Are Inverse Problems More Complex Than Direct Problems in Mathematics?

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In summary, the conversation discusses the concept of inverse problems in mathematics, particularly through the example of solving equations of different degrees. It is noted that solving inverse problems requires more computational resources and is generally considered to be unsolvable analytically. Differences between direct and inverse problems are also mentioned, such as the uniqueness of solutions and the complexity of solving. The conversation also touches on how this concept applies to programming and the use of "heavy assertions" to check for errors in algorithms.
  • #1
Spathi
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It seems to me that all mathematics, to one degree or another, is devoted to solving inverse problems. Let's take five equations as an example:

1) ax+b=0
2) ax^2+bx+c=0
3) ax^3+bx^2+cx+d=0
4) ax^4+bx^3+cx^2+dx+e=0
5) ax^5+bx^4+cx^3+dx^2+ex+f=0

To solve the first problem, you need to spend one division. The second problem is solved through the discriminant. The third problem is solved by the Cardano formula, the fourth by the Ferrari formula, and the fifth is generally “not solved”, more precisely, it is considered that it cannot be solved analytically.

In fact, all five problems are inverse problems from a simple equation of the nth degree. Those. if we write, for example, y=ax^2+b+c, then finding y from a known x is a direct problem, and finding x from a known y is an inverse problem.
Differences between direct problems and inverse ones:

1) The direct problem can be solved in for any x, and the inverse - not for any y;
2) The solution to the direct problem is unique, while there can be several solutions to the inverse problem;
3) The complexity (time) and reliability of the solution of the direct problem practically does not depend on what x is equal to or what the initial approximation was;
4) The solution of the inverse problem requires much more computational resources than the solution of the direct problem;
5) There are simple and universal methods for solving the inverse problem through a straight line, such as simple enumeration or gradient descent (although they are usually far from efficient in terms of the required resources).

Now I propose the following idea: take the five equations above and determine their complexity. The complexity of solving a problem can be measured quantitatively - the amount of computer time spent on solving it using the best algorithm. I have a strong suspicion that for the first four tasks the complexity will be described by some simple dependence on the number (for example, an exponent), and for the fifth, the complexity will clearly fall out of this dependence. And from this it will be possible to conclude that there is an unsolved problem in mathematics - to find an analytical solution to the equation of the fifth degree.

If I'm not mistaken, it is believed that decomposing a number into prime factors requires much more computational resources than multiplying these factors (although formally this is only a hypothesis). To what extent is the concept of “computing resource costs” formalized in this hypothesis?

In programming, all these terms, as I understand it, are also quite applicable. A fairly understandable example is neural networks: the work of a neural network is a solution to a direct problem, and training (training) of a neural network is a solution to a reverse one.

In my opinion, all mathematics and all computer science in one way or another comes down to optimal ways of solving inverse problems. And here you I suggest one more idea for programmers. I came to the conclusion that when programming complex algorithms, it is useful to use “heavy assertions”: these are assertions that check everything at once.

These assertions are quite easy to implement, but they greatly slow down the program, so they need to be disabled not only in the final build of the program, but in general almost always, except for the moments when you need to check critical sections of the algorithm. With heavy assertions, bugs should be easy to find. So this is all also included in my topic: assertions are also an auxiliary solution to the direct problem, which allows you to check how correctly the inverse problem is solved.
 
