedia
		Problems that are not well-posed on the sense of Hadamard are termed ill-posed.
		A measure of well-posedness of a discrete linear problem is the condition number.
		The mathematical term well-posed problem stems from a definition given by Hadamard.
	www.wikipedia.org /wiki/Well-posed_problem (255 words)

	The Well-Posedness of the Problem (FBP)
		Depending on the relation between the boundary conditions, pathological cases of degeneration or ill-posedness of the problem are possible as well as a well-posed dissolution problem.
		We prove the well-posedness of the problem in a more general case, admitting a non-monotonicity of the free boundary as a result of the interaction between dissolution and precipitation.
		In the transformed problem (FBP), the right-hand side of the differential equation, g, represents the effect of the chemical reaction causing a consumption of u, since g is nonnegative.
	www.math.tu-cottbus.de /~pawell/publications/SIAM/node5.html (365 words)

	Multi-frame reconstruction: UCLA Vision Lab Projects
		The shading problem, when posed in a multi-frame context, can result in a well-posed estimation problem and nicely integrate stereo algorithms where the radiance of the objects in the scene is uniform and therefore stereo matching is ill-posed.
		We consider the problem of estimating the surface of an object from a calibrated set of views under the assumption that the reflectance of the object is non-Lambertian.
		We pose the problem within a variational framework and use fast numerical techniques to approach the local minimum of a regularized cost functional.
	www.vision.cs.ucla.edu /projects/variational.html (458 words)

	Condition number - Wikipedia, the free encyclopedia
		In numerical analysis, the condition number associated with a numerical problem is a measure of that quantity's amenability to digital computation, that is, how well-posed the problem is. A problem with a low condition number is said to be well-conditioned, while a problem with a high condition number is said to be ill-conditioned.
		Generally, if a numerical problem is well-posed, it can be expressed as a function f mapping its data, which is an m-tuple of real numbers x, into its solution, an n-tuple of real numbers y.
		Its condition number is then defined to be the maximum value of the ratio of the relative errors in the solution to the relative error in the data, over the problem domain:
	en.wikipedia.org /wiki/Condition_number (328 words)

	Mathematica Documentation: Is the Problem Well-Posed?
		An initial or boundary value problem is said to be well-posed if a solution for it is guaranteed to exist in some well-known class of functions (for example, analytic functions), if the solution is unique, and if the solution depends continuously on the data.
		Most problems that arise in practice are well-posed since they are derived from sound theoretical principles.
		Given an ODE of order n (or a system of n first-order equations) and n initial conditions, there are standard existence and uniqueness theorems that show that the problem is well-posed under a specified set of conditions.
	documents.wolfram.com /mathematica/Built-inFunctions/AdvancedDocumentation/DifferentialEquations/DSolve/WorkingWithDSolve/AdvancedDocumentationDSolveWorkingWithDSolveIsTheProblemWell-Posed.html (412 words)

	intro_illposed.htm
		Demmel, in his 1988 paper investigating on the probability that a numerical problem is difficult, gave a geometrical interpretation of the ill posedness defining the distance from the nearest ill posed problem in terms of the reciprocal of the condition number of that problem.
		We remark that from a strictly mathematical point of view, the discrete problem is well posed, as every nonsingular matrix automatically has a continuous inverse; however, from the computational point of view we consider the discrete problem as it were an ill posed problem.
		When Hadamard wrote his 1902 paper, defining ill posed those problems whose solution does not exist or it is not unique or it if is not stable under perturbations on data, it was with the intent of saving mathematicians and computational scientists substantial time and trouble.
	www.dma.unina.it /~damore/intro_illposed.htm (657 words)

	Citations: Ill-posed problems in the natural sciences - Tikhonov, Goncharsky (ResearchIndex)
		Dense motion registration estimation is an ill posed problem since the number of variables to be recovered is larger than the number of available constraints.
		In addition, the inverse problem is technically ill posed, thus leading to solutions that may be unstable with respect to small perturbations of the projections (e.g.
		The estimation of a dense correspondence registration field is an ill posed problem.
	citeseer.ist.psu.edu /context/353723/0 (1057 words)

