Linear system with different units/dimensions

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I have a system of linear equations with a block structure



beginequation
beginpmatrix
underlineunderlineA \
underlineunderlineA^prime
endpmatrix
underlinex =
beginpmatrix
underlineb \
underlineb^prime
endpmatrix
endequation



The problem is that the blocks with and without primes have different physical units, and because of this they also have different orders of magnitude (physically the orders of magnitude are not comparable, but numerically they are), which makes the numeric solution unstable (the system is overdetermined, I minimize the residual). However $underlinex$ is just one vector that is the same for both blocks.



Is there a formalism (maybe involving tensors?) to solve two coupled linear systems?




linear-algebra




share|cite|improve this question



















  • Is $underlinex$ scalar ?
    – Yves Daoust
    Jul 17 at 11:05










  • What exactly do you mean with scalar? It is a real valued coefficient vector; each element of the vector is scalar, if that is what you mean.
    – Jakob Filser
    Jul 17 at 11:08










  • Can't you rescale one of the blocks to get more comparable magnitudes ?
    – Yves Daoust
    Jul 17 at 12:17










  • I tried that, but the outcome varies wildly and is physically unreasonable. It's basically a multiobjective problem and I'm somewhat afraid there is no unique solution to it.
    – Jakob Filser
    Jul 17 at 12:52










  • Is your solver reliable ?
    – Yves Daoust
    Jul 17 at 13:38














up vote
0
down vote

favorite












I have a system of linear equations with a block structure



beginequation
beginpmatrix
underlineunderlineA \
underlineunderlineA^prime
endpmatrix
underlinex =
beginpmatrix
underlineb \
underlineb^prime
endpmatrix
endequation



The problem is that the blocks with and without primes have different physical units, and because of this they also have different orders of magnitude (physically the orders of magnitude are not comparable, but numerically they are), which makes the numeric solution unstable (the system is overdetermined, I minimize the residual). However $underlinex$ is just one vector that is the same for both blocks.



Is there a formalism (maybe involving tensors?) to solve two coupled linear systems?




linear-algebra




share|cite|improve this question



















  • Is $underlinex$ scalar ?
    – Yves Daoust
    Jul 17 at 11:05










  • What exactly do you mean with scalar? It is a real valued coefficient vector; each element of the vector is scalar, if that is what you mean.
    – Jakob Filser
    Jul 17 at 11:08










  • Can't you rescale one of the blocks to get more comparable magnitudes ?
    – Yves Daoust
    Jul 17 at 12:17










  • I tried that, but the outcome varies wildly and is physically unreasonable. It's basically a multiobjective problem and I'm somewhat afraid there is no unique solution to it.
    – Jakob Filser
    Jul 17 at 12:52










  • Is your solver reliable ?
    – Yves Daoust
    Jul 17 at 13:38












up vote
0
down vote

favorite









up vote
0
down vote

favorite











I have a system of linear equations with a block structure



beginequation
beginpmatrix
underlineunderlineA \
underlineunderlineA^prime
endpmatrix
underlinex =
beginpmatrix
underlineb \
underlineb^prime
endpmatrix
endequation



The problem is that the blocks with and without primes have different physical units, and because of this they also have different orders of magnitude (physically the orders of magnitude are not comparable, but numerically they are), which makes the numeric solution unstable (the system is overdetermined, I minimize the residual). However $underlinex$ is just one vector that is the same for both blocks.



Is there a formalism (maybe involving tensors?) to solve two coupled linear systems?




linear-algebra




share|cite|improve this question











I have a system of linear equations with a block structure



beginequation
beginpmatrix
underlineunderlineA \
underlineunderlineA^prime
endpmatrix
underlinex =
beginpmatrix
underlineb \
underlineb^prime
endpmatrix
endequation



The problem is that the blocks with and without primes have different physical units, and because of this they also have different orders of magnitude (physically the orders of magnitude are not comparable, but numerically they are), which makes the numeric solution unstable (the system is overdetermined, I minimize the residual). However $underlinex$ is just one vector that is the same for both blocks.



Is there a formalism (maybe involving tensors?) to solve two coupled linear systems?





linear-algebra





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 17 at 10:32









Jakob Filser

1




1











  • Is $underlinex$ scalar ?
    – Yves Daoust
    Jul 17 at 11:05










  • What exactly do you mean with scalar? It is a real valued coefficient vector; each element of the vector is scalar, if that is what you mean.
    – Jakob Filser
    Jul 17 at 11:08










  • Can't you rescale one of the blocks to get more comparable magnitudes ?
    – Yves Daoust
    Jul 17 at 12:17










  • I tried that, but the outcome varies wildly and is physically unreasonable. It's basically a multiobjective problem and I'm somewhat afraid there is no unique solution to it.
    – Jakob Filser
    Jul 17 at 12:52










  • Is your solver reliable ?
    – Yves Daoust
    Jul 17 at 13:38
















  • Is $underlinex$ scalar ?
    – Yves Daoust
    Jul 17 at 11:05










  • What exactly do you mean with scalar? It is a real valued coefficient vector; each element of the vector is scalar, if that is what you mean.
    – Jakob Filser
    Jul 17 at 11:08










  • Can't you rescale one of the blocks to get more comparable magnitudes ?
    – Yves Daoust
    Jul 17 at 12:17










  • I tried that, but the outcome varies wildly and is physically unreasonable. It's basically a multiobjective problem and I'm somewhat afraid there is no unique solution to it.
    – Jakob Filser
    Jul 17 at 12:52










  • Is your solver reliable ?
    – Yves Daoust
    Jul 17 at 13:38















Is $underlinex$ scalar ?
– Yves Daoust
Jul 17 at 11:05




Is $underlinex$ scalar ?
– Yves Daoust
Jul 17 at 11:05












What exactly do you mean with scalar? It is a real valued coefficient vector; each element of the vector is scalar, if that is what you mean.
– Jakob Filser
Jul 17 at 11:08




What exactly do you mean with scalar? It is a real valued coefficient vector; each element of the vector is scalar, if that is what you mean.
– Jakob Filser
Jul 17 at 11:08












Can't you rescale one of the blocks to get more comparable magnitudes ?
– Yves Daoust
Jul 17 at 12:17




Can't you rescale one of the blocks to get more comparable magnitudes ?
– Yves Daoust
Jul 17 at 12:17












I tried that, but the outcome varies wildly and is physically unreasonable. It's basically a multiobjective problem and I'm somewhat afraid there is no unique solution to it.
– Jakob Filser
Jul 17 at 12:52




I tried that, but the outcome varies wildly and is physically unreasonable. It's basically a multiobjective problem and I'm somewhat afraid there is no unique solution to it.
– Jakob Filser
Jul 17 at 12:52












Is your solver reliable ?
– Yves Daoust
Jul 17 at 13:38




Is your solver reliable ?
– Yves Daoust
Jul 17 at 13:38















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