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  • #2
Spathi said:
It seems to me that all mathematics, to one degree or another, is devoted to solving inverse problems. Let's take five equations as an example:
1) ax+b=0
2) ax^2+bx+c=0
3) ax^3+bx^2+cx+d=0
4) ax^4+bx^3+cx^2+dx+e=0
5) ax^5+bx^4+cx^3+dx^2+ex+f=0
To solve the first problem, you need to spend one division. The second problem is solved through the discriminant. The third problem is solved by the Cardano formula, the fourth by the Ferrari formula, and the fifth is generally “not solved”, more precisely, it is considered ...
not considered, proven(!)
Spathi said:
... that it cannot be solved analytically.
In fact, all five problems are inverse problems from a simple equation of the nth degree. Those. if we write, for example, y=ax^2+b+c, then finding y from a known x is a direct problem, and finding x from a known y is an inverse problem.
It is usually considered as the problem to find zeros: ##f(x)=0## determine the set (technical term: variety) of ##x##. Calling it an inversion suggests the false assumption that it can be uniquely solved. This is not the case.
Spathi said:
Differences between direct problems and inverse ones:
1) The direct problem can be solved in for any x, and the inverse - not for any y;
2) The solution to the direct problem is unique, while there can be several solutions to the inverse problem;
3) The complexity (time) and reliability of the solution of the direct problem practically does not depend on what x is equal to or what the initial approximation was;
4) The solution of the inverse problem requires much more computational resources than the solution of the direct problem;
5) There are simple and universal methods for solving the inverse problem through a straight line, such as simple enumeration or gradient descent (although they are usually far from efficient in terms of the required resources).
Now I propose the following idea: take the five equations above and determine their complexity. The complexity of solving a problem can be measured quantitatively - the amount of computer time spent on solving it using the best algorithm. I have a strong suspicion that for the first four tasks the complexity will be described by some simple dependence on the number (for example, an exponent), and for the fifth, the complexity will clearly fall out of this dependence. And from this it will be possible to conclude that there is an unsolved problem in mathematics - to find an analytical solution to the equation of the fifth degree.
If I'm not mistaken, it is believed that decomposing a number into prime factors requires much more computational resources than multiplying these factors (although formally this is only a hypothesis). To what extent is the concept of “computing resource costs” formalized in this hypothesis?
In programming, all these terms, as I understand it, are also quite applicable. A fairly understandable example is neural networks: the work of a neural network is a solution to a direct problem, and training (training) of a neural network is a solution to a reverse one.
In my opinion, all mathematics and all computer science in one way or another comes down to optimal ways of solving inverse problems. And here you I suggest one more idea for programmers. I came to the conclusion that when programming complex algorithms, it is useful to use “heavy assertions”: these are assertions that check everything at once. These assertions are quite easy to implement, but they greatly slow down the program, so they need to be disabled not only in the final build of the program, but in general almost always, except for the moments when you need to check critical sections of the algorithm. With heavy assertions, bugs should be easy to find. So this is all also included in my topic: assertions are also an auxiliary solution to the direct problem, which allows you to check how correctly the inverse problem is solved.
Your point of view is the one a student at school might get. It is not really mathematics, as is the class called mathematics. It is calculating. And there are reasonable algorithms known that can numerically approximate zeros. One of them carries Newton's name, just to show you how old those algorithms are.
 
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  • #3
There's more to mathematics than polynomial equations!
 
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  • #4
Spathi said:
... it is considered that it cannot be solved analytically.
It is actually proven. And it is a good example of what mathematics is and can do.

Proof:
##A_n## is simple for all ##n>4.\quad \blacksquare##

What is ##A_n## and what is simple?
##A_n## is a group and simplicity means that there are no proper normal subgroups.
What is a group and what are normal subgroups?
...
What has all this group theory to do with an integer polynomial?
Because this part of group theory corresponds to certain field extensions.
What is a field and what is an extension?
A field is something like the rational numbers and extension if we add roots, e.g. ##\sqrt[3]{2}.##
...
What has field theory to do with an integer polynomial?
This part of field theory is actually Galois theory.
Those field extensions show us what can be solved and what can not.
How that?
Because a root for an integer polynomial, i.e. an explicit form for a zero ##z## must be written by ##+\, , \,-\, , \,\cdot\, , \,:\, , \,\sqrt[m]{r}## and these can be identified with field extensions of the rational numbers as the quotient field of the integers.
...
And all this to solve a polynomial equation?
No. Groups are used e.g. in crystallography, quantum physics, or biology.

Once we understood the tools, we can use them everywhere.
So

Proof:
##A_n## is simple for all ##n>4.\quad \blacksquare##

is actually a complete proof that we cannot solve exactly for the zeros of integer polynomials from degree five on. To understand that it is a proof takes a semester course in Abstract Algebra at university, or at least the understanding of Galois theory. And this is the math, not the calculation (approximation) of a zero by Newton's algorithm.
 
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  • #5
Can OP provide a TLDR?

What is wrong with current algorithms in solving "inverse" problems?

I have a somewhat hard time understand the point of your post buddy. What is that you really want to ask/discuss?
 
  • #6
Spathi said:
It seems to me that all mathematics, to one degree or another, is devoted to solving inverse problems.
Yes and no.
"to one degree or another" -- ok, if that includes zero.
"devoted to" -- no. There is a lot of mathematics that have nothing to do with inverse problems.
You can not draw an accurate conclusion about "all mathematics" until you are familiar with all mathematics. I know that I will never accomplish that.
 
  • #7
fresh_42 said:
Proof:
An is simple for all n>4.◼

is actually a complete proof that we cannot solve exactly for the zeros of integer polynomials from degree five on. To understand that it is a proof takes a semester course in Abstract Algebra at university, or at least the understanding of Galois theory. And this is the math, not the calculation (approximation) of a zero by Newton's algorithm.