	The Well-Posed Puzzle
		You can make some progress in attacking this problem, but at some point, you will find that the variables separate into two groups, and you can't come up with a relationship that allows you to finish the solution.
		The behavior of the problem is entirely determined by the properties of the matrix A.
		Posing the puzzles seems to have its own set of problems.
	www.csit.fsu.edu /~burkardt/papers/well_posed_puzzle.html (2860 words)

	situations.html
		The rhetor must be able to enter into an indeterminate situation and disclose or formulate problems therein; be must also present the problems in such a way as to facilitate their resolution by the audience engaged with him in the rhetorical process.
		The antinomy posed by Bitzer and Vatz is that either the rhetorical situation controls the acts of the rhetor or the rhetor freely creates the situation.
		In reference to the political and economic instabilities of Latin America, for example, one interest group may see the pressing "problem" to be prevention of communist expansion in the hemisphere, while another may see the problem to be one of raising the living standard of the Latin peasant.
	www.public.iastate.edu /~consigny/situations.html (3418 words)

	math lessons - Inverse problem
		Inverse problems are typically ill posed, as opposed to the well-posed problems more typical when modelling physical situations where the model parameters or material properties are known.
		In this sense the inverse problem of inferring m from measured d is ill-posed.
		The inverse problem is the task that often occurs in many branches of science and mathematics where the values of some model parameter(s) must be obtained via manipulation of observed data.
	www.mathdaily.com /lessons/Inverse_problem (679 words)

	lnotes.html
		This is a fundamental requirement when trying to address this problem numerically, where errors due to number representation and round-off means we're often solving a slightly perturbed problem.
		In other words, not only needs there to be a unique solution, but when the problem is perturbed slightly, another unique solution must exists which is close to the first.
	web.mat.bham.ac.uk /D.F.M.Hermans/msmxg6/ln/lnotes151.html (58 words)

	Well-posed, Ill-posed, and Intermediate Problems with Applications
		The authors introduce a third class of problems: intermediate ones, which are problems that change their property of being well- or ill-posed on equivalent transformations of governing equations, and also problems that display the property of being either well- or ill-posed depending on the type of the functional space used.
		Until recently, all problems in mathematics, physics and engineering were divided into two classes: well-posed problems and ill-posed problems.
		The book is divided into two parts: Part one deals with general properties of all three classes of mathematical, physical and engineering problems with approaches to solve them; Part two deals with several stable models for solving inverse ill-posed problems, illustrated with numerical examples
	www.brill.nl /product.asp?article_ID=1456&ID=24633 (250 words)

	Topic:
		Namely, the inverse problem is ill posed; it involves determining a 3-dimensional vector function of time (the current density in the heart at a given point and time) from a scalar function of 3 variables (the potential on the body's surface at the same time).
		The reason this type of problem is ill posed is that there are many possible solutions (current densities in the heart) that could give rise to the same surface potential function.
		This problem turns out to have a unique solution, and hence is considered well posed.
	www.cfm.brown.edu /people/mromeo/fall98/sam.htm (731 words)

	The Economic Approach to Artificial Intelligence
		But recognizing that AI's problems are in large part economic does help us to formulate the questions, and opens to us a variety of concepts and techniques that offer a starting point on potential solutions.
		First, that the fundamental problem to be solved is one of resource allocation.
		Inevitably, this problem bears on what we are to do, that is, some course of action to be embarked upon.
	ai.eecs.umich.edu /people/wellman/econAI.html (1319 words)

