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What is well-posed and ill-posed problems?

What is well-posed and ill-posed problems?

[edit] Mathematical description Such a system is called well-posed because it may correspond to a physical system. A system that fails to exhibit these properties is called ill-posed because it cannot be a correct and complete description of a physical system.

What is meant by ill-posed problem?

[′il ¦pōzd ′präb·ləm] (mathematics) A problem which may have more than one solution, or in which the solutions depend discontinuously upon the initial data. Also known as improperly posed problem.

What makes a problem well-posed?

A problem in differential equations is said to be well-posed if: (1) A solution exists; (2) That solution is unique; (3) The solution changes continuously with changes in the data.

What is an ill-posed problem in machine learning?

Ill-posed problems are typically the subject of machine learning methods and artificial intelligence, including statistical learning. These methods do not aim to find the perfect solution; rather, they aim to find the best possible solution and/or the solution with the least errors.

What is Cauchy problem in PDE?

A Cauchy problem in mathematics asks for the solution of a partial differential equation that satisfies certain conditions that are given on a hypersurface in the domain. A Cauchy problem can be an initial value problem or a boundary value problem (for this case see also Cauchy boundary condition).

What is well posedness in PDE?

A PDE is well-posed (in the sense of Hadamard) if (1) For each choice of data, a solution exists in some sense. (2) For each choice of data, the solution is unique in some space. (3) The map from data to solutions is continuous in some topology. In this case, u is called the solution, and u0 is the initial data.

What is mean by ill defined?

Definition of ill-defined : not easy to see or understand The property’s borders are ill-defined. an ill-defined mission.

Which of the following is are well-posed ODE initial value problem?

We will say that an initial-value problem is well posed if the linear system defined by the PDE, together with any bounded initial conditions is marginally stable. As discussed in [452], a system is defined to be stable when its response to bounded initial conditions approaches zero as time goes to infinity.

What is a well-defined problem?

Well-defined (well-structured) problems are those that contain a clear specification of three elements of the problem space: the initial state (the problem situation), the set of operators (rules and strategies) to solve the problem, and the goal state (the solution).

Which of the following is a well-posed ODE initial value problem?

What is a well-posed learning problem give some suitable examples?

Well Posed Learning Problem – A computer program is said to learn from experience E in context to some task T and some performance measure P, if its performance on T, as was measured by P, upgrades with experience E.

What is well-posed learning?

Well-Posed Learning Problems • Definition: A computer program is said to learn from experience E with respect to some class of tasks T and performance measure P, if its performance at tasks in T, as measured by P, improves with experience E.

What is the solution to the Cauchy problem?

It follows s=x−y+1, t=y−1 and that u=z0(x−y+1) is the solution of the Cauchy initial value problem.

How do you solve Cauchy problem in Ode?

If we set x = 0 we get, using the initial condition: sin y = u(0,y) = f(y). Hence a solution to the Cauchy problem is given by u(x, y) = sin(y − x). In Figure 1.1 we sketch S for this problem, along with some characteristic curves (which are of the form y = x + c).

Is Wave Equation well-posed?

The pairing of the elliptic equation with initial conditions led to an ill-posed problem. Thus the small change in the initial data leads a small change in the solution at any positive time. One expects this, since the initial value problem is well-posed for the wave equation.

What’s another word for ill-defined?

In this page you can discover 16 synonyms, antonyms, idiomatic expressions, and related words for ill-defined, like: indistinct, clouded, dim, fuzzy, murky, obscured, unclear, vague, well-defined, unspecific and imprecise.

What does the word imprecise?

Definition of imprecise : not precise : inexact, vague an imprecise estimate.

How do you know if a problem is well-posed?

A method to determine the well-posedness of a problem is the energy method. The method is based upon deriving an energy estimate for a given problem. and integrate in space over the given interval. From this energy estimate one can conclude that the problem is well-posed.

What is the difference between well-structured and ill structured problems?

Well-structured (puzzle) problems can be represented by a problem space consisting of well-defined initial and goal states that are connected by legal moves. In contrast, the initial, goal, and intermediate states of ill-structured (design) problems are incompletely specified.

What is well-defined problems and solutions?