Sorry, I don't understand this yet, partially because I am not a native English speaker. What connection has the 5th degree equation solution to simple rumbers? I thought the subject is only about real numbers, only n is simple.
The phrase "equation cannot be solved analitically" needs to be explained. Of course we can find the solution of this equation using e.g. the Taylor 's row. But maybe there must be a more efficient algorithm? As far as I understand, the phrase "cannot be solved analitically" means that we cannot write a formula for computing the answer by standard functions (roots, sinuses, logarithms, etc). But maybe we can invent a new function, something like "quasy-root", and write a formula with it for calculating the answer?
 
  • #8
malawi_glenn said:
What is wrong with current algorithms in solving "inverse" problems?

I have a somewhat hard time understand the point of your post buddy. What is that you really want to ask/discuss?
I suggest a kind of philosophical approach which can be applied in mathematics or maybe in programming ("heavy assertions" above).
Previoulsy I did research work in the field of gas phase electronography, and we had a meme "in quantum chemistry, the direct problem is solved, while in electronography, the inverse problem is solved". I suppose this helps to think in a somewhat better way.
 
  • #9
Spathi said:
Sorry, I don't understand this yet, partially because I am not a native English speaker. What connection has the 5th degree equation solution to simple rumbers? I thought the subject is only about real numbers, only n is simple.
I only wanted to demonstrate the difference between real mathematics and solving a simple equation.
Spathi said:
The phrase "equation cannot be solved analitically" needs to be explained.
I referred to "... is considered to be unsolvable." It is not "considered", it is known.
Spathi said:
Of course we can find the solution of this equation using e.g. the Taylor 's row. But maybe there must be a more efficient algorithm?
Yes. This is subject to numerical analysis. The need for efficiency lost much of its importance since blank computation power took over.
Spathi said:
As far as I understand, the phrase "cannot be solved analitically" means that we cannot write a formula for computing the answer by standard functions (roots, sinuses, logarithms, etc).
Yes.
Spathi said:
But maybe we can invent a new function, something like "quasy-root", and write a formula with it for calculating the answer?
Well, the easiest way is ##p(x)=0.##
 
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  • #10
Spathi said:
As far as I understand, the phrase "cannot be solved analitically" means that we cannot write a formula for computing the answer by standard functions (roots, sinuses, logarithms, etc).
It is better to say standard simple functions.
Spathi said:
But maybe we can invent a new function, something like "quasy-root", and write a formula with it for calculating the answer?
There are a lot of special functions that have been defined to solve important equations. But the equations must be important enough so that the special functions are used a lot and have important properties. Otherwise, the list of "special" functions would just become a junk heap.
 
  • #11
FactChecker said:
There are a lot of special functions that have been defined to solve important equations. But the equations must be important enough so that the special functions are used a lot and have important properties. Otherwise, the list of "special" functions would just become a junk heap.
So, my question is, are there any special functions invented for solving the 5th degree equations?
This is the questions of computational efficiency: what way of computing the solution for these equations is more efficient - a formula consisting of some special functions (each of them being computed as a convergent row), or one complex convergent row (the Taylor's row)?
 
  • #13
  • #14
Spathi said:
But what if we invent new types of radicals?
Which is precisely what the article says in the section pointed to.

We can't help you if you don't read our answers.
 
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Related to Are Inverse Problems More Complex Than Direct Problems in Mathematics?

1. What is the difference between direct and inverse problems?

Direct problems involve using known input data to calculate an output, while inverse problems involve using an output to determine the input.

2. How are direct and inverse problems used in scientific research?

Direct problems are often used to predict outcomes or behavior based on known factors, while inverse problems are used to determine the underlying causes or parameters of a system or phenomenon.

3. What are some applications of direct and inverse problems?

Direct and inverse problems are used in a variety of fields, such as engineering, physics, geology, and medical imaging. They can be used to model and understand complex systems, optimize designs, and make predictions.

4. What are the challenges associated with solving inverse problems?

Inverse problems can be difficult to solve because they often involve incomplete or noisy data, and there may be multiple solutions that fit the data. Additionally, they may require complex mathematical or computational methods to solve.

5. How do scientists approach solving inverse problems?

Scientists use a variety of techniques and approaches to solve inverse problems, such as regularization methods, optimization algorithms, and statistical analysis. They also rely on prior knowledge and assumptions about the system to guide their solutions.

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