	Colloquium - 4 October 2001 - Department of Mathematics - University of Montana
		Tikhonov proposed a special approach for solving ill-posed problems: for searching for stable solutions of unstable ill-posed problems it is necessary to use special regularizing operators (algorithms), if they exist, which give stable approximations to exact solution of unstable problems.
		t present, the theory of ill-posed problems is developed and is widely used to solve inverse problems in optics, spectroscopy, electrodynamics, plasma diagnostics, geophysics, astrophysics, image processing, etc.
		The methods for solving ill-posed problems now available can be successfully used in various branches of natural sciences.
	www.umt.edu /math/colloq/fall01/100401.html (344 words)

	The Mathematics Behind Imaging - Feature Story - Scientific Computing and Imaging Institute
		In the 1940s, mathematician Andrei Tikhonov, working on a theory of geophysical exploration, published a paper starting the development of “regularization theory,” which considers ill-posed problems to be solvable and proposes methods for their solution.
		Another research direction being pursued at the SCI Institute is solving “ill-posed” imaging problems by constraining the solution with focusing criterion.
		If a problem is not proven to have an existing and unique solution, it is considered “ill-posed,” or unsolvable.
	www.sci.utah.edu /stories/2001/jan_imaging.html (633 words)

	halting.txt
		Undecidability and the Halting Problem -------------------------------------- The halting problem is the prototypical example of an undecidable problem, or a well posed problem that requires a yes or no answer that is impossible to solve.
		It should be stressed that the class of undecidable problems are impossible to solve not merely because of practical reasons such as time or memory requirements.
		As a prototypical example of an undecidable problem, the halting problem sheds light on the fundamental phenomenon of undecidability.
	keck.ucsf.edu /~surya/halting.txt (577 words)

	Numerical relativity - the binary black hole problem
		An ill-posed continuum problem is numerically intractable however it is discretised.
		While it is now clear that one has to start from a well-posed continuum problem, and much remains to be done to find the best formulation, the Einstein equations are also much more complicated than other well-known hyperbolic systems such as the Euler or MHD equations.
		Coordinate choices for the binary problem that are stable for more than a fraction of one orbit have hardly been explored, partly due to the technical difficulty that they involve coupled elliptic equations.
	www.maths.soton.ac.uk /~cg/numrel/numrel.html (1684 words)

	Initial Value Problem
		In what follows we will assume that we are dealing with a well posed problem!
		Furthermore, since problems encountered in the study of the physical phenomena only approximate the actual situation, it is of interest to know whether small changes in the statement of the problem induce correspondingly small changes in the solution.
		When attempting to solve a problem of this type it is important to know whether the solution exists and whether it is unique.
	www.physics.csulb.edu /phys360/rk1/rk-doc.htm (1148 words)

	IOM Web Site - Instruction
		Ill-posed problems (for which either no solution exists, the solution is not unique or the solution fails to depend continuously on forcing, initial conditions or boundary conditions) cannot be solved satisfactorily in the usual manner.
		The equation and associated conditions comprise a "well-posed" problem if (Hadamard, 1952; Courant and Hilbert, 1962) a solution exists, the solution is uniquely determined by the forcing, initial conditions, boundary conditions, and the solution depends continuously on the forcing, initial conditions, boundary conditions.
		We therefore begin by demonstrating the well-posedness of the one-dimensional convection equation by proving existence of a solution by constructing an explicit solution using the Green's function
	www.eas.asu.edu /iom/instruction/mod1wellposedness/mod1wellposedness.html (159 words)

	Jacques Hadamard - Wikipedia, the free encyclopedia
		He introduced the idea of well-posed problem in the theory of partial differential equations.
		He also gave his name to the Hadamard inequality on volumes, and the Hadamard matrix, on which the Hadamard transform is based.
		After the Dreyfus affair, which involved him personally, he became politically active and became a staunch supporter of Jewish causes.
	en.wikipedia.org /wiki/Jacques_Hadamard (195 words)

	2 Direct and Inverse Problem Formulation
		When Hadamard wrote his 1902 paper defining well- and ill-posed problems, it was with the intent of saving mathematicians and scientists substantial time and trouble.
		If a problem does not meet one or more of these criteria the problem is considered to be ill-posed.
		Since there are so many important ill-posed problems, mathematicians, scientists, and engineers have developed numerous methods to get around the problem of ill-posedness.
	www.ccs.uky.edu /csep/BF/NODE3C.html (243 words)

	Algorithmic Composition
		The basic requirements for a well-posed problem are that (1) the known information is clearly specified; (2) we can determine when the problem has been solved; and (3) the problem does not change during its attempted solution.
		Given both the problem and the device, an algorithm is the precise characterization of a method of solving the problem, presented in a language comprehensible to the device.
		a problem and a device to be used in solving the problem.
	eamusic.dartmouth.edu /~wowem/hardware/algorithmdefinition.html (3742 words)

	Title page for ETD etd-11172004-074040
		The ill-posed problem of quantizing space-time is replaced by a more determined and well-posed problem of regularizing quantum dynamics.
		The problem is then to eliminate the Heisenberg singularity from quantum mechanics as economically as possible.
	etd.gatech.edu /theses/available/etd-11172004-074040 (209 words)

	Appropriate boundary conditions
		Problems which arise in practical applications are usually ``well-posed boundary value problems''.
		This leads to the notion of a well-posed problem.
		If a problem is not well-posed, it is useless.
	www.maths.soton.ac.uk /staff/Andersson/MA361/node38.html (168 words)

	Colloquium - Department of Mathematical Sciences - The University of Montana
		t is very well known that ill-posed problems have unpleasant properties even in the cases when there exist stable methods (regularizing algorithms) of their solution.
		Tikhonov, A.N., Leonov, A.S. and Yagola, A.G., Nonlinear Ill-Posed Problems.
		any problems of science, technology and engineering are posed in the form of operator equation of the first kind with operator and right part approximately known.
	www.umt.edu /math/colloq/fall03/112003.html (317 words)

	Introduction
		In the present work we will address the problem of interpolating a given continuous function f between N+1 distinct points
		at some point can be well approximated by a polynomial of degree n (in the sense that its derivative is also approximated well).
		In other words, for a good approximation of f'as well as f the degree of the interpolating polynomial should be at least
	plato.la.asu.edu /papers/paper86/node1.html (331 words)

	The Well-Posed Problem - Jaynes (ResearchIndex)
		It will be helpful to think of this in a more concrete way; presumably, we do no violence to the problem (i.e., it is still just as "random") if we suppose that we are tossing straws onto the circle, without specifying how they are tossed.
		Bertrand's problem (Bertrand, 1889) was stated originally in terms of drawing a straight line "at random" intersecting a circle.
		1 Fifty Challenging Problems in Probability (context) - New, pp et al.
	citeseer.ist.psu.edu /jaynes73wellposed.html (434 words)

	Charleston Proceedings--Volume 3, Section G--3807_Paillet
		A well-posed inversion problem can be formulated to solve for estimates of fracture-zone transmissivity and hydraulic head if two different steady or quasi-steady flow profiles are obtained, along with drawdown information.
		Geophysical well logs are used to identify the depth where permeable fractures intersect boreholes in fractured bedrock aquifers.
		This information is used to sample fracture populations and to develop models for fracture flow networks at field sites.
	toxics.usgs.gov /pubs/wri99-4018/Volume3/SectionG/3807_Paillet (247 words)

	Problem Reformulation
		It is very likely that a proper choice for one or both can lead to well posed problems that are easier to solve.
		Unlike the constraint PDE which governs the optimization problem under study, the parameter space as well as the cost functional are not uniquely determined for a given engineering task.
		The problem given in section 4.1 is shown to have good stability properties for the high frequencies.
	www.math.cmu.edu /~shlomo/VKI-Lectures/lecture2/node8.html (281 words)

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